How does MCMC help bayesian inference? - statistics

Question

Literature says that the metropolis-hasting algorithm in MCMC is one of the most important algorithms developed last century and is revolutional. Literature also says that it is such development in MCMC that gave bayesian statistics a second birth.
I understand what MCMC does - it provides an efficient way to draw samples from any complicated probability distribution.
I also know what bayesian inference is - it is the process by which the full posterior distribution of parameters is calculated.
I am having difficult time connecting the dots here:
Which step in the process of bayesian inference does MCMC come into play? Why is MCMC so important that people say it is MCMC that gave bayesian statistics a second birth??

Answer 1

You might want to ask a similar question on StatsExchange. However, here is an attempt for a high level "build some intuition" answer (disclaimer: I am a Computer Scientist and not a Statistician. Head over to StatsExchange for a more formal discussion).
Bayesian Inference:
In the most basic sense we follow Bayes rule: p(Θ|y)=p(y|Θ)p(Θ)/p(y). Here p(Θ|y) is called the 'posterior' and this is what you are trying to compute. p(y|Θ) is called the 'data likelihood' and is typically given by your model or your generative description of the data. p(Θ) is called the 'prior' and it captures your belief about the plausible values of the parameters before observing the data. p(y) is called the 'marginal likelihood' and using the law of total probability can be expressed as ∫ p(y|Θ)p(Θ) dΘ. That looks really neat but in reality the p(y) is often intractable to compute analytically and in high dimensions (i.e. when Θ has many dimensions) numerical integration is imprecise and computationally intractable. There are certain cases when the conjugate structure of the problem allows you to compute this analytically, but in many useful models this is simply not possible. Therefore, we turn to approximating the posterior.
There are two ways (that I know of) to approximate the posterior: Monte Carlo and Variational Inference. Since you asked about MCMC, I'll stick to that.
Monte Carlo (and Markov Chain Monte Carlo):
Many problems in Statistics deal with taking expectations of functions under probability distributions. From the Law of Large Numbers, an expectation can be efficiently approximated by a Monte Carlo estimator. Therefore, if we can draw samples from a distribution (even if we don't know the distribution itself) then we can compute a Monte Carlo estimate of the expectation in question. The key is that we don't need to have an expression for the distribution: If we just have samples then we can compute the expectations that we are interested in. But there is a catch... How to draw the samples??
There has been a lot of work which developed ways of drawing samples from unknown distributions. These include 'rejection', 'importance' and 'slice' sampling. These were all great innovations and were useful in many applications but they all suffered by scaling poorly to high dimensions. For example, rejection sampling draws samples from a known 'proposal' distribution and then accepts or rejects that sample based on a probability that needs to evaluate the likelihood function and the proposal function. This is wonderful in 1 dimension but as the dimensionality grows, the probability mass that a given sample gets rejected increases dramatically.
Markov Chain Monte Carlo was an innovation that has some super nice theoretical guarantees attached to it. The key idea was to not randomly draw samples from a proposal distribution but rather to use a known sample (with the hope that the sample is in an area of high probability mass) and then make a small random step under a draw from a proposal distribution. Ideally, if the first draw was in an area of high probability mass then the second draw is also likely to be accepted. Therefore, you end up accepting many more samples and you don't waste time drawing samples that are to be rejected. The amazing thing is that if you run the Markov Chain long enough (i.e. to infinity) and under specific conditions (the chain must be finite, aperiodic, irreducible and ergodic) then your samples will be drawn from the true posterior of your model. That's amazing! The MCMC technique is to draw dependent samples so it scales to a higher dimensionality than previous methods, but under the right conditions, even though the samples are dependent, they are as if they are drawn IID from the desired distribution (which is the posterior in Bayesian Inference).
Tying it together (and hopefully answering your question):
MCMC can be seen as a tool that enables Bayesian Inference (just as analytical calculation from conjugate structure, Variational Inference and Monte Carlo are alternatives). Apart from an analytical solution, all of the other tools are approximating the true posterior. Our goal is then to make the approximation as good as possible and to do this as cheaply as possible (in both computation cost and the cost of computing a bunch of messy algebra). Pervious sampling methods did not scale to high dimensions (which are typical of any real world problem) and therefore Bayesian Inference became computationally very expensive and impractical in many instances. However, MCMC opened the door to a new way to efficiently draw samples from a high dimensional posterior, to do this with good theoretical guarantees and to do this (comparatively) easily and computationally cheaply.
It is worth mentioning that Metropolis itself has problems: it struggles with highly correlated latent parameter space, it requires a user-specified proposal distribution and the correlation between samples can be high leading to biased results. Therefore more modern and sometimes more useful MCMC tools have been proposed to try combat this. See 'Hamiltonian Monte Carlo' and the 'No U-Turn Sampler' for the state of the art. Nonetheless, Metropolis was a huge innovation that suddenly made real world problems computationally tractable.
A last note: See this discussion by MacKay for a really good overview of these topics.

Answer 2

This post https://stats.stackexchange.com/a/344360/137466 perfectly clears my question on how MCMC sampling helps solving bayesian inference. Especially this following part from the post is the key concept that I missed:
The Markov chain has a stationary
distribution
which is the distribution that preserves itself if you run it through
the chain. Under certain broad assumptions (e.g., the chain is
irreducible, aperiodic), the stationary distribution will also be the
limiting distribution of the Markov chain, so that regardless of how
you choose the starting value, this will be the distribution that the
outputs converge towards as you run the chain longer and longer. It
turns out that it is possible to design a Markov chain with a
stationary distribution equal to the posterior distribution, even
though we don't know exactly what that distribution is. That is, it
is possible to design a Markov chain that has $\pi( \theta |
\mathbb{x} )$ as its stationary limiting distribution, even if all we
know is that $\pi( \theta | \mathbb{x} ) \propto L_\mathbb{x}(\theta)
\pi(\theta)$. There are various ways to design this kind of Markov
chain, and these various designs constitute available MCMC algorithms
for generating values from the posterior distribution.
Once we have designed an MCMC method like this, we know that we can
feed in any arbitrary starting value $\theta_{(0)}$ and the
distribution of the outputs will converge to the posterior
distribution (since this is the stationary limiting distribution of
the chain). So we can draw (non-independent) samples from the
posterior distribution by starting with an arbitrary starting value,
feeding it into the MCMC algorithm, waiting for the chain to converge
close to its stationary distribution, and then taking the subsequent
outputs as our draws.