MCMC - target distribution, proposal distribution and likelihood function?

Question

i have just started out in MCMC and I am not sure if I fully understand the concepts of MCMC with respect to the above terms. Let me try to explain that in my own words and include some thoughts / questions that I still have on it.

My understanding for MCMC (in Bayesian inference) is that it help us identify the final distribution of the samples (also known as the posterior distribution). And we can then use it to determine the sample's mean, range etc.

Example: We wanted to find out the mean test score of students. We are given 2 samples (i.e. the test scores of 2 students only and let's say the score is 65 and 75). We conduct the MCMC.

So to implement MCMC, we need the following "ingredients" - Target distribution: This is actually the posterior distribution that we actually want the MCMC to sample from. In other words, we need to determine a distribution of the mean score.

Q1: If our goal is to determine the final distribution of the samples and yet we are required to specify the target distribution that the MCMC is working on, isnt that like forcing the solution to a distribution that we already have in mind?

Q2: What if we have no idea what this distribution is supposed to be, including the mean and standard deviation. What should we do to proceed from here?

• Proposal distribution: I see this as a distribution to determine the step / noise to be added on the parameter when we sample from the target distribution. Let say we had a starting test score at 68. We use the proposal distribution to generate a random noise (e.g. 3) and this is added to the test score for testing. So the next state is 78.

Q3: Am i right in my interpretation here?

• Likelihood function: As far I understand so far, this is used in the acceptance testing and is also where we incorporate our observed data in (ie. 2 students test score we have on hand).

Q4: Did i interpret this wrongly? So should the incorporation of our observed data be when we are determining the target distribution?

I think I may be already confused. I really hope you can help me out with this. Thank you in advance.

Answer 1

The question reflects on a misunderstanding of the respective roles of MCMC algorithms and Bayesian inference principles, and calls for some prior investment in the basics or core of what constitutes Bayesian inference. The answer below cannot address this missing background and I cannot but recommend investing some time in an introductory Bayesian textbook like Albert or Kruschke.

MCMC is a tool to simulate from a posterior distribution, not a method of (Bayesian) inference per se. This means that it requires two inputs:

• the distribution of the data, written as a density $f(x;\theta)$, which also writes as the likelihood $\ell(\theta|x)$. This function has to be specified completely, possibly as a marginal across latent variables, i.e., as an integral over these variables. In which cases the latent variables are added to the parameter to define an augmented target, of which the true target is a marginal. (Pseudo-marginal Metropolis-Hastings techniques partake from the same principle.)
• the prior distribution on the parameter $\theta$, written as a density $\pi(\theta)$. Again, this function has to be specified completely. (Or again be expressed as a marginal.)

Given those two elements, MCMC aims at simulating from the distribution with density $\pi(\theta|x)$ which is proportional to $\pi(\theta)\times \ell(\theta|x)$. It can operate without the normalising constant. And without these elements, MCMC makes no sense.

Concerning the proposal distribution, this is a technical component in some of the MCMC algorithms, namely a new value for the Markov chain is proposed from a certain distribution and accepted or rejected according to an acceptance probability, computed in such a way that the stationary distribution of the Markov chain is the posterior.