Bayes Theorem And its Applications

First and foremost, Bayes Theorem is a statement about conditional probabilities. "Given this, what can we say about that?" It is used most commonly in situations where one is attempting to derive conclusions about some theory or whatever given that one has obtained a particular set of data. Going from data to theory is hard, but it is often straightforward to go from theory to data. Let's say you have a theory called Life, the Universe, and Everything. The prediction for the data is always 42. Now you conduct an experiment and get the answer 42. Does that mean that Life, the Universe, and Everything is correct? What if I have a theory called 2-3-7 that predicts the answer should be a combination of powers of those prime numbers? 42 is also possible in this theory. Bayes Theorem is no silver bullet, but it does provide a framework for inverting the theory-prediction combination such that one can make meaningful statements about theory given the results of some experiment or measurements.

Bayes Theorem is not difficult to prove. It is best done with the aid of a Venn diagram. Such a derivation is given here .

Bayes Theorem can be applied in a wide variety of situations. The following is collection of topics that have interested me over the years and which I have investigated in one way or another, in one case even writing a paper.

Dark Energy Survey Year 1 Cosmology Results from Large Scale Structure and Weak Lensing. This paper uses a Bayesian Hypothesis Test to examine whether two sets of DES data are consistent with each other under the hypothesis that the standard Lambda-CDM model is correct. I am one of the (many) coauthors.
Where is Bill Ewasko? Bill went for a day hike in Joshua Tree National Park in June of 2010 and has not been seen since. Three days later, a cell tower recorded a single ping from his cell phone. In spite of intensive searching by numerous groups and individuals, he has still not been found. I have hiked this area as well and also found nothing. This paper uses Bayes theorem to identify a few possible remaining places to search. [UPDATE] He has since been found - by chance.
Search for the SS Central America . This was a treasure hunt. Bayes theorem was used to identify where to search for a ship that, while carrying a cargo of gold from mines in California, sank during a hurricane in 1857 off the coast of Georgia. Unlike the search for Bill Ewasko, this one was successful. I have given informal talks on the story. The most important lesson to be drawn is as follows: whenever gold is involved, it brings out the worst behavior in everybody.