Bayesian vs. Frequentist Statistical Theory
The Frequentist view of probability is that a coin with a 50% probability of heads will turn up heads 50% of the time.
The Bayesian view of probability is that a coin with a 50% probabilit of heads is one on which a knowledgeable risk-neutral observer would put a bet at even odds.
The Bayesian view is better.
When it comes to statistics, the essence of the Frequentist view is to ask whether the number of heads that shows up in one or more trials is probable given the null hypothesis that the true odds in any one toss are 50%.
When it comes to statistics, the essence of the Bayesian view is to estimate, given the number of number of heads that shows up in one or more trials and the observer’s prior belief about the odds, the probability that the odds are 50% versus the odds being some alternative number.
I like the frequentist view better. It’s neater not to have a prior involved.
September 30th, 2007 at 1:34 pm
“I like the frequentist view better. It’s neater not to have a prior involved.” You just stated a prior, you preference for neatness.
October 7th, 2008 at 6:48 pm
Michael Webster’s comment is in essence an ad hominem attack on you, not a response to your statistical question. In response to the statistical question:
The frequentist is testing the null hypothesis, right? In specific, the null hypothesis being tested is “are the odds of the coin coming up heads are 1:1?” But suppose I said “Oh, I’m sorry, I forgot to tell you my coin has three sides.” The frequentist would then still test “the null hypothesis”, but the null hypothesis would now look like “the odds of coming up one particular side are 1:2 (with probability 1/3)”.
We say the frequentist is testing “the” null hypothesis because the hypothesis being tested really remains the same: it is “the probability the coin comes up heads is what we should expect (whatever that is).” So whether he wants to admit it or not, the frequentist is using prior information to determine what specific form the null hypothesis will take.
In these two cases the prior information is sufficient to determine exactly what form “the null hypothesis” should take. Suppose I only told you, though, that my “coin” (or die) has between 4 and 7 sides? In other words, how does the frequentist test “the” null hypothesis when there isn’t sufficient information to determine a specific form of the null hypothesis to test? However the frequentist solves that problem, their solution amounts to a Bayesian prior.
The Bayesian, on the other hand, acknowledges that prior knowledge about the hypothesis is required, _and includes that fact in his analysis_ in a mathematically rigorous fashion. The Bayesian accounts for selecting a specific form for the null hypothesis whether there is sufficient information to choose a specific form or not; at best, the frequentist only accounts for that prior information when there isn’t enough information to select a specific form for the null hypothesis.
The frequentist view is incomplete in a way that the Bayesian view is not; the Bayesian view is better.
October 7th, 2008 at 8:30 pm
The frequentist view does require some prior information, since it has to define what might be observed, but it doesn’t have to specify the probability of the world that is its null hypothesis. If its null hypothesis world is that either heads or tails will show up, but then a third option (belly?) shows up, the null is rejected, since it says belly has zero probability.
This does mean that a frequentist test really is testing more than one thing. If it is testing a coin toss, then it is not just testing whether Heads has a 50% chance of showing up, but whether Belly is a possibility.