In this chapter we assume that you have a method for calculating the likelihood of some data from a parameterized model. Using some prior on the parameters, Bayes’ theorem then gives the probability of the parameters given the data and model. A common goal in cosmology is then to find estimates of the parameters and their error bars. This is relatively simple when the number of parameters is small, but when there are more than about five parameters it is often useful to use a sampling method. Therefore in this chapter we focus mainly on finding parameter uncertainties using Monte Carlo methods.

Why do sampling?

We suppose that you have (i) some data, d, (ii) a model, M, (iii) a set θ of unknown parameters of the model, and (iv) a method for calculating the probability of these parameters from the data P(θ|d, M). For convenience we mostly shall leave the dependence on d and M implicit, and thus write P(θ) = P(θ|d, M).

An example we will consider throughout this chapter is the estimation of cosmological parameters from cosmic microwave background (CMB) data. For example, you could consider that (i) the data is the CMB power spectrum, (ii) the cosmological model is a Big Bang inflation model with cold dark matter and a cosmological constant, and (iii) the unknown parameters are cosmological parameters such as the matter density and expansion rate of the Universe.