Week 37: Statistical interpretations and Resampling Methods
Contents
Plans for week 37
Material from last week and relevant for the weekly exercises
Linking the regression analysis with a statistical interpretation
Assumptions made
Expectation value and variance
Expectation value and variance for \( \boldsymbol{\beta} \)
Material for lecture Thursday September 14
Deriving OLS from a probability distribution
Independent and Identically Distrubuted (iid)
Maximum Likelihood Estimation (MLE)
A new Cost Function
More basic Statistics and Bayes' theorem
Marginal Probability
Conditional Probability
Bayes' Theorem
Interpretations of Bayes' Theorem
Example of Usage of Bayes' theorem
Doing it correctly
Bayes' Theorem and Ridge and Lasso Regression
Ridge and Bayes
Lasso and Bayes
Why resampling methods
Resampling methods
Resampling approaches can be computationally expensive
Why resampling methods ?
Statistical analysis
Resampling methods
Resampling methods: Bootstrap
The Central Limit Theorem
Finding the Limit
Rewriting the \( \delta \)-function
Identifying Terms
Wrapping it up
Confidence Intervals
Standard Approach based on the Normal Distribution
Resampling methods: Bootstrap background
Resampling methods: More Bootstrap background
Resampling methods: Bootstrap approach
Resampling methods: Bootstrap steps
Code example for the Bootstrap method
Plotting the Histogram
The bias-variance tradeoff
A way to Read the Bias-Variance Tradeoff
Example code for Bias-Variance tradeoff
Understanding what happens
Summing up
Another Example from Scikit-Learn's Repository
Various steps in cross-validation
Cross-validation in brief
Code Example for Cross-validation and \( k \)-fold Cross-validation
More examples on bootstrap and cross-validation and errors
The same example but now with cross-validation
Notes on scaling with examples
Conditional Probability
The conditional probability, if \( p(Y) > 0 \), is
$$ p(X\vert Y)= \frac{p(X,Y)}{p(Y)}=\frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}. $$
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