Conditional probability -- the probability of observing one event as a result of having observed another event -- is a potentially important factor in designing intelligent Web applications. Paul Meagher (www.datavore.com) introduces Bayesian inference by discussing the basic mathematical concepts involved and demonstrating how to implement the underlying conditional probability calculations using PHP. In this article, the author discusses how Bayesian inference can be used to build an online PHP-based wizard that guides a user through the process making a medical diagnosis. This three-part series features interesting applications designed to help you appreciate the power and potential of Bayesian inference concepts.

If you examine current artificial intelligence, statistics, and data-mining journals, books, and conferences, you will notice that Bayesian inference techniques are being applied to increasingly complex problems in a growing number of application areas. Many Web developers, however, lack a constructive understanding of Bayesian inference and this prevents them from utilizing these techniques in their own software-development practice. This article, the first of three, aims to provide web developers with some necessary background.

For a Web developer, a constructive understanding of Bayesian inference means that you are able to see how Bayesian inference can be applied to the Web development problems you are facing. To achieve this level of understanding, a developer cannot be content to simply study a few examples of how the relevant math formulas work; he must also:

1. See how these math formulas might be distilled into software routines
2. Determine how these mathematical software routines can be integrated into interesting Web applications

In mathematics, we are usually concerned with declarative, or "what is", descriptions, whereas in computer science we are usually concerned with imperative, or "how to", descriptions. -- Abelson, Sussman, and Sussman in Structure and Interpretation of Computer Programs, p.22

Bayesian inference techniques have been widely used in developing various types of Artifical Intelligence (AI) systems, for instance for text retrieval, classification, medical diagnosis, data mining, troubleshooting, and more, so this article series will be of interest to anyone interested in building intelligent Web applications.

In this article, I will introduce some of the basic mathematical concepts and notations you need to know in order to properly apply the techniques of Bayesian inference. I will also demonstrate how you can implement conditional probability and Bayes theorem calculations using PHP, and how these calculations can be used to build an online medical diagnosis wizard.

In the next two articles, I will also explore the application of Bayesian inference to the design and analysis of Web surveys. This article lays some of the groundwork necessary for understanding this advanced application of Bayesian inference concepts.

The first and most important piece of groundwork to mention concerns the concept of conditional probability.

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