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

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?

• The analyst's personal experiences are not the analyst's prior; the prior is what you have before you look at any experiences, theoretically if you were a complete blank slate. Based on the analyst's experiences, Bayes' theorem says how to rationally update the posterior. The evidence you gave for "all the residents of Xanadu are racists" sounds like pretty weak evidence, so Bayes' theorem wouldn't assign a high probability to the statement. Commented Jun 23 at 18:58
• Does this answer your question? Are the priors of Bayesianism really subjective?
– g s
Commented Jun 23 at 18:59
• What is the purpose of using a loaded word like "bigotry" in a philosophical context like this? Commented Jun 23 at 19:15
• Two people with similar experiences can have very different priors, if that's what you're asking. If there's bias in their reflections, it will influence their assigning priors. And thus their bias will carry through the calculations. Seems like a feature not a bug, and I don't see what needs saving. Not sure if you're going for something more provocative though. Commented Jun 23 at 19:20
• "But suppose the analyst is looking at this hypothesis: all the residents of Xanadu are racists."What is Cromwell's rule and why is it important for Bayesians? Commented Jun 25 at 7:21

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.

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.

• As an aside, the risk to be (correctly) diagnosed with breast cancer in a randomly selected American 40 year old woman is rather 1.5/1000, not 1/100. (I guess this from the fact that the probability for the next decade is 1.55%; for the preceding decade it is 0.49% and for the succeeding one 2.4%, so for 40 it is close to the middle between 0.55 and 2.4.) This makes the benefits of screening much less clear (to the point that they are not measurable). Commented Jun 24 at 14:37
• Actually, I messed up the number somewhat: The incidence should be between 0.49/1000 (preceding decade average) and 1.55/1000 (succeeding decade average), that is, around 1/1000. Commented Jun 24 at 14:52
• Bayes' rule is a mathematical truth. As such, it is true only within the abstract world of mathematics. The numbers you quote - 80%, 9.69% - are real-world values, not true probabilities. They are just estimates, obtained from measurements, in studies, on limited samples, with all possible biases flowing in - selection bias, confirmation bias, survivor bias etc. Commented Jun 25 at 9:40

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.

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.

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.

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.

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.

• This answer is inaccurate in several important ways. 1) Bayes' Theorem is not descriptive of how people DO update their beliefs based on new data. The theorem is prescriptive, defining how one SHOULD update their beliefs. 2) The fact that an individual claims to be practicing Bayesianism does not make it true. Creationists and conspiracy theorists are especially prone to confirmation bias, ignoring inconvenient evidence rather than updating their beliefs. This is precisely what Bayes' Theorem is intended to combat. Commented Jun 25 at 17:28

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).