Bias variance tradeoff (chatgpt)

October 1, 2023

(original) https://chat.openai.com/share/fdc20be3-d70b-4c98-8815-308ae676739c

You:

in stat, when we estimate y from x, with y = f(x) we assume there’s error term such tat y = f(x) + e and we also assume e is mean 0 why do we make the assumption? do we adjust our estimator f(what’s the proper term for f?) to make e to have 0 mean?

ChatGPT:

Yes, when we use a model of the form $y = f(x) + e$y=f(x)+e, the term $e$e represents the error or residual. This error captures any discrepancies between the observed values of $y$y and the predicted values from our model $f(x)$f(x). The assumption that the errors have a mean of zero is made for several reasons:

• Interpretability: By assuming a zero mean error, we are essentially saying that the model does not systematically overpredict or underpredict the outcome on average. If the mean of the errors was not zero, then it would suggest a systematic bias in our predictions.
• Mathematical Simplicity: Many of the statistical properties and derivations that come with linear regression (and other models) are more straightforward when we assume errors have a zero mean.
• Identification: In the context of linear regression, assuming zero-mean errors helps in uniquely identifying the parameters of the model. Without this assumption, there would be infinitely many combinations of parameters that fit the data equally well.

Regarding the adjustment of our estimator:

When we fit a model like $y = f(x) + e$y=f(x)+e, the goal is often to minimize the sum of squared errors (in the case of ordinary least squares regression). By doing this, we are effectively forcing the residuals (or errors) to have a mean of zero.

If, for instance, you take a simple linear regression:

$y = \beta_0 + \beta_1x + e$y=β0​+β1​x+e

And you minimize the sum of squared residuals:

$\min \sum_{i=1}^{n} e_i^2$min∑i=1n​ei2​

Where $e_i$ei​ is the residual for observation $i$i, it can be shown mathematically that the resulting estimated residuals will always have a mean of zero.

The term $f$f in your notation represents the “functional form” of your model. In the context of linear regression, this would be the linear equation that relates $x$x to $y$y. When we talk about estimating this function, we are often referring to estimating the parameters ($\beta_0, \beta_1, \ldots$β0​,β1​,…) that define the relationship. The term “estimator” might be used to refer to a rule or algorithm that produces estimates of these parameters based on observed data.

You:

so estimator can refer to f in our y = f(x) + e model?

and so based on your description, the process of minizing error term results in error to have 0 mean, so it’s not like, our concrete f was assumed to be non biased (which will make our error term 0 mean), we do not have concrete f in the first place, in the process of minimizing error term, our f is selected to have error term to have zero mean.

ChatGPT:

Yes, you’ve got it!

• Estimator and the Function f f f:
  • The term “estimator” is usually applied to a rule or method for determining the values of parameters in a statistical model. In the context of regression, we’re often trying to estimate parameters like \beta_0, (\beta_1, and so forth.
  • However, once we’ve estimated these parameters, they can be plugged into our functional form f ( x ) f(x) f(x) to provide estimates of y y y. So in a loose sense, you can think of the entire function f ( x ) f(x) f(x) with its estimated parameters as the estimator for y y y. But typically, when we say “estimator,” we’re referring to the method or rule for estimating the parameters, not the entire function.
• Error Term and Zero Mean:
  • Correct, when we fit a model like linear regression using the ordinary least squares method, we’re trying to minimize the sum of the squared residuals (errors).
  • A consequence of this minimization process is that the residuals will have a mean of zero. We don’t start with a specific, concrete function f ( x ) f(x) f(x). Instead, we determine the best function (or more precisely, the best parameters for our function) based on our data and our goal to minimize the sum of squared residuals.
  • So it’s not that we assume our function is unbiased, but rather that the process of fitting the function to our data using methods like ordinary least squares results in residuals with a zero mean. This aligns with the assumption that our errors have a zero mean, which is a desirable property for the reasons mentioned earlier.

Your understanding is on point!

You:

what i was trying to say was that, i can’t be assured i ’ve selected good model because i have zero mean error term, the process will make it so.

ChatGPT:

You’re absolutely right. Having residuals with a zero mean, while it’s a desirable property, does not alone guarantee that you have a “good” or appropriate model for your data. There are many considerations to take into account when determining the quality or appropriateness of a model, and the zero mean of the residuals is only one of them.

To assess the quality or suitability of a regression model, you typically need to consider several diagnostic measures and assumptions:

• Linearity: The relationship between predictors and the response variable should be linear. You can check this assumption visually with scatterplots or residual vs. fit plots.
• Homoscedasticity: The variance of the residuals should be consistent across all levels of the independent variables. Again, this can be checked with residual plots.
• Independence of Errors: The residuals should be independent, especially in time series data where autocorrelation might be a concern.
• Normality of Errors: For certain hypothesis tests and confidence intervals to be valid, the residuals should be approximately normally distributed. This can be assessed using histograms, Q-Q plots, or statistical tests like the Shapiro-Wilk test.
• Lack of Multicollinearity: In multiple regression, the predictors should not be too highly correlated with each other. Variance Inflation Factors (VIFs) can help assess this.
• Outliers: Outliers can greatly influence the regression line and conclusions. Residual plots can help in identifying extreme values.
• Influential Points: These are points that have a big impact on the regression equation. Cook’s distance is one way to detect influential points.
• Model Fit: Measures like R 2 R^2 R 2 or adjusted R 2 R^2 R 2, AIC, BIC, etc. can be used to get an idea of how well your model explains the variation in the data.

A model that results in zero mean residuals but violates many of the other assumptions might not be a good fit for the data. Always consider multiple diagnostic measures and criteria when evaluating the appropriateness of a regression model.

You:

so, like the other properties you mentioned, “zero mean error” is a desired property, but the process forces it to be the case , leaving me not being able to test if my model is good regarding if it has zero error mean. do i have a way to test that?

ChatGPT:

You’re correct in noting that the ordinary least squares (OLS) estimation method for linear regression inherently results in residuals with a zero mean. Consequently, merely observing a zero mean in residuals is not, by itself, evidence that the model is correctly specified.

However, while you can’t test for zero mean residuals (since it’s given by the OLS process), you can test for functional form mis-specification and other issues which would, in a broader context, mean that your model is not capturing the relationship correctly, and thus effectively leaving a biased error term.

Here are ways to test for some common issues:

• Functional Form Mis-specification:
  • Added Variable (or Residual) Plots: If the relationship is indeed linear, there shouldn’t be any discernible pattern in the plots.
  • Harvey-Collier multiplier test for linearity: This test checks the null hypothesis that the relationship is linear.
• Heteroskedasticity:
  • Breusch-Pagan test and White’s test can be used to detect non-constant variance in the residuals (heteroskedasticity).
• Autocorrelation (for time series data):
  • Durbin-Watson test: Used to detect autocorrelation in the residuals from a regression analysis.
• Model Specification:
  • Ramsey RESET test: This test is used to detect nonlinearity and omitted variables in a regression model. If this test indicates issues, it implies that the model is misspecified, and the error term might not only have a problem with its mean.

While you can’t directly test for a non-zero mean in residuals (since OLS ensures it), these tests can provide indications of model mis-specification, which would imply that your model might not be correctly capturing the underlying relationship, leading to errors that don’t just stem from random fluctuations but from more systematic issues. If these systematic issues are present, then while the error might have a zero mean due to the mathematical properties of OLS, it wouldn’t be the “right” or unbiased error term from the perspective of capturing the true relationship between variables.

You:

too high level concepts. let’s briefely talk about autocorrelation, what do you mean by “detec autocorrelation” ?

ChatGPT:

Certainly! Let’s break down autocorrelation in a simpler manner.

Autocorrelation refers to the correlation of a time series with its own past and future values. In the context of linear regression residuals, autocorrelation means that the value of the error (or residual) at any given time is correlated with the error at a previous time.

For instance, think about daily temperatures. If today is hotter than average, it’s quite likely that tomorrow will also be hotter than average. This means the temperature today is correlated with the temperature tomorrow. This is a form of autocorrelation.

Why is it a problem in regression?

• Inefficient Estimates: When residuals are autocorrelated, the OLS estimates remain unbiased, but they are no longer efficient. This means that they do not have the smallest possible variance and other estimators might give a better (more precise) estimate.
• Inaccurate Standard Errors: The standard errors of your regression coefficients can be biased. This means hypothesis tests (like t-tests for the regression coefficients) can be invalid.
• Misleading Model Fit: Measures of model fit, like R 2 R^2 R 2, can be misleading.

Detecting Autocorrelation:

One of the most common ways to detect autocorrelation is by looking at the residuals.

• Graphical Analysis: A plot of the residuals against time can sometimes show patterns indicating autocorrelation.
• Durbin-Watson Test: This is a formal statistical test that can help detect autocorrelation in the residuals. The test statistic ranges from 0 to 4, where a value of 2 suggests no autocorrelation. Values below 2 suggest positive autocorrelation, and values above 2 suggest negative autocorrelation.
• ACF and PACF Plots: These plots can help determine the nature and degree of autocorrelation. The Autocorrelation Function (ACF) plot shows the correlation between the time series with a lagged version of itself. The Partial Autocorrelation Function (PACF) plot similarly provides correlations of the time series with its lags but controls for shorter lags.

In essence, autocorrelation is about patterns in time. If the errors from your regression are not truly random but follow a discernible pattern across time, you may have autocorrelation issues, which can affect the reliability of your regression results.

You:

ok, trying to understand core part first, so with temparature example, what are we trying to estimate? from which (X) to what (Y)? i guess time is something we are not considering to have effect on our model of prediction and it’s affecting our model badly, but I can’t tell because what we are up to

ChatGPT:

Great, starting with the core concept will help. Let’s break down the temperature example further.

Imagine we want to predict the daily temperature based on a number of factors, like humidity, air pressure, and wind speed. Here:

• Y (Dependent Variable): Daily temperature.
• X (Independent Variables): Humidity, air pressure, and wind speed.

So, our regression model might look something like this:

$\text{Temperature} = \beta_0 + \beta_1 \times \text{Humidity} + \beta_2 \times \text{Air Pressure} + \beta_3 \times \text{Wind Speed} + e$Temperature=β0​+β1​×Humidity+β2​×Air Pressure+β3​×Wind Speed+e

Where $e$e is the error term.

Now, when we talk about autocorrelation in this context, we’re referring to the residuals (the error terms, $e$e). We’re suggesting that the error term from one day might be correlated with the error term from the previous day.

Why could this happen with temperature?

• Omitted Variable: We might be missing an important variable in our model, like the temperature from the previous day, which is likely a good predictor of today’s temperature.
• Inherent Time Series Nature: Many natural processes, like weather, have a time component inherent in them. Even if we account for all other variables, the temperature today will be influenced by the recent past due to the inherent nature of weather patterns.

So, when we say there’s autocorrelation in our residuals, we’re noticing a pattern: maybe when we underestimate today’s temperature, we also tend to underestimate tomorrow’s. Or, conversely, when we overestimate today’s temperature, we also tend to overestimate tomorrow’s. This pattern suggests our model isn’t capturing everything it needs to about how temperature behaves over time.

In summary, while our original model might be trying to predict temperature based on variables like humidity, air pressure, and wind speed, there’s a time-based component (like temperature from the previous day) that we’re missing out on, leading to patterns in our errors.

You:

so autocorelation is not , our model is autocorelating, it’s saying OLS is autocorelating and if we have better model, OLS will have smaller number,

ChatGPT:

Almost. Let’s clarify a few things:

• Autocorrelation refers to the residuals, not the model or OLS: When we say there’s autocorrelation, we mean that the residuals (errors) from our model exhibit a pattern over time. In other words, the error term at one point in time is correlated with the error term at another point in time.
• OLS does not autocorrelate: Ordinary Least Squares (OLS) is a method used to estimate the coefficients of your model. OLS does not “autocorrelate”. Instead, if the true relationship between your variables has a time component that your model doesn’t capture, then the residuals produced by an OLS regression can show autocorrelation.
• If there’s autocorrelation, it suggests model misspecification: Autocorrelation in the residuals is an indication that there might be some important time-based factor that your model is missing. If you could capture that factor in your model, the autocorrelation in your residuals would likely decrease, and your model would provide better predictions.
• A “better model” would exhibit less autocorrelation: If you could adjust your model to account for the time-based patterns (for example, by including the temperature from the previous day as an additional predictor), then the residuals from this new model would show less autocorrelation. This would be considered an improvement in the model because it’s capturing more of the underlying patterns in the data.

So, in summary, when we say there’s autocorrelation, we’re referring to a time-based pattern in the residuals, not the model itself or OLS. This pattern is an indication that the model might be missing some important time-based factor. Adjusting the model to capture this factor would lead to residuals with less autocorrelation.

You:

ok i guess autocorelation can refer to non time based effect as well? when we are missing important feature, we can improve by incoporating that feature, .. autocorelation specifically refers to missing “time” related feautre in our model?

ChatGPT:

Autocorrelation is most commonly discussed in the context of time series data, where the order of observations matters and where the term specifically refers to the correlation of a series with its own lags (past values). But you’re right in recognizing that the fundamental concept is broader. Let’s clarify:

• Autocorrelation in Time Series: In time series analysis, “autocorrelation” refers to the correlation of a variable with its own past (or sometimes future) values. The idea is that today’s value might be influenced by yesterday’s value, the day before that, and so on.
• Spatial Autocorrelation: The concept can be extended beyond time. For example, in spatial data (like geographical data), you might find that the value of a variable at one location is influenced by values at nearby locations. This is called spatial autocorrelation.
• Sequential Data: Even in non-time-based data, if there’s a logical sequence or order to your observations, you could potentially see a form of autocorrelation. For instance, in manufacturing, the quality of one item on an assembly line might be influenced by the quality of the item before it.
• Missing Important Feature: The presence of autocorrelation in residuals can sometimes be an indication of missing important features in the model. If the residuals exhibit autocorrelation in a time series model, it often suggests that there’s some time-related feature or pattern that the model isn’t capturing. However, the missing feature doesn’t always have to be time-related; it’s just that in the context of time series data, time-related features are common culprits.

In summary, while “autocorrelation” most commonly refers to time-based relationships in time series data, the core concept can apply in other contexts as well. The key idea is that an observation is influenced by its preceding observations (whether those “preceding” observations are in time, space, or some other logical order).

You:

ok, but unless we have some incling whether we might have time related missing feature, we won’t be able to test if autocorelation is happening..

ChatGPT:

Yes, you’re on the right track.

When we suspect autocorrelation, it’s typically because we have reason to believe that sequential observations might influence one another. This suspicion most often arises in contexts where time is a factor, but as mentioned, it can also occur in spatial data or other ordered data.

Here’s how it typically unfolds:

• Domain Knowledge: Often, our first hint about the possibility of autocorrelation comes from understanding the subject matter. For example, if you’re analyzing economic data, you know economies have cycles, and therefore monthly data might be influenced by prior months. If you’re analyzing daily website visits, you might suspect that traffic patterns could have day-of-week effects where Mondays are influenced by Sundays, etc.
• Exploratory Data Analysis (EDA): Plotting data is a common first step. For time series, a simple time plot can sometimes reveal patterns that suggest autocorrelation. For instance, you might see cycles, trends, or seasonality in the data.
• Residual Analysis: Once you fit an initial model, you’d examine the residuals. Patterns in residuals can indicate model misspecification, including missing time-related factors. For time series, plotting residuals over time or using tools like the autocorrelation function (ACF) plot can reveal autocorrelation.
• Statistical Tests: Formal tests, like the Durbin-Watson test for time series data, can be used to test for autocorrelation once you suspect it might be present.

It’s important to remember that not all datasets will have autocorrelation, and not all situations warrant checking for it. However, for datasets where observations have a sequential order (especially time series data), checking for autocorrelation is a standard diagnostic step when building and validating models.

You:

so next question would be, when we dont’ have enough knowledge (inkling), we don’t know where to look (we won’t know which dimension we will plot our data against, so that we can recognize if some autocorrelation is going on) , then what do we do? how do we know if we have some missing feature? i guess we never know? it’s like you randomly mix time based data, and you won’t be able to detect there’s time related feature missing.

ChatGPT:

Your observation is insightful. Indeed, when you don’t have domain knowledge or any indication about potential structures in your data, identifying things like autocorrelation or missing features becomes challenging. But there are some general strategies and steps you can consider:

• Exploratory Data Analysis (EDA): Even if you don’t know what specific patterns to look for, visualizing your data across different dimensions, creating scatterplots, and looking at distributions can often reveal underlying structures or anomalies.
• Residual Analysis: After fitting a preliminary model, always check the residuals. Any pattern or structure in the residuals is an indication that the model is not capturing some aspect of the data. This is a general principle and not just for autocorrelation.
• Cross-Validation: Using cross-validation techniques can help you identify if your model is overfitting or potentially missing important features. If a model performs very well on training data but poorly on unseen (validation) data, it could be an indication that there are unaccounted complexities.
• Feature Importance & Selection: Some machine learning algorithms can rank features based on their importance. While this doesn’t directly tell you what’s missing, if some features you’d expect to be significant rank low, it might prompt further investigation.
• Anomaly Detection: There are algorithms designed to identify anomalies in data. If your dataset has anomalies, understanding them can sometimes lead to insights about missing features or patterns.
• Leverage External Expertise: If you’re unfamiliar with the domain, consulting someone who has expertise can provide insights. They might know typical structures or patterns in such data.
• Iterative Process: Model building is often iterative. As you gather more data or as the system you’re modeling changes, you’ll refine and adapt your models. Each iteration provides an opportunity to identify and address gaps.
• Learning Algorithms: Some algorithms, especially in the realm of deep learning, can capture complex patterns without explicit feature engineering. While this doesn’t replace domain knowledge, it’s a tool in the toolbox.

You’re right that, in some cases, without the proper context or domain knowledge, it’s possible to miss important features or patterns. That’s one reason why domain expertise is so valued in data science. However, the combination of thorough exploratory analysis, rigorous validation, and iterative modeling can help mitigate these risks.

You:

1, 2, i guess , the proess is essentially, try our best guesses on what dimension we are gonna plot the residuals to see if there’s a pattern, we just try many of them to see the pattern (like time, space, and so on)

ChatGPT:

Absolutely. Let me emphasize and expand on those two points a bit:

• Exploratory Data Analysis (EDA):
  • This is a crucial step in any data analysis process, precisely because it’s our primary way to “get to know” the data without making any strong assumptions. By visualizing data across different dimensions, you can often spot patterns, trends, or anomalies that you might not have known to look for.
  • Even without domain knowledge, simple visualizations can sometimes reveal a lot. For instance, if you plot data points sequentially and notice cyclical patterns, you might infer a time-based component. Or if you’re dealing with geographical data and notice clusters or gradients, it could suggest spatial patterns.
• Residual Analysis:
  • After you have some model, even if it’s a very basic or preliminary one, you can examine the residuals (the differences between the observed values and the values predicted by the model).
  • Patterns in the residuals often indicate that the model isn’t capturing some important aspect of the data. This is a general principle that’s used across all types of regression and modeling.
  • For instance, if you’re not sure whether time is an important factor, you could plot the residuals in the order the data was collected. If you see patterns or trends in that plot, it’s a hint that time (or some other sequential factor) might be playing a role.

Your summary is on point. A significant part of the early modeling process is trying out different approaches, checking the residuals in various ways, and looking for patterns across different potential dimensions. Over time, this iterative process can help refine the model and identify important features or dimensions, even if you started without a clear idea of what to look for.

You:

yeah, i was specifically asking, if i don’t have domain knowledge, they my procedure of using 1 /2 would be to try many dimensions hoping to discover patterns. right? domain knowledge gives better choice on “what dimension” to try

ChatGPT:

Yes, you’ve captured the essence of the process correctly. Without domain knowledge, the approach becomes more exploratory and somewhat trial-and-error:

• Try Many Dimensions: Without prior knowledge, you’d likely visualize the data across as many reasonable dimensions and combinations as you can think of to discover potential patterns. This could mean plotting against time, different features, various aggregations of features, combinations of features, and so on.
• Residual Analysis Across Dimensions: Similarly, you’d plot residuals against different dimensions to see if there’s a systematic pattern that suggests some aspect of the data isn’t being captured by the current model.
• Leverage Statistical and Machine Learning Techniques: There are various statistical tests and algorithms that can provide insights or detect patterns even when you don’t have specific domain knowledge. For instance, clustering algorithms might reveal groupings in your data that you weren’t aware of.

Having domain knowledge certainly streamlines this process. When you know the domain, you often have a good idea of which dimensions are most likely to be informative, which potential patterns to look for, and which variables are likely to be influential. Without that knowledge, the process is more exploratory, and you rely more heavily on general data analysis techniques and tools to uncover patterns and structures in the data.

You:

ok, this is question on “how to ask question” for you to specifically address what i’m asking, in this incident, i was trying to answer like “Yes, you’ve captured the essence of the process correctly. Without domain knowledge, the approach becomes more exploratory and somewhat trial-and-error”, or “No,.. “, but I often get answers that are related to the question but not specifically addressing what I’m asking, (which most times actually are good, because you expand on many stuff), but when I want to get direct answers to my specific question, how should i ask better?

ChatGPT:

I appreciate your feedback, and I understand your point. Here are a few tips to get more direct responses:

• Start with a Direct Prompt: Phrases like “Yes or No:”, “Briefly explain:”, or “In one sentence:” can be helpful to signal that you’re looking for a concise and direct answer.
• Be Specific: The more specific your question, the easier it is to give a focused answer. For instance, “Is autocorrelation specifically about time-related patterns?” is a direct question that can be answered succinctly.
• Ask for Clarification: If the initial answer isn’t quite what you were looking for, feel free to ask for clarification or specify that you’d like a shorter, more direct response.
• Limit the Scope: If you’re asking multiple questions in one go, it can sometimes lead to broader answers. Focusing on one key point at a time can help in getting a more specific response.
• Feedback: If an answer feels too broad or tangential, let me know! Feedback helps, and I can adjust my responses accordingly.

Remember, my goal is to provide information and clarity, so guiding me with your preferred response style will certainly help cater the answers to your needs!

You:

in statistics, what does “inference” mean ? (explain it as if you are explaining to first stat course student)

ChatGPT:

Certainly!

In statistics, “inference” means making educated guesses or conclusions about a larger group based on a smaller sample from that group.

Imagine you want to know the average height of all students in your school. Instead of measuring everyone (which would take a lot of time), you might measure just a few students (a sample) and then make an “inference” or educated guess about the average height of all students in the school.

So, in short, inference is about using sample data to make conclusions or predictions about a larger group (often called the population).

You:

ok, so inference seems to be overloaded in many ways, doesn’t inference mean something different (or specific) in hypothesis testing context?

ChatGPT:

Yes, you’re correct. In the context of hypothesis testing, “inference” takes on a more specific meaning.

In hypothesis testing, “inference” refers to the process of deciding, based on sample data, whether a certain statement or claim about a population is likely to be true.

Here’s a simple breakdown:

• Null Hypothesis (H0): This is a statement that there is no effect or no difference. It’s what we’re trying to test against.
• Alternative Hypothesis (Ha): This is what we want to prove — that there is an effect or difference.
• Test Statistic: Based on our sample data, we calculate a value (like a t-statistic or z-statistic) to help us make our decision.
• P-value: This tells us how likely we would observe our sample data (or something more extreme) if the null hypothesis were true. A small p-value suggests that our sample data is unlikely under the null hypothesis, leading us to reject it.
• Decision: Based on the p-value and a chosen significance level (often 0.05), we decide whether to reject the null hypothesis or not.

In this context, “inference” is about drawing a conclusion from our hypothesis test. For instance, if we reject the null hypothesis, we might infer that a certain treatment has an effect, or that there’s a difference between two groups.

So, while “inference” broadly means drawing conclusions based on data, in the context of hypothesis testing, it’s specifically about deciding whether to believe a certain claim or statement about a population based on sample data.

You:

ok, and i see another use of inference when talking about statistical learning in bigger context, prediction vs inference means different thing, and here inference means?

ChatGPT:

You’re right. In the context of statistical learning or machine learning, when comparing “prediction” and “inference,” the terms have specific meanings:

• Prediction:
  • Here, the main goal is to make accurate predictions for new, unseen data.
  • We’re primarily concerned with how well our model performs in terms of its predictive accuracy.
  • We might not be deeply interested in understanding why the model is making a particular prediction or the exact relationship between variables.
• Inference:
  • In this context, inference refers to understanding the relationships between variables in our model.
  • We want to know which variables are influential, how they’re related, and the nature of their relationships. Is the relationship linear? Quadratic? How does one variable change as another increases?
  • The aim is to gain insights about the underlying process or system from which the data originates.
  • For example, in a linear regression model, we might be interested in the coefficients of predictors, as they give us insight into the relationships between predictors and the response variable.

In summary, in the context of statistical learning:

• Prediction is about making accurate forecasts without necessarily understanding the intricacies of the relationships.
• Inference is about understanding the relationships between variables, even if it doesn’t always lead to the most accurate predictions.

This distinction is especially important when choosing models. Some models might be great for prediction but difficult to interpret (like some complex neural networks). Others might be less predictive in certain situations but provide clearer insights into variable relationships (like linear regression).

You:

so inference, is about ‘human understandability’?

ChatGPT:

Yes, in the context of statistical learning, “inference” often centers around “human understandability.” It’s about deriving insights and understanding the relationships between variables, which humans can interpret and make sense of.

While predictive models provide outputs based on input data, inferential models also aim to explain why those outputs are as they are. The emphasis is on understanding the structure of the data, the importance and role of different variables, and the relationships between them.

In many real-world scenarios, especially in domains like medicine, economics, or policy-making, understanding the “why” behind predictions is crucial. It’s not enough to have a black-box model that makes accurate predictions; decision-makers also need to understand the reasons behind those predictions to make informed choices.

So yes, inference in statistical learning is closely tied to human interpretability and comprehension of the model and its workings.

You:

MSE seems similar to Variance, can one be interpreted as using another?

ChatGPT:

Yes, Mean Squared Error (MSE) and variance are conceptually related, both involving the squaring of differences. Let’s break down each one and then discuss their relationship.

• Mean Squared Error (MSE):
  • The MSE measures the average squared difference between the actual values and the predicted values.
  • Mathematically: M S E = 1 n ∑ i = 1 n ( y i − y ^ i ) 2 MSE = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 MSE = n 1​ ∑ i=1 n​ ( y i​ − y ^​ i​ ) 2
    • Where y i y_i y i​ is the actual value, y ^ i \hat{y}_i y ^​ i​ is the predicted value, and n n n is the number of observations.
  • Variance:
  • Variance measures how far individual numbers in a dataset are from the mean of that dataset.
  • Mathematically: V a r ( X ) = 1 n ∑ i = 1 n ( x i − x ˉ ) 2 Var(X) = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 Va r( X) = n 1​ ∑ i=1 n​ (x i​ − x ˉ) 2
    • Where x i x_i x i​ is an individual data point and x ˉ \bar{x} x ˉ is the mean of all the data points.

Relationship and Interpretation:

• Both MSE and variance involve squaring differences. For MSE, it’s the difference between predicted and actual values. For variance, it’s the difference between each data point and the mean.
• If you think of a regression problem where you’re trying to predict some values based on certain inputs, the variance of the target variable (the one you’re trying to predict) gives a measure of the “spread” or variability of that variable. Meanwhile, the MSE gives a measure of the accuracy of your predictions.
• In the context of regression, if your predictions are just the mean of the target variable (i.e., a very naive model that always predicts the mean), then the MSE will be equal to the variance of the target variable.

So, while MSE and variance are conceptually related and involve similar calculations, they serve different purposes. MSE is a measure of prediction error, while variance is a measure of data spread. However, understanding one certainly gives intuition about the other.

You:

as with variance (where the formular we defined for variance to capture the notion of spread is somewhat arbitrary, arbitrary in a sense that, i don’t think there’s a way to tell variance better captures the notion of spread than sum of absolute difference) using MSE for measuring our model is arbitrary? (in a sense that one can’t argue MSE better captures the notion of accurary of model than others)

ChatGPT:

Your observation touches on a deeper point in statistics and model evaluation: the choices of metrics often come with some level of arbitrariness, and they’re influenced by both mathematical convenience and the specific goals or challenges of a given problem.

• Variance vs. Absolute Differences:
  • Variance uses squared differences, which emphasize larger deviations more than smaller ones. Squaring ensures all differences are positive, and it’s mathematically convenient, especially when working with Gaussian distributions.
  • Absolute differences (used in metrics like Mean Absolute Error or Median Absolute Deviation) give a linear penalty to deviations. They can be more robust to outliers because they don’t square the deviations.
  • There isn’t an absolute sense in which one is “better” than the other; it depends on the specific application and the properties you desire in a measure of spread or error.
• MSE vs. Other Metrics:
  • MSE (Mean Squared Error) gives more weight to larger errors. This might be desirable in some situations where larger errors are particularly undesirable.
  • MAE (Mean Absolute Error) penalizes all errors linearly. It’s less sensitive to outliers than MSE.
  • Huber Loss is a combination of MSE and MAE, which is quadratic for small errors and linear for large errors. It tries to balance the properties of MSE and MAE.
  • Log-cosh Loss, Quantile Loss, and others are alternative metrics that can be more appropriate in specific situations.

Which metric to use depends on the specific problem. For instance:

• If outliers are a concern and you don’t want to overly penalize them, MAE might be preferred over MSE.
• If it’s crucial to minimize large errors and they are more “costly” than small errors, then MSE might be more appropriate.
• In finance, metrics that capture the asymmetry of loss, like the quantile loss, can be valuable since the cost of underestimating risk might be much higher than overestimating it.

In Summary: The choice of metric, whether for measuring spread or model accuracy, often comes with trade-offs. While there is an element of arbitrariness, the choice should ideally be guided by the specific characteristics and requirements of the problem at hand. Mathematical convenience also plays a role, especially when optimizing models or when theoretical properties are derived.

You:

tell me more about quantile loss, does it give more weight to loss than profit?

ChatGPT:

Certainly! The quantile loss, often used in quantile regression, is designed to capture asymmetric penalties for over- and under-predictions. It’s particularly useful in situations where the costs associated with overestimation and underestimation are not the same.

Let’s dive into the concept:

Quantile Loss: Given a quantile $\tau$τ (a value between 0 and 1), the quantile loss for an individual prediction is defined as:

<span class=“katex-error” title=“ParseError: KaTeX parse error: Can’t use function ‘]’ in math mode at position 127: …} \end{cases} \̲]̲

Where:

• ( y…” style=“color: rgb(204, 0, 0);">\begin{cases} \tau (y - \hat{y}) & \text{if } y > \hat{y} \ (1-\tau) (\hat{y} - y) & \text{if } y \leq \hat{y} \end{cases} ] Where:
• ( y ) is the true value.
• ( \hat{y} ) is the predicted value.
• ( \tau ) is the quantile of interest. Interpretation:
• If ( y > \hat{y} ) (under-prediction), the loss is ( \tau (y - \hat{y}) ).
• If ( y \leq \hat{y} ) (over-prediction), the loss is ( (1-\tau) (\hat{y} - y) ). So, how the loss is weighted depends on the choice of ( \tau ):
• For ( \tau = 0.5 ), the quantile loss becomes the Mean Absolute Error (MAE). There’s no asymmetry.
• For ( \tau > 0.5 ), under-predictions are penalized more heavily than over-predictions.
• For ( \tau < 0.5 ), over-predictions are penalized more than under-predictions. Application: Quantile loss is especially useful in scenarios where asymmetric costs are involved. For example:
• Finance: If a financial institution underestimates the risk (like potential loan defaults), it might face substantial losses. But if it overestimates the risk, it might only miss out on some potential profits. Hence, it might want to use a ( \tau ) that gives more weight to underestimations.
• Supply Chain: If a retailer underestimates demand for a product, it may lose sales opportunities. But if it overestimates demand, it might end up with excess inventory. Depending on which cost is higher (lost sales or holding excess inventory), an appropriate ( \tau ) can be chosen. In essence, the quantile loss allows modelers to tailor predictions to specific quantiles and asymmetrically weigh errors, making it more sensitive to over-predictions or under-predictions based on the chosen quantile.

You:

format is messed up, all red fonts..

ChatGPT:

I apologize for that. Let me explain the concept without the mathematical notation, so it’s clear and straightforward.

Quantile Loss Explained:

Imagine you’re predicting something, and there are different penalties depending on whether you predict too high or too low. The quantile loss lets you adjust the penalty based on the type of mistake.

• Equal Penalties: If the penalty for predicting too high is the same as predicting too low, then you’re just looking at the average absolute difference between your predictions and the actual values.
• Higher Penalty for Under-Predicting: Say, for example, in finance, if you underestimate the risk, it might cost you a lot. But if you overestimate the risk, you might just miss out on some opportunities. So, you’d want a higher penalty for under-predicting. The quantile loss lets you set this by picking a value (let’s call it “tau”) more than 0.5.
• Higher Penalty for Over-Predicting: On the other hand, consider a situation where overestimating something is more costly than underestimating it. Here, you’d pick a “tau” value less than 0.5.

So, in simple terms, the quantile loss is a way to adjust the penalties for over-predicting versus under-predicting based on the specific needs of a situation.

You:

ok tau 0.5 means what, threshold to decide over / underestimate? or weight to penalize either over or under that you want to penalize?

ChatGPT:

Great question. The value of $\tau$τ (tau) in the context of quantile loss serves as both:

• A Threshold to Decide Over/Underestimation: When τ \tau τ is set to 0.5 (the median), the model aims to minimize the absolute differences between the predicted values and the actual values, without favoring over or underestimations.
• A Weight to Penalize Over/Underestimation:
  • For τ \tau τ values greater than 0.5, the model penalizes underestimations more heavily than overestimations. The higher the τ \tau τ, the more the emphasis on penalizing under-predictions.
  • For τ \tau τ values less than 0.5, the model penalizes overestimations more than underestimations. The closer τ \tau τ is to 0, the more the emphasis on penalizing over-predictions.

So, in essence:

• If τ \tau τ = 0.5: Equal weight is given to over-predictions and under-predictions. This is akin to minimizing the Mean Absolute Error (MAE).
• If τ \tau τ > 0.5: Under-predictions are more heavily penalized.
• If τ \tau τ < 0.5: Over-predictions are more heavily penalized.

The value of $\tau$τ specifies the quantile of interest and determines how the asymmetric penalties are applied in the quantile loss.

You:

so there’s single variable tau for 1 and 2? not separate variable for those two concepts?

ChatGPT:

Yes, in the context of quantile loss, there’s a single variable, $\tau$τ, that simultaneously determines both:

• The quantile of interest.
• The asymmetry in how over-predictions and under-predictions are penalized.

This single $\tau$τ value captures the essence of both concepts. It’s a concise way of encoding the desired quantile and the corresponding asymmetry in penalties. There’s no need for separate variables because the choice of quantile inherently brings with it the associated penalty structure for over- and under-predictions.

You:

bias-variance trade off ,

expected test MSE can be decomposed into sum of var , bias, var(e) .. in my book it’s said to be out of scope of this book.

is it hard to prove it for first -2nd year college student level?

ChatGPT:

The bias-variance decomposition of the expected test Mean Squared Error (MSE) is a foundational concept in statistical learning. While the derivation can be a bit involved, it’s definitely understandable for someone at a first or second-year college level, especially with a basic understanding of statistics and expectations. Let’s go through a high-level overview of the decomposition.

Bias-Variance Tradeoff:

The expected test MSE for a given point $x_0$x0​ can be expressed as:

$E(y_0 - \hat{f}(x_0))^2$E(y0​−f^​(x0​))2

Where:

• y 0 y_0 y 0​ is the true output value for x 0 x_0 x 0​ .
• f ^ ( x 0 ) \hat{f}(x_0) f ^​ (x 0​ ) is the predicted value for x 0 x_0 x 0​ .

This expression can be decomposed into three parts:

• Variance of the model.
• Squared Bias of the model.
• Irreducible Error (which is the variance of the error term ϵ \epsilon ϵ).

Mathematically:

$E(y_0 - \hat{f}(x_0))^2 = Var(\hat{f}(x_0)) + [Bias(\hat{f}(x_0))]^2 + Var(\epsilon)$E(y0​−f^​(x0​))2=Var(f^​(x0​))+[Bias(f^​(x0​))]2+Var(ϵ)

Where:

• V a r ( f ^ ( x 0 ) ) Var(\hat{f}(x_0)) Va r( f ^​ (x 0​ )) is the variance of the model predictions.
• B i a s ( f ^ ( x 0 ) ) Bias(\hat{f}(x_0)) Bias( f ^​ (x 0​ )) is the bias of the predictions, which is the difference between the expected prediction and the true value.
• V a r ( ϵ ) Var(\epsilon) Va r(ϵ) is the irreducible error, which can’t be reduced regardless of the model used.

Derivation: The derivation involves expressing the test MSE as the expected value of the squared difference between the true value and the prediction. You then manipulate this expression by adding and subtracting the expected prediction, and applying properties of expectations and variances to break it down into the bias, variance, and irreducible error.

The essence of the bias-variance tradeoff is this:

• Models with high complexity might fit the training data very well (low bias) but can produce different predictions for different training sets (high variance).
• Simpler models might not fit the training data as closely (high bias) but are more consistent in their predictions across different training sets (low variance).

Is it hard to prove? For someone with a grounding in basic statistical concepts, it’s more about following the steps closely. The real challenge often lies in deeply understanding and internalizing the implications of the tradeoff for model building and selection.

If you’re interested in a more detailed step-by-step derivation or further explanation, please let me know!

You:

define Bias(f(x)) bias of the prediction , more mathematically for me, (Bias() would be a name for another mathematical form)

ChatGPT:

Certainly! The bias of a prediction for a given point $x_0$x0​ can be mathematically defined in terms of the expected value of our model’s predictions and the true underlying function we’re trying to estimate.

Let:

• f ( x 0 ) f(x_0) f(x 0​ ) be the true underlying value (or function value) at the point x 0 x_0 x 0​ .
• f ^ ( x 0 ) \hat{f}(x_0) f ^​ (x 0​ ) be our model’s prediction for x 0 x_0 x 0​ .

The bias for the prediction at $x_0$x0​ is then defined as:

$\text{Bias}(\hat{f}(x_0)) = E[\hat{f}(x_0)] - f(x_0)$Bias(f^​(x0​))=E[f^​(x0​)]−f(x0​)

Where $E[\hat{f}(x_0)]$E[f^​(x0​)] is the expected value of our model’s predictions at $x_0$x0​.

In words:

• The bias at x 0 x_0 x 0​ is the difference between the average prediction our model gives for x 0 x_0 x 0​ and the true value f ( x 0 ) f(x_0) f(x 0​ ).

If the bias is positive, it means, on average, our model overestimates the true function at $x_0$x0​. If it’s negative, our model underestimates it. If the bias is zero, on average, our model is spot on at $x_0$x0​.

You:

notation wise, i used to see random variable as X without subscript 0, X0 used to refer a data point, and you are using X0 as “given point” (as if indicating a data point) and using it as random variable (because u r using E() on it) , so why “data point” is being refered here?

ChatGPT:

You’re right, and I apologize for the oversight. In statistics and machine learning, notation can sometimes vary between resources, which can lead to confusion. Let’s clarify the concepts and the notation.

Let’s define things more rigorously:

• Let X X X be a random variable representing the input space.
• Let Y Y Y be a random variable representing the output space.
• We assume there’s some true underlying relationship f f f such that Y = f ( X )

ϵ Y = f(X) + \epsilon Y #

f( X) + ϵ, where ϵ \epsilon ϵ is a random error term.

• We also have a predictive model, f ^ \hat{f} f ^​ , that tries to estimate or mimic this true function f f f.

For a specific value $x$x from the domain of $X$X (a specific data point or instance):

• f ( x ) f(x) f(x) is the true output value.
• f ^ ( x ) \hat{f}(x) f ^​ (x) is the predicted value given by our model.

The bias at this specific point $x$x is:

$\text{Bias}[\hat{f}(x)] = E[\hat{f}(x)] - f(x)$Bias[f^​(x)]=E[f^​(x)]−f(x)

Where:

• E [ f ^ ( x ) ] E[\hat{f}(x)] E[ f ^​ (x)] represents the expected prediction of our model for the input x x x.
• f ( x ) f(x) f(x) is the true function value at x x x.

I hope this clarifies the notation and the concepts. Your observation is keen, and it’s essential to be precise with notation, especially in fields like statistics and machine learning, to avoid ambiguity.

You:

so here x is taken as a data point where it has fixed value such as =3 or =5, and f(x) is the random variable because our f can output different value for value x=3 ?

ChatGPT:

I see where the confusion is coming from. Let’s break it down more clearly:

• The Input x x x:
  • This is a specific value, or a realization, from the domain of the random variable X X X. It’s fixed, as you’ve noted. For instance, x = 3 x = 3 x = 3 or x = 5 x = 5 x =
• The True Function f f f:
  • This is a deterministic function that maps from the input space to the output space. For any given x x x, f ( x ) f(x) f(x) produces a fixed value. There’s no randomness in f f f itself.
• The Random Error ϵ \epsilon ϵ:
  • This is what introduces randomness. So, while f ( x ) f(x) f(x) is deterministic, the observed output Y Y Y for an input x x x is random because Y = f ( x )

  ϵ Y = f(x) + \epsilon Y #

  f(x) + ϵ. The randomness in Y Y Y comes from ϵ \epsilon ϵ, not f f f.
• **The Predictive Model f ^ \hat{f} f ^​ **:
  • This is an estimate of f f f. When trained on different datasets or under different conditions, f ^ \hat{f} f ^​ might give different predictions for the same x x x. Hence, the predictions f ^ ( x ) \hat{f}(x) f ^​ (x) can be considered random variables (due to the randomness in the training data and the learning process).

So, for a given $x$x:

• f ( x ) f(x) f(x) is deterministic and fixed.
• Y Y Y, the observed output for x x x, is random because of the error term ϵ \epsilon ϵ.
• f ^ ( x ) \hat{f}(x) f ^​ (x) can be considered a random variable because it varies based on the randomness in the training data and the learning process.

The expression $E[\hat{f}(x)]$E[f^​(x)] represents the expected prediction of our model for the input $x$x over different possible datasets or conditions under which the model might be trained.

You:

in the last sentence, the input “x” over .. here “x” is considered fixed value?

ChatGPT:

Yes, in the last sentence, the input $x$x is considered a fixed value. When we refer to $E[\hat{f}(x)]$E[f^​(x)], we’re looking at the expected value of the predictions our model would make for that specific fixed input $x$x over different possible datasets or conditions under which the model might be trained.

So, while the input $x$x remains constant, the model’s prediction $\hat{f}(x)$f^​(x) can vary due to randomness in the training data, noise, or other stochastic aspects of the learning process.

You:

ok so when u used X0 in the bias definition, you were implying X0 is fixed and F^(x0) is random variable

ChatGPT:

Correct, I used $x_0$x0​ in the earlier discussion to refer to a specific, fixed point in the input space. The notation was meant to emphasize that we’re evaluating bias at a particular input value. At this fixed input $x_0$x0​:

• f ( x 0 ) f(x_0) f(x 0​ ) is the true and deterministic output for x 0 x_0 x 0​ .
• f ^ ( x 0 ) \hat{f}(x_0) f ^​ (x 0​ ) is the model’s prediction for x 0 x_0 x 0​ , which can be thought of as a random variable because its value might vary depending on factors like the training data used, randomness in the training process, etc.

Therefore, the bias at this specific point $x_0$x0​ is:

$\text{Bias}(\hat{f}(x_0)) = E[\hat{f}(x_0)] - f(x_0)$Bias(f^​(x0​))=E[f^​(x0​)]−f(x0​)

This captures the difference between the average (or expected) prediction of the model for $x_0$x0​ and the true function value at $x_0$x0​.

You:

ok, then hard to interet what E[F^(X0)] is, we don’t know what our space is for F^(), how would one go about calculating E of something we don’t know how it looks, i guess we don’t calculate so, it’s just there as one of 3 components to make up right side to match left (test MSE)

ChatGPT:

You’re touching on a very insightful point. The decomposition of the test MSE into bias and variance is largely conceptual. It helps us understand the sources of error in our predictions, but in many practical scenarios, directly calculating each component isn’t straightforward.

Let’s delve deeper into $E[\hat{f}(x_0)]$E[f^​(x0​)]:

• Conceptual Meaning: E [ f ^ ( x 0 ) ] E[\hat{f}(x_0)] E[ f ^​ (x 0​ )] represents the average prediction our model would make at x 0 x_0 x 0​ if we were to train it over numerous different training datasets drawn from the same overall population. Each of these training datasets would give rise to a potentially different learned function f ^ \hat{f} f ^​ . The expected value is then the average of all these predictions.
• Practical Challenges:
  • In a real-world setting, we typically don’t have access to infinite samples from our population to create countless training datasets.
  • The space of all possible functions f ^ ( ) \hat{f}() f ^​ () is vast and isn’t explicitly known.
• Bootstrap Methods: One approach to approximate E [ f ^ ( x 0 ) ] E[\hat{f}(x_0)] E[ f ^​ (x 0​ )] and the variance term is to use bootstrapping. By repeatedly resampling our single training dataset and training our model on these resamples, we can obtain different versions of f ^ \hat{f} f ^​ and thereby approximate the expected value and variance of our model’s predictions. This doesn’t fully capture the true underlying variability across all possible training datasets from the entire population, but it gives a practical estimate.
• Main Takeaway: The bias-variance decomposition is a powerful conceptual tool. It helps us understand the trade-offs in model complexity. Simple models might have high bias (systematic errors in prediction) but low variance (consistent predictions across different training sets). Complex models might have low bias but high variance, meaning they fit training data closely but may vary wildly in their predictions based on which specific data they’re trained on. While directly calculating bias and variance isn’t always feasible, the conceptual understanding guides decisions in model selection and regularization.

To summarize, you’re right that in many cases, the bias-variance decomposition is more of a conceptual framework than a formula we compute directly. It provides insight into the sources of prediction error and informs model-building strategies.

You:

bias-variance decomposition is also using X0, so it also states relationship on single point,

ChatGPT:

Yes, you’re right. The bias-variance decomposition is often presented with respect to a specific point $x_0$x0​ in the input space. Let’s clarify the decomposition in that context:

Given a regression problem, suppose:

• f ( x 0 ) f(x_0) f(x 0​ ) is the true function value at x 0 x_0 x 0​ .
• f ^ ( x 0 ) \hat{f}(x_0) f ^​ (x 0​ ) is the predicted value at x 0 x_0 x 0​ based on our model trained on a particular dataset.
• Y Y Y is the observed value at x 0 x_0 x 0​ , which would be f ( x 0 )

ϵ f(x_0) + \epsilon f(x 0​ ) + ϵ where ϵ \epsilon ϵ is random noise.

The expected test Mean Squared Error (MSE) at $x_0$x0​ can be decomposed as:

$E[(Y - \hat{f}(x_0))^2] = \text{Bias}^2[\hat{f}(x_0)] + \text{Var}[\hat{f}(x_0)] + \text{Var}[\epsilon]$E[(Y−f^​(x0​))2]=Bias2[f^​(x0​)]+Var[f^​(x0​)]+Var[ϵ]

Where:

• Bias [ f ^ ( x 0 ) ] = E [ f ^ ( x 0 ) ] − f ( x 0 ) \text{Bias}[\hat{f}(x_0)] = E[\hat{f}(x_0)] - f(x_0) Bias[ f ^​ (x 0​ )] = E[ f ^​ (x 0​ )] − f(x 0​ ) represents the difference between the average prediction of the model at x 0 x_0 x 0​ and the true function value at x 0 x_0 x 0​ .
• Var [ f ^ ( x 0 ) ] \text{Var}[\hat{f}(x_0)] Var[ f ^​ (x 0​ )] represents the variance of the model’s predictions at x 0 x_0 x 0​ , reflecting how much the predictions fluctuate for different training datasets.
• Var [ ϵ ] \text{Var}[\epsilon] Var[ϵ] represents the irreducible error, capturing the inherent variability in the data not explained by the model.

It’s important to note that while the decomposition is expressed at a specific point $x_0$x0​, the concepts of bias and variance are applicable across the input space. The decomposition helps understand the error sources at a given point, but similar considerations apply more broadly.

In practice, one might be interested in the average bias, variance, and MSE across all points in a dataset or domain, leading to aggregated measures that provide a holistic view of model performance.

You:

i guess you can take another E over entire thing on both sides, treating the entire thing as random variable, and we take expected value of that, then we get.. aggregate result.. which would be expressed as..

ChatGPT:

Yes, you’re on the right track! Taking an expectation over the entire input space will give you an aggregate measure of the bias and variance, which summarizes the overall performance of the model across all input points.

To be more explicit:

Given the earlier decomposition for a specific point $x_0$x0​:

$E[(Y - \hat{f}(x_0))^2] = \text{Bias}^2[\hat{f}(x_0)] + \text{Var}[\hat{f}(x_0)] + \text{Var}[\epsilon]$E[(Y−f^​(x0​))2]=Bias2[f^​(x0​)]+Var[f^​(x0​)]+Var[ϵ]

To aggregate over the entire input space, take an expectation with respect to $X$X:

$E_X[E[(Y - \hat{f}(X))^2]] = E_X[\text{Bias}^2[\hat{f}(X)] + \text{Var}[\hat{f}(X)] + \text{Var}[\epsilon]]$EX​[E[(Y−f^​(X))2]]=EX​[Bias2[f^​(X)]+Var[f^​(X)]+Var[ϵ]]

This can be decomposed into:

$E_X[E[(Y - \hat{f}(X))^2]] = E_X[\text{Bias}^2[\hat{f}(X)]] + E_X[\text{Var}[\hat{f}(X)]] + E_X[\text{Var}[\epsilon]]$EX​[E[(Y−f^​(X))2]]=EX​[Bias2[f^​(X)]]+EX​[Var[f^​(X)]]+EX​[Var[ϵ]]

Where:

• E X [ Bias 2 [ f ^ ( X ) ] ] E_X[\text{Bias}^2[\hat{f}(X)]] E X​ [Bias 2[ f ^​ ( X)]] is the average squared bias across the input space.
• E X [ Var [ f ^ ( X ) ] ] E_X[\text{Var}[\hat{f}(X)]] E X​ [Var[ f ^​ ( X)]] is the average variance across the input space.
• E X [ Var [ ϵ ] ] E_X[\text{Var}[\epsilon]] E X​ [Var[ϵ]] is the average irreducible error across the input space.

This gives an overall measure of the model’s bias, variance, and error across all possible input values. It’s a more holistic view of the trade-offs the model makes between fitting the training data (bias) and generalizing to new data (variance), as well as capturing the inherent noise in the data (irreducible error).

You:

ok, so derivation is not that hard, but after deriving it, it looks like something i’ve seen before.

i think it was something like, variance of many groups can be computed as sum of two things, one how far my group average is from the average of the whole and another what my group’s variance is, you add those two, you get variance of whole group. I think it was eve’s law or something,

and the bias variance trade off reminds of me that formular.. cant describe the connection precisely though

ChatGPT:

You’re thinking of the “law of total variance” or sometimes referred to as “Eve’s law” (though “law of total variance” is the more common term). And yes, there is a conceptual similarity between the bias-variance decomposition and the law of total variance.

Law of Total Variance:

Given two random variables $X$X and $Y$Y, the variance of $Y$Y can be decomposed as the sum of the expected variance of $Y$Y given $X$X and the variance of the expected value of $Y$Y given $X$X:

$\text{Var}(Y) = E[\text{Var}(Y|X)] + \text{Var}(E[Y|X])$Var(Y)=E[Var(Y∣X)]+Var(E[Y∣X])

The first term $E[\text{Var}(Y|X)]$E[Var(Y∣X)] represents the average variance of $Y$Y within each “group” defined by the values of $X$X, while the second term $\text{Var}(E[Y|X])$Var(E[Y∣X]) represents the variance between the expected values of these groups.

Connection to Bias-Variance Decomposition:

The bias-variance decomposition conceptually breaks down the expected test error (or MSE) into components that explain different sources of error. Similarly, the law of total variance breaks down the total variance of a random variable into components that explain variability within groups and between groups.

The parallel can be seen in how we decompose the variability:

• In the bias-variance decomposition, the “bias” captures systematic errors (how far our model’s average prediction is from the truth), while “variance” captures the model’s variability over different training datasets.
• In the law of total variance, Var ( E [ Y ∣ X ] ) \text{Var}(E[Y|X]) Var( E[ Y∣ X]) captures the variability between the group averages, while E [ Var ( Y ∣ X ) ] E[\text{Var}(Y|X)] E[Var( Y∣ X)] captures the average variability within each group.

Both frameworks offer a decomposition that helps provide insight into the sources of variability or error, even though they address slightly different problems. Your intuition to see a connection between the two is quite keen, as they both use similar mathematical and conceptual tools to break down complex phenomena into more interpretable components.

You:

can you prove law of total variance, doenst need to be mathematically rigrous, i want to see why it make sense intuitively

ChatGPT:

Certainly! The Law of Total Variance can be understood intuitively by thinking of variance as the “spread” or “dispersion” of data, and considering this spread at two different levels: within groups and between groups.

Intuitive Explanation of the Law of Total Variance:

• Imagine Grouping Data: Suppose you have a large dataset of people’s heights, and you decide to group them by nationality. Now, each nationality will have its own average height and its own spread of heights around that average.
• Variance Within Groups: Within each nationality (or group), there will be a certain amount of variance in height. Some people will be taller than the average for their group, and some will be shorter. This variance represents the spread of heights within each specific group. When you average these variances across all groups, you get E [ Var ( Y ∣ X ) ] E[\text{Var}(Y|X)] E[Var( Y∣ X)], the expected variance of Y Y Y within the groups defined by X X X.
• Variance Between Groups: If you were to look only at the average height for each nationality and ignore individual heights, you’d notice that these averages also vary. Some nationalities might have higher average heights, while others have lower averages. This variance represents the spread of the group averages, and is captured by Var ( E [ Y ∣ X ] ) \text{Var}(E[Y|X]) Var( E[ Y∣ X]).
• Combining the Two: The total variance in height across all individuals (ignoring nationality) will be a combination of:
  • The variance within each nationality (how much individuals vary from their own group’s average).
  • The variance between nationalities (how much each group’s average varies from the overall average).

Mathematically, this is represented by:

$\text{Var}(Y) = E[\text{Var}(Y|X)] + \text{Var}(E[Y|X])$Var(Y)=E[Var(Y∣X)]+Var(E[Y∣X])

In Simple Terms: The total variability in a dataset (like people’s heights) can be thought of as the sum of the variability within specific groups (variance of individual heights around their group’s average) and the variability between those groups (variance of the group averages around the overall average).

This intuitive explanation paints a picture of how data can vary at different levels, and the Law of Total Variance provides a mathematical way to capture and quantify this idea.

You:

good, let’s go little deeper, why it’s true, i guess we need some math here

ChatGPT:

Alright, let’s dive a bit deeper into the mathematics of the Law of Total Variance. I’ll try to balance rigor with clarity.

Proof of the Law of Total Variance:

We want to prove:

$\text{Var}(Y) = E[\text{Var}(Y|X)] + \text{Var}(E[Y|X])$Var(Y)=E[Var(Y∣X)]+Var(E[Y∣X])

• Start with the Definition of Variance:

$\text{Var}(Y) = E[Y^2] - (E[Y])^2$Var(Y)=E[Y2]−(E[Y])2

• Decompose Y 2 Y^2 Y 2 using Conditional Expectation:

Using the law of iterated expectations (which states that $E[E[Y|X]] = E[Y]$E[E[Y∣X]]=E[Y]), we can write:

$E[Y^2] = E[E[Y^2|X]]$E[Y2]=E[E[Y2∣X]]

• Now, use the Definition of Variance Again, but Conditioned on X X X:

$\text{Var}(Y|X) = E[Y^2|X] - (E[Y|X])^2$Var(Y∣X)=E[Y2∣X]−(E[Y∣X])2

From which:

$E[Y^2|X] = \text{Var}(Y|X) + (E[Y|X])^2$E[Y2∣X]=Var(Y∣X)+(E[Y∣X])2

Plugging this into our expression for $E[Y^2]$E[Y2]:

$E[Y^2] = E[\text{Var}(Y|X) + (E[Y|X])^2]$E[Y2]=E[Var(Y∣X)+(E[Y∣X])2]

Breaking this down, we get:

$E[Y^2] = E[\text{Var}(Y|X)] + E[(E[Y|X])^2]$E[Y2]=E[Var(Y∣X)]+E[(E[Y∣X])2]

• Combine the Expressions:

Now, let’s combine our expressions to get the total variance:

$\text{Var}(Y) = E[\text{Var}(Y|X)] + E[(E[Y|X])^2] - (E[Y])^2$Var(Y)=E[Var(Y∣X)]+E[(E[Y∣X])2]−(E[Y])2

Notice that $E[(E[Y|X])^2] - (E[Y])^2$E[(E[Y∣X])2]−(E[Y])2 is the variance of $E[Y|X]$E[Y∣X] (because variance is defined as the expected value of the square minus the square of the expected value). So, we can rewrite it as:

$\text{Var}(Y) = E[\text{Var}(Y|X)] + \text{Var}(E[Y|X])$Var(Y)=E[Var(Y∣X)]+Var(E[Y∣X])

And that completes the proof!

Intuitively, this proof essentially formalizes our earlier discussion. It breaks down the total variance of $Y$Y into two parts:

• The average variance of Y Y Y within each group defined by X X X (the “within-group” variability).
• The variance between the averages of these groups (the “between-group” variability).

The mathematics here confirms our intuitive understanding: the total variability in a dataset is a combination of the variability within groups and the variability between those groups.

You:

ok, i need refresher on what it means by E(Y|X)

ChatGPT:

Absolutely, let’s break it down.

The notation $E[Y|X]$E[Y∣X] refers to the conditional expectation of $Y$Y given $X$X. In simple terms, it’s the average or expected value of $Y$Y when we have some specific information about $X$X.

Intuitive Understanding:

Suppose $Y$Y represents the weight of a person and $X$X represents the height of a person.

• E [ Y ] E[Y] E[ Y] would represent the average weight of all people in the dataset, without considering any other information.
• E [ Y ∣ X = x ] E[Y|X = x] E[ Y∣ X = x] would represent the average weight of people who are of height x x x. In other words, it’s the average weight but only considering individuals who have a specific height x x x.
• E [ Y ∣ X ] E[Y|X] E[ Y∣ X] represents a function that gives the expected value of Y Y Y for every possible value of X X X. So, as X X X changes, E [ Y ∣ X ] E[Y|X] E[ Y∣ X] gives the expected value of Y Y Y corresponding to that specific X X X.

Mathematical Understanding:

Formally, the conditional expectation $E[Y|X]$E[Y∣X] is defined as:

$E[Y|X] = \sum_{y} y \cdot P(Y = y|X)$E[Y∣X]=∑y​y⋅P(Y=y∣X)

for discrete random variables, and as:

$E[Y|X] = \int y \cdot f_{Y|X}(y|x) , dy$E[Y∣X]=∫y⋅fY∣X​(y∣x)dy

for continuous random variables, where $f_{Y|X}(y|x)$fY∣X​(y∣x) is the conditional probability density function of $Y$Y given $X = x$X=x.

Key Takeaway:

The conditional expectation, $E[Y|X]$E[Y∣X], tells us the expected (or average) value of $Y$Y for a given value of $X$X. It allows us to refine our predictions or expectations by incorporating additional information.

You:

So E(Y|X) is actually E(Y|X=x) , where Y is averaged over possible values, but not averaged over X values

ChatGPT:

Yes, you’ve captured the essence correctly. Let’s clarify it a bit further:

• ** E [ Y ∣ X = x ] E[Y|X=x] E[ Y∣ X = x]**:
  • This is the expected value of Y Y Y when X X X is specifically known to be x x x. It’s a conditional average of Y Y Y values for this specific x x x, without considering other X X X values. It gives you a single number, the expected value of Y Y Y for that specific x x x.
• ** E [ Y ∣ X ] E[Y|X] E[ Y∣ X]**:
  • This is a more general notation. It’s not just a single number but a function over X X X. For every possible value x x x that X X X can take, E [ Y ∣ X ] E[Y|X] E[ Y∣ X] gives you E [ Y ∣ X = x ] E[Y|X=x] E[ Y∣ X = x]. So, it describes how the expected value of Y Y Y changes as X X X changes.

To make this clearer, consider the example I provided about height and weight:

• E [ Weight ∣ Height = 180 cm ] E[ \text{Weight} | \text{Height} = 180\text{cm}] E[Weight∣Height = 180cm]: This gives the average weight of individuals who are exactly 180cm tall. It’s a specific number, say 75kg.
• E [ Weight ∣ Height ] E[ \text{Weight} | \text{Height}] E[Weight∣Height]: This gives you a function. For every possible height, it tells you the average weight of individuals of that height. So, you might learn from this function that people who are 160cm tall average 60kg, people who are 170cm tall average 68kg, and so on for all possible heights.

In summary, $E[Y|X=x]$E[Y∣X=x] provides a specific expected value for a given $x$x, while $E[Y|X]$E[Y∣X] describes how these expected values change for all possible values of $X$X.

You:

in previous my statement, i wanted emphasize X is not averaged over, and X=x denotes X is not fixed, but can take any x,

ChatGPT:

Understood. You’re highlighting the essence of conditional expectation. Let’s clarify based on your emphasis.

The expression $E[Y|X=x]$E[Y∣X=x]:

• Is the expected value of Y Y Y when we observe a particular value of X X X, denoted by x x x.
• X X X is not “fixed” in the sense of its value being predetermined; instead, the x x x in the expression is a placeholder, denoting that we’re conditioning on whatever specific value X X X takes.
• This means that X X X can indeed take on any value from its domain, and for each such value, E [ Y ∣ X = x ] E[Y|X=x] E[ Y∣ X = x] gives us a corresponding expected value for Y Y Y.
• We are, as you rightly pointed out, averaging over possible outcomes of Y Y Y but not averaging over X X X values.

When you look at $E[Y|X]$E[Y∣X] without the $=x$=x:

• This represents a function over all values of X X X.
• For each value x x x that X X X might take, this function gives us E [ Y ∣ X = x ] E[Y|X=x] E[ Y∣ X = x].

Your emphasis is spot-on. The whole idea behind conditional expectation is to refine our understanding of one variable (in this case, $Y$Y) given specific knowledge about another variable (in this case, $X$X), without “averaging out” the latter.

You:

what would be the intuitive explanation on why E(Y|X) is a projection of Y onto plane of function of X ?

ChatGPT:

Alright, let’s delve into this conceptually.

The idea of a “projection” in this context borrows from the geometric interpretation common in linear algebra. When you project a vector onto a subspace, you’re finding the closest representation of that vector within that subspace. The difference between the original vector and its projection is the error or the orthogonal distance to the subspace.

Let’s translate this into our context of $E[Y|X]$E[Y∣X]:

• Function Space: Think of all possible functions of X X X as defining a “function space”. Any function you can come up with that’s based on X X X belongs in this space.
• “Projection” of Y onto Functions of X: When we talk about E [ Y ∣ X ] E[Y|X] E[ Y∣ X] being a projection, we’re saying it’s the “closest” function of X X X that represents Y Y Y, in a mean-squared-error sense. In other words, out of all possible functions of X X X, E [ Y ∣ X ] E[Y|X] E[ Y∣ X] is the one that, on average, is the closest to Y Y Y.
• Orthogonality of the Error: The error term, Y − E [ Y ∣ X ] Y - E[Y|X] Y − E[ Y∣ X], is “orthogonal” to our space of functions of X X X. This means that the error doesn’t show any systematic relationship with X X X. It’s the part of Y Y Y that cannot be explained by X X X.

To connect back with the geometric intuition:

• Think of Y Y Y as a vector in some high-dimensional space.
• The space of functions of X X X represents a subspace within this larger space.
• E [ Y ∣ X ] E[Y|X] E[ Y∣ X] is the projection of Y Y Y onto this subspace: it’s the best representation of Y Y Y within the subspace defined by functions of X X X.
• The error Y − E [ Y ∣ X ] Y - E[Y|X] Y − E[ Y∣ X] is orthogonal to this subspace: it represents the component of Y Y Y that can’t be captured by functions of X X X.

Intuitive Picture:

Imagine you’re trying to approximate the height of individuals (Y) based on their age (X). Now, there’s a general trend: as age goes from infancy to adulthood, height increases. This trend can be captured by some function of age, say a curve.

If $E[Y|X]$E[Y∣X] represents this curve, then projecting the actual heights of individuals onto this curve would give you the predicted heights based on age. The vertical distances between the actual heights and the curve would represent the errors. And these errors wouldn’t show any systematic trend with age—they’d be random variations around the curve.

In this way, $E[Y|X]$E[Y∣X] gives the “best” approximation of Y using X, in terms of minimizing the squared differences (errors) between the actual and predicted values.

You:

ok, so we are saying E(Y-E[Y|X])^2 is minimum when we use E[Y|X] (among all function of X, so here E(Y|X) is considered a function of X), is the proof approachable to 1-2 year college student?

ChatGPT:

Absolutely. The idea that $E[Y|X]$E[Y∣X] minimizes $E[(Y - g(X))^2]$E[(Y−g(X))2] over all functions $g(X)$g(X) of $X$X is foundational to understanding conditional expectation. The proof for this statement involves some calculus and basic probability, but it should be approachable to someone with 1-2 years of college-level mathematics. Here’s a simplified version of the proof:

Goal: To show that $E[Y|X]$E[Y∣X] minimizes the expected squared difference, we’ll differentiate with respect to $g(X)$g(X) and set the result to zero (a common approach to finding minima or maxima).

Setup: Consider any function $g(X)$g(X). The difference between $Y$Y and $g(X)$g(X) is given by: $\text{Error} = Y - g(X)$Error=Y−g(X)

The expected squared difference is: $E[(Y - g(X))^2]$E[(Y−g(X))2]

Proof:

• Expand the squared term: E [ Y 2 − 2 Y g ( X )

g ( X ) 2 ] E[Y^2 - 2Yg(X) + g(X)^2] E[ Y 2 − 2 Y g( X) + g( X) 2]

• Consider each term separately:
  • E [ Y 2 ] E[Y^2] E[ Y 2] is not affected by our choice of g ( X ) g(X) g( X), so when differentiating, it will drop out.
  • E [ g ( X ) 2 ] E[g(X)^2] E[ g( X) 2] is a positive constant with respect to Y Y Y. The term to focus on is E [ 2 Y g ( X ) ] E[2Yg(X)] E[2 Y g( X)]. We want to maximize this term because it will help minimize the negative term in the expected squared difference.
• To find the function g ( X ) g(X) g( X) that maximizes E [ 2 Y g ( X ) ] E[2Yg(X)] E[2 Y g( X)], differentiate with respect to g ( X ) g(X) g( X).

This leads to the conclusion that g ( X ) g(X) g( X) should be proportional to the conditional expectation E [ Y ∣ X ] E[Y|X] E[ Y∣ X].

• So among all functions g ( X ) g(X) g( X), E [ Y ∣ X ] E[Y|X] E[ Y∣ X] is the one that minimizes E [ ( Y − g ( X ) ) 2 ] E[(Y - g(X))^2] E[( Y − g( X)) 2].

While this outline is a bit hand-wavy, it captures the essence of the argument. A more rigorous proof would involve deeper integration and differentiation properties and possibly the use of functional analysis, but for an intuitive understanding, this approach should be suitable for someone with 1-2 years of college-level mathematics.

You:

what do you mean by differentiate with respect to g(X), i’m familiar with.. “with respect to X, Y .. or some variable), but you are using it with some function not variable, you are treating g(X) as a variable ?

ChatGPT:

I apologize for the oversight. You’re right; my explanation was hand-wavy and skipped over some essential steps. Let’s take a more direct and clear approach.

Instead of considering the function $g(X)$g(X) directly, think of $g$g as a family of functions parameterized by some parameters $\theta$θ. For instance, if you’re thinking about linear functions, $\theta$θ could represent the slope and intercept.

Now, the idea is to optimize over the parameter space $\theta$θ, rather than the entire function space. This way, differentiation with respect to these parameters becomes straightforward.

Let’s re-approach the problem:

Objective: Find the function $g_\theta(X)$gθ​(X) in a family of functions parameterized by $\theta$θ that minimizes: $E[(Y - g_\theta(X))^2]$E[(Y−gθ​(X))2]

Proof:

• Consider the quadratic loss for a given θ \theta θ: L ( θ ) = E [ ( Y − g θ ( X ) ) 2 ] L(\theta) = E[(Y - g_\theta(X))^2] L( θ) = E[( Y − g θ​ ( X)) 2]
• To minimize L ( θ ) L(\theta) L( θ), differentiate it with respect to θ \theta θ. You’ll set this derivative (or gradient, if θ \theta θ is multi-dimensional) to zero and solve for θ \theta θ.
• If g θ ( X ) g_\theta(X) g θ​ ( X) is flexible enough (like neural networks or polynomial regressions with high degree), the best g θ ( X ) g_\theta(X) g θ​ ( X) in terms of minimizing the expectation will approach E [ Y ∣ X ] E[Y|X] E[ Y∣ X].

However, it’s essential to realize that this is still a simplification, and the full proof in its most general form, considering arbitrary functions, would require functional analysis and is beyond standard undergraduate-level material.

To reiterate: you’re right, and I should have been clearer. Differentiating with respect to a function is a concept in functional analysis, and the casual way I approached it previously was not rigorous. The correct way to handle this problem at the undergraduate level is to think in terms of parameterized families of functions, as outlined above.

You:

I think (without going too rigourous) if you can delve deeper with differentiate with respect to g(X), how would you differentiate E(2Y*g(X)) ? I guess that will bring the condition |X somehow, and it might be easier for me to think (even though the differntiate with respec to g(X) is somewhat non rigorous)

ChatGPT:

Alright, I understand your curiosity. Let’s try to sketch out a more intuitive, if not fully rigorous, approach using the idea of differentiating with respect to $g(X)$g(X).

Objective: We want to find the function $g(X)$g(X) that minimizes: $E[(Y - g(X))^2]$E[(Y−g(X))2]

We focus on the term $-2E[Yg(X)]$−2E[Yg(X)] because optimizing this term will inherently optimize our objective.

Let’s consider a small perturbation $\delta$δ to $g(X)$g(X). That is, we’re changing our function a tiny bit and want to see how our objective changes in response.

The change in our term would be:

$\delta E[Yg(X)] = E[Y(g(X) + \delta)] - E[Yg(X)]$δE[Yg(X)]=E[Y(g(X)+δ)]−E[Yg(X)]

Now, for the perturbation $\delta$δ to make no difference, we would want:

$E[Y(g(X) + \delta)] \approx E[Yg(X)]$E[Y(g(X)+δ)]≈E[Yg(X)] or $E[Y\delta] \approx 0$E[Yδ]≈0

However, to make $E[Y\delta] = 0$E[Yδ]=0 for all possible functions $\delta(X)$δ(X), we need the condition:

$E[Y|X] = g(X)$E[Y∣X]=g(X)

This is because, given $X$X, the “best” (in terms of minimizing expected squared difference) guess for $Y$Y is its expected value given $X$X, which is $E[Y|X]$E[Y∣X].

So, intuitively, by trying to maximize the term $E[Yg(X)]$E[Yg(X)], or minimize the expected squared difference, you end up with the condition that $g(X)$g(X) should be equal to $E[Y|X]$E[Y∣X] to be optimal.

Again, this is a sketch of the idea rather than a rigorous proof. The rigorous proof would involve functional analysis and considerations of Hilbert spaces, but hopefully, this provides some intuition about why $E[Y|X]$E[Y∣X] emerges as the optimal “function” of $X$X.

You:

i guess it’s more like.. for all possible functions delta(g(X)) , we need condition E(Y|X) = g(X), (although I don’t see why that is true, the delta is on g(X) since we are differentiating with respect to g(X) i thnk?)

ChatGPT:

Yes, your understanding aligns with the general intuition. The idea is to think about how small perturbations in $g(X)$g(X) would affect the objective function, and then to set constraints based on these perturbations to find the optimal $g(X)$g(X).

Let’s reframe the explanation in terms of the perturbation:

Objective: We want to minimize the function: $L(g) = E[(Y - g(X))^2]$L(g)=E[(Y−g(X))2]

Perturbing $g(X)$g(X): Consider a small perturbation $\delta$δ to $g(X)$g(X), such that our function becomes $g(X) + \delta(X)$g(X)+δ(X). This perturbation $\delta(X)$δ(X) represents a small change in the function $g(X)$g(X) for every possible value of $X$X.

The change in the loss due to this perturbation would be (ignoring higher-order terms): $\Delta L = E[2(Y - g(X))(-\delta(X))] = -2E[\delta(X)(Y - g(X))]$ΔL=E[2(Y−g(X))(−δ(X))]=−2E[δ(X)(Y−g(X))]

To find the $g(X)$g(X) that minimizes $L(g)$L(g), this change $\Delta L$ΔL should be zero for all possible perturbations $\delta(X)$δ(X). Thus: $E[\delta(X)(Y - g(X))] = 0$E[δ(X)(Y−g(X))]=0 for all possible $\delta(X)$δ(X).

Now, for the above equation to hold true for any arbitrary function $\delta(X)$δ(X), the term inside the expectation must be zero almost everywhere, leading us to: $Y - g(X) = 0$Y−g(X)=0 or $g(X) = E[Y|X]$g(X)=E[Y∣X]

This gives us the intuition that the best guess for $Y$Y given $X$X, in terms of minimizing expected squared error, is the conditional expectation $E[Y|X]$E[Y∣X].

To reiterate, the approach uses the concept of a perturbation to get an understanding of how changing the function $g(X)$g(X) affects the loss, and then setting conditions based on these changes to determine the optimal $g(X)$g(X). This is a non-rigorous but intuitive way to understand why $E[Y|X]$E[Y∣X] emerges as the optimal prediction for $Y$Y given $X$X.

You:

ok, conceptually hard part is, that we are trying to optimized function, and we are looking at , all possible values of X, intuitively, it looks like, we are trying .. optimizing locally, to achieve global property. (this is just guesswork because i’m not understanding much.. or not understanding that part specifically)

ChatGPT:

You’ve actually touched on a deep and insightful observation. Let’s try to parse this out further.

In many optimization problems, you might try to find the maximum or minimum of a function by looking at each point in its domain and adjusting it. In simpler terms, you’re doing “local” optimizations at each point, and the hope is that by doing the best thing at every single point, you’ll end up doing the best thing overall, which is a “global” optimization.

When we’re trying to find the best function $g(X)$g(X) that predicts $Y$Y, we’re indeed looking at all possible values of $X$X and seeing what function does the best job, on average, across all these values. In a sense, the function $E[Y|X]$E[Y∣X] is giving us the “best” prediction for $Y$Y for every single value of $X$X. So for every specific $X = x$X=x, $E[Y|X = x]$E[Y∣X=x] is the best guess for $Y$Y. It’s a function that locally optimizes the prediction for each value of $X$X.

But here’s the neat part: this local optimization for every $X = x$X=x also ends up being the best global choice when you average over all possible values of $X$X. In other words, by doing the best thing for each specific value of $X$X, the function $E[Y|X]$E[Y∣X] also does the best thing on average across all $X$X.

This alignment between local and global optimization is quite special. In many optimization problems, what’s best locally isn’t always best globally. But in the case of finding the function that predicts $Y$Y from $X$X in the least squares sense, the local predictions $E[Y|X = x]$E[Y∣X=x] combine to give the globally optimal prediction function $E[Y|X]$E[Y∣X].

Your intuition is correct in recognizing that there’s a mix of local and global optimization happening here.

You:

ok, so it was not too much off the course. thanks for explanation

ChatGPT:

You’re welcome! Your intuition and curiosity are spot on. These types of questions and explorations are foundational to a deep understanding of the subject. Always feel free to ask and delve deeper into topics; it’s a great way to solidify knowledge. If you have any more questions, whether they’re about statistics, another subject, or anything else, I’m here to help!

You:

So sayng E(Y|X) = g(X) means, we are assuming any distribution is inherently fixed although we don’t know what it is. it seems so because only X remains

this question can be raised against just plain E(X) as well. When we treat it as constant, we are assuming the distribution is fixed. I guess, interpretation can be, we only deal with distribution that we assume fixed. so if there’s a family of distribution that changes over time, then we assume we know the effect of time, and the entire distribution of current / past / future, and do calculate E(X) considering time.

ChatGPT:

Your observations are astute. Let’s delve into them.

• Fixed Distribution Assumption:
  • When we talk about expectations like E [ Y ∣ X ] E[Y|X] E[ Y∣ X] or E [ X ] E E[ X], we’re assuming a fixed (but possibly unknown) joint distribution of the random variables involved. This means we’re operating under the belief that there’s some underlying probability distribution that governs the behavior of these variables.
  • For E [ Y ∣ X ] E[Y|X] E[ Y∣ X], the statement that E [ Y ∣ X ] = g ( X ) E[Y|X] = g(X) E[ Y∣ X] = g( X) means that the best prediction of Y Y Y for a given value of X X X, under the mean-squared error criterion, is some function g ( X ) g(X) g( X) of that value. Again, this prediction is based on the assumed underlying joint distribution of X X X and Y Y Y.
• Time-Varying Distributions:
  • In many real-world scenarios, the distribution of a random variable might change over time. This is often seen in financial markets, weather patterns, etc.
  • When dealing with time-varying distributions, one common approach is to include time explicitly as a variable. So instead of E [ X ] E E[ X], you’d look at E [ X t ] E[X_t] E[ X t​ ] where t t t denotes time. If the distribution changes with time, then E [ X t ] E[X_t] E[ X t​ ] will vary with t t t.
  • Another approach is to use a “sliding window” to estimate expectations. For example, to estimate today’s expected value, you might only use data from the past N N N days, effectively updating your estimate as new data comes in.
• Model Assumptions:
  • All statistical models, whether explicitly stated or not, operate under certain assumptions. The assumption of a fixed distribution is just one of them.
  • Recognizing and understanding these assumptions is crucial. It helps determine the applicability of a model to real-world data and its potential limitations.
• Changing Distributions:
  • When distributions change in ways that aren’t captured by our models (like in unexpected external shocks or “black swan” events), our models can break down.
  • In these scenarios, recognizing the breakdown and adapting is crucial. This might involve incorporating more variables, changing the model structure, or seeking new modeling techniques.

In essence, while many classical statistical techniques assume fixed distributions, there’s an increasing awareness and methodology in handling time-varying or changing distributions, especially in fields where this is the norm rather than the exception.