Quizmaster

Learning objectives

What you will be able to explain

Monte Carlo expectation
Monte Carlo estimator properties
Importance sampling identity
Compute an importance estimate
Designing an importance proposal
Markov-chain Monte Carlo language

Section 01

Monte Carlo expectation to Importance sampling identity

Monte Carlo expectation

The Monte Carlo estimate is

(2+5-1+4)/4=2.5

Guided checkpoint

Samples of $f(x)$ are $[2,5,-1,4]$ . Estimate $E[f(x)]$ with the sample mean.

Source: Chapter 17, section 17.1, pp. 590-592

Monte Carlo estimator properties

Standard error scales as

1/\sqrt n

. High dimensionality often hurts through variance and poor sampling rather than changing this basic rate.

Guided checkpoint

Judge each statement for iid samples with finite variance.

Source: Chapter 17, section 17.1, pp. 590-592

Importance sampling identity

E_q[f(x)p(x)/q(x)]=E_p[f(x)]

provided the proposal covers the target's support.

Guided checkpoint

To estimate $E_p[f(x)]$ using samples from $q(x)$ , which weight is required where $p(x)>0$ ?

Source: Chapter 17, section 17.2, pp. 592-595

Section 02

Compute an importance estimate to Markov-chain Monte Carlo language

Compute an importance estimate

The estimate is

(0.5(3)+1.5(1))/2=3/2=1.5

Guided checkpoint

Two proposal samples have $(f,w)=(3,0.5)$ and $(1,1.5)$ . Using the ordinary unnormalized importance estimator $n^{-1}\sum_i w_if_i$ , compute the estimate.

Source: Chapter 17, section 17.2, pp. 592-595

Designing an importance proposal

Importance sampling succeeds when the proposal covers and tracks important target regions; self-normalization introduces finite-sample bias but can be practical.

Guided checkpoint

Select all correct principles.

Source: Chapter 17, section 17.2, pp. 592-595

Markov-chain Monte Carlo language

MCMC replaces independent target samples with correlated states from a chain designed to preserve the target distribution.

Guided checkpoint

Match each term to its role.

Source: Chapter 17, section 17.3, pp. 595-599

Section 03

MCMC correctness conditions to Gibbs tradeoffs

MCMC correctness conditions

Stationarity addresses correctness of the limiting distribution, while mixing rate determines practical efficiency.

Guided checkpoint

Evaluate each statement.

Source: Chapter 17, section 17.3, pp. 595-599

Gibbs sampling update

Sequential or blocked conditional updates preserve the joint target distribution when the conditionals are compatible.

Guided checkpoint

What does one single-site Gibbs update do?

Source: Chapter 17, section 17.4, p. 599

Gibbs tradeoffs

Gibbs is attractive when conditionals are easy, but its local moves can struggle with strong dependencies.

Guided checkpoint

Judge each statement.

Source: Chapter 17, section 17.4, p. 599

Section 04

Why separated modes are hard

Mode separation creates metastability. Diagnostics and advanced transitions address, but do not magically eliminate, the problem.

Guided checkpoint

Select all correct statements.

Source: Chapter 17, section 17.5, pp. 599-604

Knowledge check

Turn understanding into recall.

The quiz now follows the same concepts in scored form. You can return to this lesson from the quiz whenever a gap appears.

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