Why is it necessary to sample from the posterior distribution if we already KNOW the posterior distribution?

Question

My understanding is that when using a Bayesian approach to estimate parameter values:

• The posterior distribution is the combination of the prior distribution and the likelihood distribution.
• We simulate this by generating a sample from the posterior distribution (e.g., using a Metropolis-Hasting algorithm to generate values, and accept them if they are above a certain threshold of probability to belong to the posterior distribution).
• Once we have generated this sample, we use it to approximate the posterior distribution, and things like its mean.

But, I feel like I must be misunderstanding something. It sounds like we have a posterior distribution and then sample from it, and then use that sample as an approximation of the posterior distribution. But if we have the posterior distribution to begin with why do we need to sample from it to approximate it?

Answer 1

This question has likely been considered already on this forum.

When you state that you "have the posterior distribution", what exactly do you mean? "Having" an available$-$in the sense I can compute it everywhere$-$function of $\theta$ that I know to be proportional to the posterior density, namely$$\pi(\theta|x) \propto \pi(\theta) \times f(x|\theta)$$as for instance with the completely artificial target$$\pi(\theta|x)\propto\exp\{-\|\theta-x\|^2-\|\theta+x\|^4-\|\theta-2x\|^6-100\|\theta\|^5\},\ \ x,\theta\in\mathbb{R}^{18}\tag{1},$$does not tell me what is

1. the posterior expectation of a function of $\theta$, e.g., $\mathbb{E}[\mathfrak{h}(\theta)|x]$, posterior mean that operates as a Bayesian estimator under standard losses;
2. the optimal decision under an arbitrary utility function, decision that minimizes the expected posterior loss;
3. a 90% or 95% range of uncertainty on the parameter(s), a sub-vector of the parameter(s), or a function of the parameter(s), aka HPD region$$\{h=\mathfrak{h}(\theta);\ \pi^\mathfrak{h}(h)\ge \underline{h}\}$$
4. the most likely model to choose between setting some components of the parameter(s) to specific values versus keeping them unknown (and random).

For instance, the fact that the rhs of (1) is known does not tell how to solve $$\int_{\mathcal H} \exp\{-\|\theta-x\|^2-\|\theta+x\|^4-\|\theta-2x\|^6-100\|\theta\|^5\}\,\text d\theta=\qquad\\0.95\int_{\mathbb R^{18}} \exp\{-\|\theta-x\|^2-\|\theta+x\|^4-\|\theta-2x\|^6-100\|\theta\|^5\}\,\text d\theta$$ and optimize over all such $\mathcal H$'s.

These items are only examples of many usages of the posterior distribution. In all cases but the simplest ones, one cannot provide answers by solely staring at the posterior distribution density as an available function and one need proceed through numerical resolutions like Monte Carlo and Markov chain Monte Carlo methods.

Answer 2

Yes you might have an analytical posterior distribution. But the core of Bayesian analysis is to marginalize over the posterior distribution of parameters so that you get a better prediction result both in terms of accuracy and generalization capability. Basically, you want to obtain a predictive distribution which has the following form.

$p(x|D)=\int p(x|w) p(w|D)dw$

where $p(w|D)$ is the posterior distribution which you might have an analytical form for. But in many cases, $p(w|D)$ has a complex form that does not belong to any known distribution family nor in conjugacy with $p(x|w)$. This makes the above integrand impossible to calculate analytically. Then you have to resort to sampling approximation of the integrand which is the entire purpose of the advanced sampling technique such as markov chain monte carlo

Answer 3

A short answer:

Sampling is not only used to estimate a distribution function*, it is also used to perform computations with a density function and MCMC is just one of many ways of sampling. Often such computations are a way of Monte Carlo integration.

Computing a posterior average, a 95% highest density interval, marginal distributions*, etcetera, if the density function is a complex function then such values can be difficult to derive analytically and manual integration might require a lot of computations to perform.

A sampling method is an alternative to approximate the desired quantity.

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*In the case of nuisance parameters, then Bayes rule doesn't give the posterior of the parameter, but of a joint distribution with the nuisance parameter. So the statement "if we already KNOW the posterior distribution?" must be nuanced and the marginal posterior distribution is not even known. In these cases the MCMC can also be used to compute the density function (which is unknown).