What principle protects the objective nature of the prior and the conclusion in Bayes’s theorem?

Question

The Bayesian analysis begins with the "prior": some assumption about the world and the probability that the assumption is true.

But the prior seems to be based on nothing. The hypothesis and the probability of its truth are formed before any evidence is gathered. So the hypothesis is a collection of things that the analyst thinks are true.

The theorem works fine in experiments that minimize moral judgment: why insects behave a certain way, or which diet and exercise regimen will enable an athlete to run faster.

But suppose the analyst is looking at this hypothesis: regardless of the job applicant’s qualifications, employers in Xanadu tend to hire Xanaduvians rather then equally-qualified applicants of Erewhonian ancestry. How to determine, reasonably, the content of the prior?

Here, the prior is tainted. The experimenter themselves may be a resident of Xanadu, might have one loudmouth racist Xanaduvian neighbor, or is a resident of Erewhon and so hates those from Xanadu, and so forth. These conditions can give the experimenter a personal self-interest which can affect the probability assumed in the opening statement.

What is supposed to rescue the theorem is that the subsequent analysis is confined to relevant evidence, thus preserving the objective nature of the ultimate conclusion. Theoretically, and ideally, all the analyses, regardless of the content of the priors, will converge on a single set of ideas. But here, the content of the initial assumption partially directs what evidence is relevant to it and what is not.

So what are the set of principles that protect the objective nature of the prior, and ultimately, the conclusion?

Answer 1

Bayes' theorem is just that- a theorem. It is no more based on prejudice than Pythagoras' theorem. All calculations are subject to the 'garbage in, garbage out' rule. If I make wild guesses about the lengths of the shorter sides of a right-angled triangle, I should not be surprised if the application of Pythagoras' theorem then gives the wrong value for the length of the hypotenuse- that is not the fault of the theorem itself, but of how I have misapplied it. Likewise with Bayes' theorem.

Answer 2

So does Bayes’ theorem reduce to ordinary prejudice?

In some circumstances, you could say that, you could say that priors are arguably just rooted in personal biases, but not all applications of bayes theorem have that aspect of generating priors from personal bias.

For example, this problem:

The probability of breast cancer is 1% for a woman at age forty who participates in routine screening. If a woman has breast cancer, the probability is 80% that she will get a positive mammography. If a woman does not have breast cancer, the probability is 9.69% that she will also get a positive mammography. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?

Bayesian reasoning can help us answer this question easily, but no subjective personal biases come in - the priors that go into the calculation can all come from reliably gathered statistics.

Answer 3

Bayes' theorem does not dictate how one selects the prior probabilities. Certainly one can fill a Bayesian model with bigotry and unjustified biases, but this is not necessary. One can even use what is called a "non-informative" or "uninformative" prior.

Answer 4

A mathematical theorem that can be proved to hold true inside a mathematical context, becomes part of the theory itself. In that case you can see it as tool; what you do with that tool, is up to you. Can you ascribe hostility to physics equations because of usage in building the atomic bomb? In the same way you cannot ascribe prejudice to a provable mathematical theorem.

Answer 5

I suspect that you are confusing Bayes's Theorem with Bayesian inference. As @Marco Osram pointed out, Bayes's Theorem is an actual theorem, with a proof. Frequentists don't question Bayes's Theorem; even Fisher didn't question Bayes's Theorem.

Bayesian inference is different matter. While I wouldn't say that it is "based on prejudice", it lets us manage prejudice. Let us say that we have a difference of opinion on some matter. We collect evidence, calculate likelihoods, and then derive new posterior probabilities. If we have followed Cromwell's rule, our posteriors should be closer to each other than the priors were; Bayesian inference allows us to reduce the gap caused by prejudice.

Answer 6

Short answer

NO! What Bayes' Theorem does, is allow one to do statistical analysis of what happens to one's prior belief, given new data.

To do that, one must specify one's prior, and its relative confidence vs. the new data.

Both a prior value, and relative weighting, are JUDGEMENT CALLS, and hence not "objective". Bayesian advocates tend to falsely claim that Bayesian statistics are objective. They are not, they explicitly are subjective, which allows one in principle to examine and correct one's subjective assumptions.

Longer answer

Yes, sometimes.

We humans are known for overstating the confidence in our priors, and dismissal of refuting data. Bayesianism is in many ways more subject to rationalizations in favor of one' prejudices as other reasoning processes, while providing a gloss of "objectivity" to these rationalizations.

Note that most creationists are good Bayesians, who have very high confidence in their prior of Biblical inerrancy. Most ideologues and conspiracy theorists also follow Bayesian thinking.

So -- Bayesianism potentially enables prejudice and bigotry. BUT -- when used properly, it is a very useful tool for sorting out how to combine past views with new data.

Answer 7

Yes it is. It is only as good as its priors. But all priors are fundamentally subjective. It is an attempt to try to formalize, using mathematics, heuristic reasoning.

But there’s no need to put fake numbers on things when reasoning between different hypotheses. You simply go with what you think is plausible. It’s quite literally useless for the things we care about. And in the cases where it works, it is more a function of standard probability theory than anything specific to Bayes’ theorem (such as applying it to HIV tests).