Likelihood Sampling

In the previous section, we looked at finding the maximum likelihood for some data given a model with parameters. However, the broad aim of looking at probabilistic modelling is to give us tools to understand those parameters. The data have uncertainty associated with them. Therefore, the parameters of the model should be the same. While it is possible to develop estimates of these parametric uncertainties from optimisation routines, a more rigorous approach is sampling.

In many ways, to compute the uncertainties, we want to sample the unknown probability distribution that describes the parameters of the model. While this may sound trivial, probability distributions over many dimensions–the number of dimensions is the number of parameters–can become very computationally intensive. Therefore, various algorithms have been developed to address this; here, we will look at Markov chain Monte Carlo sampling, one of the most popular approaches to likelihood sampling. The result of this sampling algorithm is a set of values for the parameters, and a histogram of these will provide an estimate of the probability distributions.