10) Bayesian learning in practise

Question 1

What is the pdf of the beta distribution

Answer 1

Question 2

What are the parameters of the prior known as

Answer 2

Hyperparameters, to distinguish them from the parameters of the likelihood function

Question 3

In terms of mean parameterization Beta(𝜇₀, 𝑘₀), how are the prior concentration parameter and the prior mean parameter set

Answer 3

• 𝑘0 = 𝛼1 + 𝛼2
• 𝜇0 = 𝛼1/𝑘0

Question 4

What are the prior mean and prior variance for the beta-binomial model

Answer 4

Question 5

What is the posterior distribution for the beta-binomial model

Answer 5

Question 6

What are the posterior updates in the beta-binomial model

Answer 6

Question 7

What does pseudodata mean in the context of Bayesian inference

Answer 7

Pseudodata refers to treating the prior distribution as if it represents additional data points. Specifically:
* α1 andα2 act as pseudocounts that influence both the posterior mean and variance
* These pseudocounts influence the posterior distribution similarly to how real observed data does.
* The prior adds “virtual” observations, stabilizing estimates when actual data is sparse

Question 8

What is shrinkage intensity in the Beta-Binomial model

Answer 8

Shrinkage intensity, λ, is a factor in the Beta-Binomial model that determines the weight of the prior mean in the posterior mean calculation

Question 9

What does the shrinkage intensity indicate

Answer 9

This factor indicates how much the prior information influences the posterior mean compared to the observed data
When λ =0: The posterior mean is equal to the ML estimate
When λ→1: The posterior mean corresponds closely to the prior mean (𝜇0)

Question 10

What is shrinkage

Answer 10

The adjustment of the ML estimate towards the prior mean is called “shrinkage,” as 𝜃^ ML is “shrunk” towards 𝜇0 which is often the target mean

Question 11

What is a conjugate prior

Answer 11

If the prior and posterior belong to the same distributional family the prior is a conjugate prior

Question 12

Why are conjugate priors useful

Answer 12

• Conjugate priors are computationally convenient
• They allow Bayesian learning by only updating prior parameters
• This avoids complex calculations

Question 13

What happens to the posterior mean and variance in the Beta-Binomial model as the sample size n becomes very large

Answer 13

Question 14

What is the Bayesian Central Limit Theorem

Answer 14