Probability Distributions and Conditional Distribution

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What is the main purpose of statistical modeling?

To simulate natural processes exactly as they occur

To represent the nature of reality through various lenses (correct)

To create artificial constructions without any abstraction

To ignore extraneous details in the model

In statistical modeling, what do probability distributions help with?

Assigning probabilities to all possible outcomes

Modeling natural processes exactly

Estimating parameters from the data (correct)

Visualizing the data flow

What do mean (μ) and parameter σ control in probability distributions?

Both μ and σ control the centering of the distribution

Both μ and σ control the shape of the distribution

μ controls where the distribution is centered, and σ controls how spread out it is (correct)

σ controls where the distribution is centered, and μ controls its spread

What is one key factor to consider when analyzing a model for overlooked details?

Reviewing all abstracted details that were removed during model creation Signup and view all the answers

Why are probability distributions considered as building blocks of statistical models?

Because they approximate shapes of natural processes with few parameters Signup and view all the answers

What helps draw a picture of the underlying process in statistical modeling?

A mathematical diagram of data flow Signup and view all the answers

What does the conditional distribution p(x|y) represent?

The density function of x given a particular value of y Signup and view all the answers

In the context of modeling, what does fitting a model involve?

Estimating the parameters of the model using observed data Signup and view all the answers

What is meant by overfitting a model?

The model captures noise in the data as if it were a pattern Signup and view all the answers

When fitting a model, what are the estimators obtained from the data?

Functions of data used to estimate the model parameters Signup and view all the answers

Which optimization methods are commonly used in fitting models?

Gradient Descent and Newton's Method Signup and view all the answers

In a multivariate setting, what does the joint distribution represent?

The distribution involving all variables simultaneously Signup and view all the answers

What is the process to find the best r-dimensional approximation to a dataset D?

Computing the eigenvalues of Σ, sorting them, and selecting the r largest eigenvalues with their corresponding eigenvectors Signup and view all the answers

How is the r-dimensional subspace chosen in principal component analysis?

By selecting the dimensions that capture at least a specified variance threshold α Signup and view all the answers

What does the value α represent when choosing the dimensionality in principal component analysis?

A desired variance threshold Signup and view all the answers

In practice, what is a common value used for the variance threshold α?

0.9 or higher Signup and view all the answers

Which step helps in selecting the fewest number of dimensions that captures at least α fraction of the total variance?

Adding additional principal components until the desired variance threshold is reached Signup and view all the answers

In multivariate functions, what role do eigenvalues and eigenvectors play?

Determining the best r-dimensional approximation Signup and view all the answers

Probability Distributions and Conditional Distribution

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