Metropolis Algorithm: Example (R code)

- An  MCMC algorithm 
- Goal: to draw samples from an unknown target distribution 
- Uses a random walk approach through the distribution space, and it accepts or rejects a sample based on the likelihood of the proposed sample. 
- The likelihood is commonly decided by a probability density function 

Limitations: 
- The proposal distribution must be symmetric (equal prob of proposing a move from A to B, as from B to A)
- Metropolis Hastings Algorithm - A newer version -  resolves this limitation and allows fro asymmetric distributions. 

New York University

- A class of techniques or methods that use a stochastic process to estimate a posterior probability – often used to generate samples/ observations from a posterior 
- STAN – Most common tool to run MCMC algorithms 
- Rstan and Rethinking package – can be used for simple models

Common MCMC Algorithms - 
- Metropolis/ Metropolis Hastings
- Gibbs Sampling 
- Hamiltonian Monte Carlo


Markov Chain Monte Carlo

McElreath, R. (2020). Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd ed.). Chapman and Hall/CRC.

Statistical Rethinking: A Bayesian Course with Examples in R and STAN (2nd edition)

Metropolis Algorithm

- A variant of Metropolis Hastings algorithm
- More general and more efficient than standard Metropolis – uses fewer steps to get a good estimate of the posterior distribution
- Uses adaptive proposals:
- - o	The proposed parameter values adjust intelligently, depending upon the current values. 
- - o	Uses conjugate pairs to compute adaptive proposals (prior distributions and likelihoods) 

Limitations: 
- Becomes very inefficient as the models become more complex. 
- The model getting stuck in regions of high correlation in the posterior. Complex models often have highly correlated parameters, these parameters cause a narrow ridge of high probability combinations – thus the model will get stuck in these regions for a long time. 
- Concentration of Measure -  Any Markov chain approach that samples individual parameters in individual steps is going to get stuck, once the number of parameters grows sufficiently large.


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