MAST30001 Chap.7 Markov Chain Monte Carlo
Markov Chain Monte Carlo
Markov chain Monte Carlo constructs an ergodic chain whose stationary distribution is a target that may be difficult to sample directly. Metropolis-Hastings proposes a candidate move and accepts it with a probability that corrects target and proposal imbalance.
This chapter shows why unknown normalising constants cancel, how detailed balance establishes stationarity, where rejected probability becomes a self-transition, and why mixing quality matters after correctness is proved. The method separates two obligations: mathematical validity requires the target to be stationary and reachable, while practical usefulness requires the chain to explore it efficiently.
A correct transition rule can still yield a poor estimator when successive states are strongly dependent.
What this chapter covers
- 01
Sampling a target through long-run Markov-chain visits
- 02
Unnormalised target weights and proposal probabilities
- 03
Metropolis-Hastings acceptance ratio
- 04
Detailed balance and stationary distributions
- 05
Rejected proposals and self-transition probability
- 06
Mixing, autocorrelation and proposal tuning
Worked example · free
Calculate asymmetric proposal acceptance
- 2The acceptance ratio is 1(0.50)/[3(0.25)]=2/3.
- 1Acceptance probability is min(1,2/3)=2/3.
- 1Accepted move probability is 0.25(2/3)=1/6.
- 1Any rejected proposal mass remains at the current state.
Key terms
- Target Distribution
- The probability distribution whose expectations or samples the Markov chain is designed to approximate.
- Proposal Kernel
- The conditional distribution used to generate a candidate next state from the current state.
- Acceptance Probability
- The probability of moving to the proposed state after correcting target and proposal imbalance.
- Self-Transition
- A transition that remains at the current state, including the result of a rejected proposal.
- Mixing
- The speed and effectiveness with which a chain explores the relevant regions of its target distribution.
- Burn-In
- An initial segment sometimes discarded to reduce sensitivity to a deliberately chosen starting state.
- Effective Sample Size
- An estimate of how many independent draws would provide information comparable to a correlated chain run.
Markov Chain Monte Carlo FAQ
Why can the target be unnormalised?
The Metropolis-Hastings ratio divides one target weight by another. Any common unknown normalising constant appears in both numerator and denominator and cancels, so relative positive weights are enough to construct the accepted transitions.
Why must rejected proposals remain in the chain?
A rejection means the next recorded value equals the current state. Removing repeats changes the long-run visit share assigned to states and generally destroys the intended stationary distribution. The resulting self-transition can also remove periodic oscillation.
Does a high acceptance rate prove good mixing?
No. Tiny proposals may be accepted almost always but move so slowly that successive values are highly correlated. Huge proposals may be rejected repeatedly. Diagnose exploration and dependence, then tune proposal scale rather than optimising acceptance rate alone.
When may proposal terms cancel from the ratio?
They cancel when the proposal kernel is symmetric between the current and proposed states. Check the actual transition probabilities, including boundary behaviour, rather than relying on a verbal label such as random walk. Truncation, reflection or state-dependent scales can make a superficially symmetric proposal asymmetric.
What does detailed balance prove?
Detailed balance equates stationary probability flow in both directions for every state pair. Summing those equations establishes stationarity. It is a sufficient reversible construction, not a claim that every chain with the desired stationary distribution must satisfy pairwise reversible flow.
Why are chain averages less efficient than independent samples?
Successive MCMC values are correlated, so a run of a given length often contains less information than the same number of independent target draws. Autocorrelation and effective sample size describe this loss. Longer runs help only after convergence and adequate exploration have been considered.
Exam move
Memorise the Metropolis-Hastings calculation as four actions: propose, form the target-and-reverse-proposal ratio, accept with the capped probability, and put rejection mass on the current state. Recompute one complete transition row and check it sums to one. Practise identifying when proposal symmetry genuinely cancels and when boundary effects make it asymmetric.
Separate correctness from efficiency: detailed balance and ergodicity establish the target, while traces, movement and autocorrelation diagnose mixing. Always state the target quantity estimated by a chain average. For finite-state exercises, build the full transition matrix from proposal and acceptance pieces and verify each row, including its rejection entry.
Then check detailed balance numerically with unnormalised target weights. For diagnostic questions, compare a sticky trace, a rapidly moving local trace and a trace that traverses all important regions. Explain which defect each reveals instead of treating a single acceptance percentage as a universal target.
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