# 20 Proportionality Constants

When evaluating a likelihood or prior as part of the log density computation in MCMC, variational inference, or optimization, it is usually only necessary to compute the functions up to a proportionality constant (or similarly compute log densities up to an additive constant). In MCMC this comes from the fact that the distribution being sampled does not need to be normalized (and so it is the normalization constant that is ignored). Similarly the distribution does not need normalized to perform variational inference or do optimizations. The advantage of working with unnormalized distributions is they can make computation quite a bit cheaper.

There are three different syntaxes to work with distributions in Stan. The way to select between them is by determining if the proportionality constants are necessary. If performance is not a problem, it is always safe to use the normalized densities.

The first two syntaxes use unnormalized densities (dropping proportionality constants):

x ~ normal(0, 1);
target += normal_lupdf(x | 0, 1); // the 'u' is for unnormalized

The final syntax uses the full normalized density (dropping no constants):

target += normal_lpdf(x | 0, 1);

For discrete distributions, the target += syntax is _lupmf and _lpmf instead:

y ~ bernoulli(0.5);
target += bernoulli_lupmf(y | 0.5);
target += bernoulli_lpmf(y | 0.5);