*Conjugate priors*-

We discussed this topic in Sect. 8.3, giving a couple of typical simple examples and references for a more detailed list of famous conjugate distributions. We want to remark here that a conjugate prior is a special case of the class of priors that simplify the calculation of the posterior (the uniform prior is the simplest of this kind of prior). *Gaussian approximation*-

For reasons that are connected with the central limit theorem, when there is a large amount of consistent data the posterior tends to be Gaussian, practically independently of the exact shape of the prior. The (multi-variate) Gaussian approximation, which we encountered in Sect. 5.10, has an important role for applications, either as a reasonable approximation of the `true' posterior, or as a starting point for searching for a more accurate description of it. We also saw that in the case of practically flat priors this method recovers the well-known minimum chi-square or maximum likelihood methods. *Numerical integration*-

In the case of low dimensional problems, standard numerical integration using either scientific library functions or the interactive tools of modern computer packages provide an easy solution to many problems (thanks also to the graphical capabilities of modern programs which allow the shape of the posterior to be inspected and the best calculation strategy decided upon). This is a vast and growing subject, into we cannot enter in any depth here, but we assume the reader is familiar with some of these programs or packages. *Monte Carlo methods*-

Monte Carlo methodology is a science in itself and it is way beyond our remit to provide an exhaustive introduction to it here. Nevertheless, we would like to introduce briefly some 'modern' (though the seminal work is already half a century old) methods which are becoming extremely popular and are often associated with Bayesian analysis, the so called*Markov Chain Monte Carlo*(*MCMC*) methods.