Likelihood

Here the likelihood for \(q\) is \[L(q):= \Pr(D | q) \propto q^{30} (1-q)^{70}\] where \(D\) here denotes the data. This expression comes from the fact that the data consist of 30 “1” alleles (each of which occur with probability \(q\)) and 70 “0” alleles (each of which occur with probability \(1-q\)), and we assume that the samples are independent. (You might have heard this likelihood called the “binomial likelihood”, because it arises when the data come from a binomial distribution.)

Prior

Recall that the prior distribution is a distribution that is supposed to reflect what we know about \(q\) prior to seeing the data. For the purposes of illustration we will assume a uniform prior on \(q\): \(q \sim U[0,1]\). That is \[p(q) = 1 \qquad (q \in [0,1]).\]

This \(U[0,1]\) prior says many things. For example, it says that before seeing the data the idea that \(q<0.5\) is just as plausible as \(q>0.5\). And it says that \(q<0.1\) is just as plausible as \(q>0.9\), or \(0.4<q<0.5\). If for some reason these are not equally plausible then you should use a different prior, at least in principle. However, in practice it is sometimes (but not always) the case that the results of Bayesian inference are robust to the choice of prior distribution, so in such cases is common not to worry too much about minor deviations between what you believe and what the prior implies.

For now, we are simply aiming to show how the Bayesian calculations are done under this prior specification.

Posterior calculation

Using Bayes Theorem to combine the prior distribution and the likelihood we obtain: \[p(q | D) \propto p(D|q) p(q) = q^{30} (1-q)^{70} \qquad (q \in [0,1]).\]

Here, because \(q\) is a continuous parameter, this is referred to as the “posterior density”" for \(q\).

Now the final “trick” is to notice that this density, \(q^{30} (1-q)^{70}\) is exactly the density of a Beta distribution. Indeed, specifically it is the density of a Beta(31,71) distribution. So the posterior distribution for \(q\) is Beta(31,71), and we might write \(q | D \sim \text{Beta}(31,71)\).

This kind of “trick” is common in Bayesian inference: you look at the posterior density and “recognize” it as a distribution you know. It turns out that the number of distributions in common use is relatively small, so you only need to learn a few distributions to get sufficiently good at this trick for practial purposes. For example, it is a good start to be able to reconize the following distributions: exponential, binomial, Poisson, Gamma, Beta, Dirichlet, and Normal. If your posterior distribution does not look like one of these, then you may well be in a situation where you need to use computational methods (like Importance Sampling or Markov chain Monte Carlo) to do your computations.

So, in this case we are lucky: the posterior distribution is a nice distribution that we recognize, and this means we can do lots of calculations very easily: R has lots of built-in functions to deal with the Beta distribution, and many analytic properties have been derived (e.g. Wikipedia.) We can use this to summarize and interpret the posterior distribution, as illustrated here.

Session Information

sessionInfo()

R version 3.3.2 (2016-10-31)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: OS X El Capitan 10.11.6

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] backports_1.0.5 magrittr_1.5    rprojroot_1.2   htmltools_0.3.5
 [5] tools_3.3.2     rstudioapi_0.6  yaml_2.1.14     Rcpp_0.12.9    
 [9] stringi_1.1.2   rmarkdown_1.3   knitr_1.15.1    git2r_0.18.0   
[13] stringr_1.1.0   digest_0.6.10   workflowr_0.3.0 evaluate_0.10