Unit 12: Markov Chain Monte Carlo
Turning the last unit's convergence guarantee around: build a chain whose stationary distribution IS the hard distribution you actually wanted to sample from.
4 topics · 13 questions
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12.1 Why Sample From a Hard Distribution With a Markov Chain
- How it worksHow does building a chain whose stationary distribution matches the target turn sampling into just running the chain?
- Common mistakeWhy doesn't MCMC need to know the target distribution's normalizing constant to work?
- Going furtherWhat happens to MCMC sampling accuracy when the chain hasn't run long enough to reach stationarity?
Your turn1 of 2
A target distribution has unnormalized weights w(x) = 3.63 and w(y) = 3.42 (both proportional to the true probabilities by some unknown constant Z). Find the ratio π(y)/π(x) that MCMC can compute WITHOUT knowing Z.12.2 The Metropolis-Hastings Algorithm
- How it worksHow does the acceptance ratio use detailed balance to guarantee the target distribution is stationary?
- Worked exampleFind the acceptance probability of a move from x to y when π(x)=0.2 and π(y)=0.5 with a symmetric proposal.
- Common mistakeWhy doesn't Metropolis-Hastings require the target distribution to be easy to sample from directly?
- Going furtherWhat happens to Metropolis-Hastings when a proposed move is always rejected because it makes π much smaller?
Your turn1 of 2
Find the acceptance probability of a move from x to y when π(x) = 0.718 and π(y) = 0.206 with a symmetric proposal (py less than px).12.3 Sampling a Discrete Distribution With MCMC
- How it worksHow does proposing a random neighboring state and applying the Metropolis rule build the chain step by step?
- Worked exampleFind the long-run fraction of time an MCMC chain spends on state 3 when its target stationary probability is 0.35.
- Going furtherWhat happens to an MCMC chain's samples in the first few steps before it has mixed to stationarity?
Your turn1 of 2
An MCMC chain targets a uniform distribution over 15 states. Find the long-run fraction of time spent on any single specific state.12.4 Where MCMC Shows Up in Practice
- How it worksHow does MCMC turn an intractable multi-dimensional integral into a sampling problem a computer can just run?
- Going furtherWhat happens to MCMC-based Bayesian inference when the number of parameters being estimated grows into the hundreds?
- Ties togetherHow does MCMC's use of a stationary distribution connect back to the convergence guarantee from ordinary Markov chains?
Your turn1 of 2
Why do Bayesian statisticians reach for MCMC the moment a posterior distribution has no clean formula?