Introduction

Welcome to Edition 9 of our Bayesian series: “Making Bayesian Statistics Accessible to Everyone”. This edition is dedicated to reinforcing your understanding through a 20-question quiz, combining true/false and multiple choice formats.

The questions assess your knowledge from Editions 1 to 4, covering:

  • Bayesian probability and the posterior formula

  • Beta-Binomial conjugate analysis

  • Prior elicitation and belief updates

  • Basic examples of conjugate models

At the end, you’ll find professional explanations and correct answers to enhance learning.

Quiz: Questions 1 to 20

True/False Questions

Q1

The posterior distribution is proportional to the product of the prior and the likelihood.

  • True

  • False

Q2

The Beta distribution is a conjugate prior for the Normal distribution.

  • True

  • False

Q3

In Bayesian statistics, the parameter \(\theta\) is treated as a random variable.

  • True

  • False

Q4

A uniform prior is considered a noninformative prior for parameters defined on \((0,1)\).

  • True

  • False

Q5

The likelihood function describes the distribution of the parameter given the data.

  • True

  • False

Q6

In a Beta(1,1) prior, the mean is 0.5.

  • True

  • False

Q7

Bayesian credible intervals and frequentist confidence intervals always have the same interpretation.

  • True

  • False

Q8

The Binomial-Beta model is an example of a conjugate model.

  • True

  • False

Q9

Posterior mean always equals the maximum likelihood estimate (MLE).

  • True

  • False

Q10

In Bayesian updating, the influence of the prior diminishes with increasing sample size.

  • True
  • False

Multiple Choice Questions

Q11

Which of the following is the correct posterior mean for a Beta($\alpha$,$\beta$) prior with \(y\) successes in \(n\) Binomial trials?

  • \(\displaystyle \frac{y}{n}\)

  • \(\displaystyle \frac{\alpha + y}{\alpha + \beta + n}\)

  • \(\displaystyle \frac{\alpha}{\alpha + \beta}\)

  • \(\displaystyle \frac{\beta + y}{n + 1}\)

Q12

Which of the following priors is conjugate for a Poisson likelihood?

  • Beta

  • Normal

  • Gamma

  • Uniform

Q13

If the prior for \(\theta\) is Beta($2,2$) and the data are 8 successes in 10 trials, what is the posterior?

  • Beta(2,2)

  • Beta(10,10)

  • Beta(8,2)

  • Beta(10,4)

Q14

Which of the following is NOT a conjugate pair?

  • Binomial – Beta

  • Poisson – Gamma

  • Normal – Normal

  • Exponential – Beta

Q15

Which value of \(\theta\) maximizes the posterior distribution in the Beta($\alpha$, \(\beta\)) case (assuming \(\alpha,\beta > 1\))?

  • \(\displaystyle \frac{\alpha – 1}{\alpha + \beta – 2}\)

  • \(\displaystyle \frac{\alpha}{\alpha + \beta}\)

  • \(\displaystyle \frac{\beta}{\alpha + \beta}\)

  • \(\displaystyle \frac{1}{2}\)

Q16

In the absence of prior knowledge, what is a common choice for a noninformative prior in a Binomial model?

  • Beta(0,0)

  • Beta(1,1)

  • Normal(0,1)

  • Gamma(1,1)

Q17

What does the area under the posterior distribution represent?

  • Probability of the sample

  • Normalization constant

  • Total posterior probability (should be 1)

  • Mean of the prior

Q18

Which of the following expressions is the correct posterior distribution in the Binomial-Beta case?

  • \(\text{Beta}(\alpha + n, \beta + y)\)

  • \(\text{Beta}(\alpha + y, \beta + n – y)\)

  • \(\text{Gamma}(\alpha + y, \beta + n)\)

  • \(\text{Normal}(\mu, \sigma^2)\)

Q19

What is the Jeffreys prior for a Binomial proportion \(\theta\)?

  • Beta(1,1)

  • Beta(0.5,0.5)

  • Uniform(0,1)

  • Improper prior

Q20

Which principle justifies the use of \(p(\theta) \displaystyle \propto [J(\theta)]^{½}\) as a noninformative prior?

  • Bayesian Principle

  • Frequentist Invariance

  • Jeffreys Invariance Principle

  • Likelihood Principle

Solutions and Explanations

QuestionAnswerExplanation
1TrueBayes’ theorem: posterior ∝ prior × likelihood.
2FalseThe conjugate prior for Normal is also Normal, not Beta.
3TrueIn Bayesian analysis, parameters are treated as random variables.
4TrueBeta(1,1) is the uniform distribution over (0,1).
5FalseThe likelihood describes the distribution of the data given parameters, not the other way around.
6TrueMean of Beta(1,1) = 1 / (1 + 1) = 0.5.
7FalseCredible intervals are probability statements about parameters. Confidence intervals are about sampling variability.
8TrueBy definition of conjugacy: Beta prior + Binomial likelihood → Beta posterior.
9FalsePosterior mean ≠ MLE in general. Only coincide in special cases.
10TrueAs more data are collected, the prior has less influence.
11(b)Posterior mean in Binomial-Beta: (α + y) / (α + β + n).
12©Gamma is conjugate prior for Poisson mean parameter.
13(d)Posterior: Beta(α + y, β + n − y) = Beta(2 + 8, 2 + 2) = Beta(10, 4).
14(d)Exponential – Gamma is conjugate. Exponential – Beta is not.
15(a)Posterior mode of Beta(α, β) is (α − 1)/(α + β − 2), when α, β > 1.
16(b)Beta(1,1) is the uniform prior over (0,1).
17©Posterior must integrate to 1 ⇒ total probability.
18(b)Correct posterior: Beta(α + y, β + n − y).
19(b)Jeffreys prior for Binomial: Beta(0.5,0.5).
20©Jeffreys Invariance Principle: Prior derived from Fisher Information.

Thank You

This concludes Edition 9. In the next edition, we continue with advanced quiz material focusing on hierarchical models, Jeffreys priors, location/scale parameters, and practical modeling scenarios.

Stay engaged with 3 D Statistical Learning as we continue making Bayesian thinking second nature!