Bayesian inference

1 · What it is

Bayesian inference treats an unknown quantity, such as the share of people who click an ad, as uncertain rather than fixed. You begin with a prior, a distribution that says how plausible each value seems before looking at the data. The likelihood scores how well each possible value explains the data you actually saw. Multiplying the two and rescaling so the total is 1 gives the posterior, your updated belief. That step is Bayes’ theorem, which the probability entry introduces.

The result is a whole distribution, not one number. You can summarise it with a mean and a credible interval, a range that holds a chosen share of the posterior probability, such as 90%. Because the unknown is treated as random, that interval is a direct probability statement about it. Classical, or frequentist, statistics instead treats the unknown as a fixed constant. Its confidence interval describes how often the method’s intervals would capture the true value, which is a different kind of statement.

Some pairs of prior and likelihood fit together so neatly that the posterior has the same form as the prior, a property called conjugacy. A Beta prior on a rate, updated with success and failure counts, is the classic case. The posterior is again a Beta, with the successes added to its first number and the failures to its second. Its mean always lies between the prior mean and the observed rate. As the counts grow, the prior matters less and less.

Once a model gets complicated, there may be no conjugate prior to use at all. Software then draws many random samples from the posterior and summarises those instead. A common family of such methods is Markov chain Monte Carlo, or MCMC. The Stan software, for example, runs an MCMC sampler called NUTS by default. In PyMC, another such tool, the model is ordinary Python, not a separate modelling language. Variational inference swaps sampling for optimisation. It fixes a set of candidate distributions, then searches that set for the one nearest the posterior. It usually finishes sooner than sampling. One review of the method says its behaviour is still only partly understood.

Bayesian ideas turn up across machine learning. Bayesian regression, for example, learns how strong its regularisation should be from the data instead of having it set by hand. The usual penalty in ridge regression matches a Bayesian estimate under a Gaussian prior. Bayesian optimisation tunes a model’s hyperparameters by treating how well the model performs as a draw from a Gaussian process. The posterior from past runs then guides which settings to try next. A 2012 study reported that it could match or beat human experts at tuning. The approach also suits A/B tests, where results arrive over time and a choice must be made.

Choosing the model, prior included, is a big hurdle in much Bayesian work. Part of checking the work is asking how much the answer shifts if those modelling choices change. Computation is the other cost, since fitting can be time consuming.