Does Bayesian epistemology provide mechanisms to prevent arbitrariness in the selection of one's priors and belief update rules?

Question

Does Bayesian epistemology provide mechanisms to prevent arbitrariness in the selection of priors and belief update rules, especially when such choices could render posterior evidence effectively powerless to alter one's beliefs?

To provide some context and motivation for this question, in light of my recent question Can Bayes' theorem be used non-fallaciously to argue for miracles?, one commenter offered the following perspective:

Your probability P(M) is based on human reports (which are highly subjective), but does not incorporate scientific viewpoint (which is a result of rigorous studies.) Thus, if we consider a probability of a thermodynamic system adopting a highly improbable configuration - like all the atoms gathering in half of a container or a decomposing human body coming back to life, we have to deal with numbers like P(M)~ 10^{-N_A}, where the Avogadro constant itself is N_A=6 * 10^{23}. This is prohibitive.

This appears to reflect a case of assigning priors in a way that makes certain outcomes virtually impossible (prohibitive is the actual word used by the commenter), conveniently insulating beliefs from contrary evidence. Does Bayesian epistemology include safeguards against such practices, ensuring priors and updates remain open to meaningful revision in light of new evidence?

Answer 1

What you are referring to are called strong priors in that they are highly concentrated around a particular value.

This is not always bad: In the classic example of medical testing, the base rate is important to consider.

The trouble is when we don't have a good, objective justification for a strong prior. We don't necessarily need evidence but a strong theoretical rationale is also good (provided it is an accepted theory).

In practice, we don't use strong priors that often in Bayesian data analysis. Instead we rely on what are called "uninformative priors", or "weakly informative priors" such as:

1. Maximum entropy priors -- seek least informative distribution
2. Invariant priors -- e.g., our derived prior should be invariant to particular transformations (e.g., Jeffrey's prior)
3. Diffuse priors: Basically any unimodal distribution f(x) scaled to be very wide g(x)=1/s*f(x/s) for some large s

Andrew Gelman, in his popular book Bayesian Data Analysis, tends to prefer option 3, since you avoid potential pathologies of 1/2 and often we do have some idea of where the true parameter may be (the real number line is awfully long after all ;-).

Answer 2

No because there is no objective way to define your priors. Bayes’ theorem is a human constructed system for updating your belief in a hypothesis after coming across certain evidence. But it tells you nothing about what your initial credence should be.

Many have tried to get around this by saying that the priors don’t matter. After all, if you update your evidence consistently, then two people with different priors will eventually arrive at the same conclusion. The priors will be “washed away” so to speak.

But this is also false. Different people can see the same piece of data as evidence for completely different theories. For example, UFO footage may be seen as evidence for a person who believes in aliens visiting earth, and thus he may increase his probability of that hypothesis. A person who has the opposite viewpoint may increase his probability of the hypothesis that aliens have not visited earth and that this was the result of advanced technology from China (after all, you may expect to see things that look like flying saucers if foreign countries were experimenting on tech you haven’t come across). He may then increase his probability for the latter hypothesis.