Data assimilation is a procedure for combining observations and forecasts of a system into a single, improved description of the system state. Because observations and forecasts are uncertain, they are each best described by probability distributions. The problem of combining these two distributions into a new, updated distribution that summarizes all our knowledge is solved by Bayes theorem. If the distributions are Gaussian, then the parameters of the updated distribution can be written as an explicit function of the parameters of the observation and forecast distributions. The assumption of Gaussian distributions is tantamount to assuming linear models for observations and state dynamics. The purpose of this chapter is to provide an introduction to the essence of data assimilation. Accordingly, this chapter discusses the data assimilation problem for Gaussian distributions in which the solution from Bayes theorem can be derived analytically. Practical data assimilation usually requires modifications of this assimilation procedure, a special case of which is discussed in the next chapter.

This chapter describes how Bayesian inference is applied in Hierarchical Modelling of Species Communities (HMSC). The chapter starts by summarising the structure of the core HMSC model. It then briefly recalls some of the fundamentals of Bayesian inference, aimed primarily for those readers who are not very familiar with it. The core part of the chapter describes the structure of the prior distribution of HMSC and explains in particular how the default prior has been chosen. The chapter also briefly discusses how posterior sampling is conducted in HMSC through Markov chain Monte Carlo. The chapter uses the R-package HMSC-R to demonstrate how the prior distribution can be sampled, and to illustrate that samples from the prior distribution are identical to posterior samples if the model does not have any data. Finally, the chapter discusses how the computational time needed to fit an HMSC model depends on the size and type of the data.