I am reading a book example (diagram from p10) in which a person scores 9/10 on which we assumed a uniform prior. The posterior distribution could be easily worked out analytically, but the book gives an example of estimating the posterior distribution by MCMC methods. I can see from the bottom part of the diagram that about about 95% of samples had a theta of between 0.59 and 0.98, where theta represents 'ability' in the sense of accuracy rate on these tests. However, I don't understand how a theta value is obtained for each sample in the chain, and what the relationship is between a sample in a chain and the proceeding sample. The book also does not explain why three chains are chosen or where the initial values for these chains come from.