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4.3.5 Bayesian Reasoning in Everyday Lif... > 4.3.5.1 Holmesian Deduction - Pg. 68

68 CH A P T E R 4: Bayes' Rule analytical mathematics. Yet another method is to numerically approximate the integral. When the parameter space is small, then it can be covered with a comb or grid of points and the integral can be computed by exhaustively summing across that grid. But when the parameter space gets even moderately large, there are too many grid points, and therefore other methods must be used. A large class of random sampling methods have been developed, which can be referred to as Markov chain Monte Carlo (MCMC) methods, that can numerically approximate probability distributions even for large spaces. It is the development of these MCMC methods that has allowed Bayesian statisti- cal methods to gain practical use. The next major part of this book explains these various methods in some detail. For applications to complex situations, we will ultimately focus on MCMC methods. Another potential difficulty of Bayesian inference is determining a reasonable prior. What distribution of beliefs should we start with, over all possible param- eter values or over competing models? This question may seem daunting, but in practice it is typically addressed in a straightforward manner. As we will dis- cuss more in Chapter 11, it is actually advantageous and rational to start with an explicit prior. Prior beliefs should influence rational inference from data,