The Basics of Bayes' Theorem

I'd like to introduce you to the "basics" of Bayes' theorem. The basics are very simple (well, at least for the mathematician, statistician, or physicist). My focus in this article is not require much expertise. In future articles, I'll cover all the details.

Basically, Bayes' theorem relates one conditional probability to another. A conditional probability_ is the probability that one thing is true given another thing is true. For example, the probability that you have a winning poker hand given you have four aces, a relatively larger number, is related to the probability that you have four aces given you have winning poker hand, a relatively smaller number.

Thomas Bayes introduced his theorem using a thought experiment. [This blog post](https://www.tcbegley.com/blog/posts/bayesian-billiards) quotes from The Art of Statistics by David Spiegelhalter.

Suppose a white ball is thrown at random onto a billards table, its position along the table marked with a line, and then the white ball is removed. A number of red balls are then thrown at random on to the table, and you are told only how many lie to the left and how many to the right of the line. Where do you think the line might be, and what should be your probability of the next red ball falling to the left of the line.

Thomas Bayes' reasoning in his paper eventually led to the proposition, described much more succinctly using mathematical notation, that the probability of A given B (is true) is proportional to the product of the probability of B (independent of anything else) and the probability of B given A (is true). A more concrete example in words would be that the probability of rain when I see a wet sidewalk is proportional to the the product of the probability of a wet sidewalk (under any conditions) and the probability of a wet sidewalk when it has rained.

That seems messy. Basically, Bayes' theorem relates two probabilities: the probability of something being true when another thing is true and the probability of another thing being true when something is true. The relationship is a proportion or a fraction, and the constant of proportionality is another probability.

"So what?" you may ask. The value of Bayes' theorem is that calculating these different probabilities may take vastly different amounts of effort. In general, Bayes' theorem allows us to calculate a "difficult probability" in terms of two other probabilities which may be easier to calculate.

Whew. I don't know that I've clarified the issue. Let me know. And thanks.