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Implement Bayesian inference using PHP: Part 2
By Paul Meagher - 2004-05-19 Page:  1 2 3 4 5 6 7 8 9 10

Conclusions and Resources

This article demonstrated the use of Bayesian methods to solve parameter estimation problems. I discussed several important concepts relevant to using Bayesian methods to solve parameter estimation problems, including maximum likelihood estimators, binomial random variables, the Bernoulli process, the beta distribution, and conjugate priors. I hope the discussion provided you with a better general understanding of Bayesian inference techniques and also helped you to understand some concepts that play important roles in statistics and probability.

I highlighted the important role played by probability distributions in representing the likelihood and prior terms in Bayes theorem. Recall that in the first article of this series, I computed the posterior distribution without resorting to the use of theoretical probability distributions. You can conclude from this that full mastery of Bayes methods involves learning a more theoretically-oriented probability distribution approach to Bayesian inference along with a more empirically-oriented joint frequency approach.

In my next article, I will move from analyzing simple binary surveys to the concepts and techniques that are useful for analyzing simple binary classification surveys and multivariate classification surveys. In doing this, I will examine another classic inference problem that Bayes methods are particularly good at solving - - classification problems.


  • Download the source code used in this article. Updates to article code will be made available at

  • Go deeper into Bayes parameter estimation in these lecture notes on Bayes Networks by James Cussens.

  • Learn probability concepts by exploring the Virtual Laboratories in Probability and Statistics.

  • Visit Radford Neal's site for research on Bayesian neural networks and essays on the philosophy of Bayesian inference.

  • Check out the Jsci Project, which aims is to encapsulate scientific methods/principles in the most natural way possible using Java. It provided the probability distributions code used in this article.

  • Try The BUGS Project (Bayesian inference Using Gibbs Sampling), a piece of computer software for the Bayesian analysis of complex statistical models using Markov chain Monte Carlo (MCMC) methods.

  • Read the other articles in the author's series on Bayesian inference:
  • Learn to craft Web data-gathering applications in "Apply probability models to Web data using PHP".

  • Explore a Bayesian method for detecting structural changes in a long-range dependent process in "Bayesian Methods for Change-point Detection in Long-range Dependent Processes" (IBM Research, April 2001).

  • For the PHP developer, find out how to write more efficient code "Writing Efficient PHP" (developerWorks, July 2002).

  • Learn how to construct a user-modeling platform with PHP "Web site user modeling with PHP" (developerWorks, December 2003).

  • Read Statistics: Probability, Inference, and Decision, 2nd ed. (International Thomson Publishing; 1975), by Winkler and Hayes, a source that the author relied on for this article.

  • Check out Artificial Intelligence: A Modern Approach (Prentice Hall; 2003), by Russell and Norvig, for an excellent discussion of Bayes inference methods.

  • Browse the developerWorks bookstore for titles on this and other related subjects.

  • Visit developerWorks Web Architecture zone for a range of articles on the topic of Web architecture and usability.

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First published by IBM developerWorks

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