Cross Validated
Cross Validated Meta
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up.
What is the difference and relationship between posterior distribution function and likelihood function in MCMC?
I am learning MCMC in class, and I encounter one question about the relationship between posterior probability and likelihood function.
In our lecture, the professor asked us to take samples from the posterior distribution, however, he used the likelihood function as the target distribution $P(X)$ in the Metropolis-Hastings algorithm. So I am confused why the sample generated by the likelihood using MCMC can be seen as a sample from the posterior distribution?
mcmc likelihood posterior metropolis-hastings
Rax Yao
Rax YaoRax Yao
The probability density function (PDF) for the posterior distribution is proportional to the product of the likelihood and the prior distribution: $$ p(\theta|y) \propto p(y|\theta)\,p(\theta), $$ where $y$ denotes the observed data and $\theta$ denotes the unobserved parameter. If the prior is flat, $$ p(\theta) \propto 1, $$ and the likelihood is normalizable, $$ \int p(y|\theta)\,d\theta < \infty, $$ then the posterior is proportional to the likelihood: $$ p(\theta|y) \propto p(y|\theta). $$ The Metropolis-Hastings algorithm can be applied to the likelihood in this case to produce draws from the posterior.
mefmef
Thanks for contributing an answer to Cross Validated!
Not the answer you're looking for? Browse other questions tagged mcmc likelihood posterior metropolis-hastings or ask your own question.
What is the conceptual difference between posterior and likelihood?
Understanding MCMC and the Metropolis-Hastings algorithm
Confused with MCMC Metropolis-Hastings variations: Random-Walk, Non-Random-Walk, Independent, Metropolis
How to find the support of the posterior distribution to apply Metropolis-Hastings MCMC algorithm?
Proposal function for MCMC - prior distribution NORMAL
Correlation between posterior samples of the parameters
How can i sample from posterior distribution which contains integration using MCMC?
Compute the likelihood in Metropolis–Hastings: How does it relate to a posterior in Bayesian Analysis?
MCMC - target distribution, proposal distribution and likelihood function?