Two on Bayesian Probability

Bayesian probability holds, among other things, that probability is sticky: once the probability of an event rises to 1 or drops to 0, it stays there forever. Your weather forecaster defies this when they tell you that the probability of rain is 95% when it is already raining. But there is a lot more to Bayes, whose theories underlie much of our contemporary algorithms and science. Here's an introduction to his life:
For most of the two and a half centuries since the Reverend Thomas Bayes first made his pioneering contributions to probability theory, his ideas were side-lined. The high priests of statistical thinking condemned them as dangerously subjective and Bayesian theorists were regarded as little better than cranks. It is only over the past couple of decades that the tide has turned. What tradition long dismissed as unhealthy speculation is now generally regarded as sound judgement.
And here is a piece on application.
Bayesian statistics is two things: a useful technology and a bundle of mythology. A Bayesian data analyst almost never, and I mean almost never, inquires as to her degrees of belief: she makes mathematically convenient and not absurd assumptions and goes on. She tests the resilience of the outcomes she obtains by varying those assumptions—the prior probabilities, the penalties in a model score, etc.. Essentially, her “prior probabilities” are just a measure to guide through a search space of alternative possible values for parameters in a model or models. The measure is adaptive, in the sense that it alters (by Bayes Rule) as data are acquired. It is subjective, in the sense that there is no best adaptive measure for guiding search, but there are better and worse adaptive measures. Generally, the measures are nobody’s degrees of belief.

1 comment:

Tom Grey said...

The big break for Bayes is Decision Analysis. Given any decision under uncertainty, there is some probability of the result being Good, Medium, or Bad -- or some other set of outcomes.

The probabilities of achieving these results are dependent on the decision taken; and usually one of the possible decisions is to gather more info.

The math behind using probabilities these ways is from Statistics, where they developed such math with random processes and the Law of Large Numbers. It is Bayes Theorem which allows the math to be used in unique, one-time decision events.

Bayesians look at probability as a measure of information about an events; Statisticians look at frequencies and derive probabilities from the data.

Consider flipping a coin in the air and stepping on it. What is the probability that it is heads? The statistical answer: if there were enough flips, it would have a 50% chance.
But we know that in this case it is either 1 (heads) or 0 (tails).
The Bayesian says: our prior probability estimate is 50%, so for this first flip it's 50%.