Computational Bayesian Statistics Including Markov Chain Monte Carlo
William M. Bolstad, James Michael Curran · 2016
This chapter offers a brief introduction to modern computational Bayesian statistics, explaining how computational Bayesian statistics relies heavily on being able to efficiently sample from potentially complex distributions. The histogram of the random sample from the posterior will approach the posterior density as the sample size increases towards infinity. Thus statistics calculated from the random sample will approach the parameters of the posterior distribution. A graph in the chapter shows Chiara's samples from the posterior distribution for sample sizes 1,000,10,000,100,000, and 1,000,000 respectively. A closed form for the integral and hence for the posterior can only be found in a limited number of special cases. The difficulty of evaluating the posterior in the general case left Bayesian statistics out of mainstream applied statistical practice. Statisticians were aware from their studies in decision theory that Bayesian statistics offered real advantages in theory. The statistician can focus on the statistical aspects of the model without worrying about calculability. This allows the applied statistician to use realistic models that are based on the underlying situation instead of being restricted to models that are mathematically easy to work with.