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MCMC Theory
Astro 193, March 2


Today
· Reminder from the previous classes:
· Likelihood and Maximum Likelihood · Bayesian Inference

· Markov Chains · Basic MCMC · Jumping Rules
· The Metropolis Sampler · The Metropolis-Hastings Sampler · Gibbs


Likelihood and ML
· Likelihood Functions:
· P(data | ) - The distribution of the data given the model parameters. E.g., Y ~ Poisson(): likelihood() = Ye- / Y!

· Maximum Likelihood:
Y=3


Bayesian Inference
posterior ( | d) likelihood (d | ) * prior () d- data - parameters
· Combine current and previous knowledge · Define the likelihood function · Describe the posterior probabilty distribution via MCMC simulations. => Summary of Inference by Simulations


MCMC Markov Chain Monte Carlo Simulations
· General method based on drawing values of from approximate distributions and then correcting those draws to better approximate the target posterior distribution.

· Markov Chain:
A sequence of random variables 0, 1, 2 .... Such that p(k | k-1,
k-2

, ..., 0) = p(k|k-1)

· Key: the approximate distributions are improved at each step in the simulations - converging to the target distribution.


The Metropolis Sampler
· Draw 0 from the some starting distribution

David Van Dyk - MCMC Lecture at CfA 2014


The Metropolis-Hastings Sampler


The Gibbs Sampler
Alternating conditional sampling:
- divided into d components = (0, 1, ..., d) Sampler cycles through components, drawing each subset conditional on the values of all the others. There are d steps in each iteration t

Each j is updated conditional on the latest values of the other components of , which are the iteration t values for the components already updated and the iteration t-1 for the others.


Metropolis vs. Gibbs Comparison

Bayesian Data Analysis, Gelman et al Chapter 11