Solving parameter estimation problems

While the first article in this series discussed building intelligent Web applications through conditional probability, this Bayesian inference article examines how you can use Bayes methods to solve parameter estimation problems. Relevant concepts are explained in the context of Web survey analysis using PHP and JPGraph.

Statistics consists of a set of classic problems and knowledge about the methods that might be used to solve those problems. Classic problems in statistics include parameter estimation, hypothesis testing, prediction, model selection, and decision making. Researchers in artificial intelligence have added classification, clustering, and learning to this list of inference problems.

Bayes inference methods are distinguished from other inference methods (such as least squares, maximum likelihood, maximum entrophy, minimum description length) by the fact that a Bayes inference method can be applied to every type of problem in this list and often represents the optimal method to use. Statisticians, philosophers, and computer scientists have proposed numerous guides to reliable inference, but Bayes inference methods reign supreme as the most versatile and often the most effective set of methods to use for a vast range of inductive inference problems.

In this article, I will discuss the classic problem of parameter estimation and develop Bayesian parameter estimation code using the popular Web scripting language PHP. I am specifically concerned with developing parameter estimation code that might be used in the analyses of simple Web surveys.