### Issues Regarding Bayesianism

My assessment of the plausibility of the resurrection uses the same tools that Richard Swinburne did when he put the probability of the resurrection at 97% in his book the Resurrection of God Incarnate. I obviously think he has erred in his approach, but I don't think his error was to frame his arugment in Bayesian terms. Before critiquing his argument, I will discuss some of the controversy associated with Bayesianism in general.

There are a wide variety of Bayesians, so my descriptions will not accurately describe all those who call themselves Bayesian. Bayesians think contingent statements (or hypothesis) plausibilities can be mapped to probabilities. Generally, the initial probabilities are considered to be subjective. Further Bayesians state that the degree of belief should be updated according to Bayes' Theorem given below. Here the term H refers to the hypothesis, and E refers to the evidence.

P(H|E) =  P(E|H) P(H)
P(E)

The denominator is often rewritten using an identity is probability,
P(E) = P(E|H1)P(H1) + P(E|H2)P(H2)
+ . . . + P(E|Hn)P(Hn
)
where H1, through Hn are mutually exclusive hypothesis. In the case of two hypotheses, ~H is considered the negation of the primary hypothesis, P(~H) = 1 - P(H)

Bayes’ Theorem itself is not considered controversial. What is considered controversial is what values can be put into it. Bayesians would readily put values into the theorem that cannot be interpreted as some sort of frequencies.

Arguments for Bayesianism
Bayes theorem seems to allow quite a bit of room for subjective preferences while implying that one is meeting all the demands of rationality. Despite this, there are good reasons to think that Bayesianism is the normative standard for evaluating contingent beliefs.

Argument from Cox's Theorem
E.T. Jaynes argues that Bayes’ Theorem in probability is simply a special case of a more general form of inductive logic. He argues that can be derived from desirable postulates of plausible reasoning. The postulates make no reference to relative frequency and thus there is no need to restrict the values to probability as defined by frequentists.

The postulates are I) The first postulate is "Degrees of belief can be represented by real numbers," II)The second is “Qualitative correspondence with common sense.” This postulate indicates the “direction” that evidence moves reasoning, this postulate does not quantify how much the reasoning moves. For example if A and B are two events, and AB is the conjunction of those events, if data D makes A more likely (and doesn’t affect the likelihood of B), then D does not make the conjunction less likely. III) The third postulate is "Consistency." It entails that the order in which evidence is evaluated should not matter, equivalently plausible scenarios should be represented by the same plausibility measure, and evidence should not be ignored on an ad hoc basis.

To deny that Bayes' Theorem should be used to update beliefs, one would have to deny one of the postulates (or perhaps find an error in the proof). For example, some deny that one can in fact order their beliefs. I will address this later in more detail.

Diachronic Dutch Book
A Dutch book is a series of bets that can be offered to one who holds a series of inconsistent probability beliefs. If the person holding those beliefs acted on them by betting, they would be guaranteed to lose money no matter what events transpired. That is, if a Dutch book can be made against you, your beliefs are incoherent.

Jeffrey Kasser in a lecture on the Philosophy of Science from the Teaching Company offers the following example. Suppose that you think the probability of it raining tomorrow is 0.6 and you simultaneously think the probability it does not rain is 0.6. In betting terms, you should agree to pay \$6 for a payoff of \$10 if it rains due to the first belief and simultaneously to pay \$6 for a payoff of \$10 if it doesn't rain due to the second belief. If you accept those bets you will pay \$12 for a payoff of \$10 regardless of the outcome. It can be shown that a person's beliefs are consistent with the axioms of probability then no Dutch book can be made against them.

The classical Dutch book arguments apply to states of belief at a given time. The diachronic (across time) Dutch book argument applies this same sort of an argument to situations where beliefs are updated based upon new evidence. It can be shown that if one's beliefs aren't updated according to Bayes' Theorem, a Dutch book can be made against one's beliefs. Conversely, no Dutch book can be made against one who updates their beliefs according to Bayes’ Theorem.

Arguments against Bayesianism
Tolerance for subjective probabilities
As I understand it, this is the most influential objection to Bayesianism. It seems that Bayes' Theorem does not have the power to exclude some very unusual beliefs provided beliefs are updated properly. There are several aspects of this objection. One aspect can be illuminated by examining what motiviated logical positivists to try to banish metaphysics from science.

Prior to Einstein, the notion of the relativity of unaccelerated motion, and the constantness of the speed of light with respect to all observers seemed contradictory. Einstein was able to develop his special theory of relativity by critically examining some metaphysical concepts. He challenged the notion of simultenatity at a distance and length. It is thought that the reason other physicists couldn't find Einstein's theory was that thought they had clear understanding of the metaphisics of space and time.

It was clear to some philospophers that some of the metaphysical notions we had, were blinding us to understanding how nature works. Eventually, it became thought that removing metaphysics from science was not entirely possible, yet the problems that motivated them remain influential. To many philosophers, having subjective priors seem analogous to allowing metaphysics to influence science.

Bayesians do have some responses. Often it is possible to develop an uniformative prior by using maximum entropy distributions when possible. However, this is not alway possible or approriate. Secondly, Bayesians can claim that with sufficient evidence, the prior information can be washed away. A complication is that the amount of required may be massive, and it would require those with the different prior to assess the evidence identically. Further Jaynes has gives examples where some evidence can actually cause divergence of beliefs in the probability scale. For example in my ESP example, those predisposed to belief in ESP would likely be convinced of its reality after the Soal, and Bateman's report, while those predisposed to believe in deception will be convinced of deception after the same report.

However, it is not clear how much of a problem this is. Kuhn argued that some difference of opinion due to subjective factors is healty for science. It seems that this is likely true for the pursuit of knowledge in general.

The catch-all hypothesis
When the denominator P(E) is expanded into two term P(E|H)P(H) + P(E|~H)P(~H), the alternative is often not well defined. There are usually an endless number ways the alternative could be false. Thus Bayes' Theorem does little to present us the truth of any particular hypothesis.

I think this problem is more a "feature" than a bug. To quote Jaynes, "Unless the observed facts are absolutely impossible on hypothesis H, it is meaningless to ask how much those facts tend in themselves to confirm or refute H. Not only the Mathematics, but also our innate common sense (if we think about it for a moment)tells us that we have not asked any definite, well{posed question until we specify the possible alternatives to H0. Then as we saw in Chapter 4, probability theory can tell us how our hypothesis fares relative to the alternatives that we have specified; it does not have the creative imagination to invent new hypotheses for us."

This is an aspect of Bayesianism that does seem anti-metaphisical. Bayes' Theorem doesn't necessarily tell us how the world really is. It merely helps us sort our beliefs.

The Problem of Old Evidence
Some object that Bayes' Theorem does not allow one to use old evidence to confirm a new hypothesis. The reason given is that if the evidence is already know than P(E) should be one, and so should P(E|H). If you plug these values into Bayes' theorem, you can see that the probability of P(H|E) will not be changed. However, this does not strike me as that stong of an objection. Many examples of Bayesianism will not simply presume that P(E|H) is one just because the evidence is observed.

Consequences of rejecting Bayesianism
If the objections hold enough force, one must conclude that the ranking of one's beliefs is an untenable project. For example, in classical statistics, the only term considered is the evidence term P(E|H). However, one cannot get to the assessment P(H|E) without including all the terms in Bayes' Theorem. Instead of the subjective criteria, classical statistics reject or accept hypothesis based upon a p value of 0.05 which is not seems like another aribitary standard. Jaynes argues that the aribitary standard can and has been abused which has damaged both science and the public good.

The consequence of holding to Bayes' Theorem is that subjective and/or metaphysical considereation cannot be excluded when ranking one's beliefs. It doesn't appear that this is too high price to pay if one thinks their beliefs can in fact be ranked. One making a Bayesian argument will often have to submit their metaphysical consideration to scrutiny. It seems to me that those who try to avoid Bayes Theorem are even more succeptible to errors due to subjectivity since the subjectivity is still there, yet unacknowledged.

The fact that Richard Swinburne used Bayes' Theorem to frame his argument has made it easier for me to see 1) how he is framing the alternative hypothesis 2) what he thinks reasonable prior beliefs should be 3) his assessment of how much the evidence should move his belief. I think that he is to be commended for making his argument so scrutible. My next post address the shortcommings of his argument.

It seems to me that conformance to Bayesian reasoning is a necessary, but not sufficient criteria for updating contingent beliefs.

Joe Otten said...

I think Swinburne's big crime was not using Bayes, nor even using a subjective prior, but in fact using entirely subjective "data" - the P(E|H) values.

If you think, as I do, that the Problem of Evil is a sound argument against most popular kinds of God, you would use a P(E|H) of 0 or 1, giving a P(H) of 0 or 1.

If on the other hand you are inclinded to dismiss the Problem of Evil, you might use a P(Evil|God) of 0.2 or 0.5 or something, and if you think evil is theologically necessary you might even suggest P(Evil|God) = 0.9, perversely making the existence of Evil count for the existence of God.

My point is that these are all numbers plucked out of the air, whereas the P(E|H) numbers should be a function of experimental data.

Compared to this crime against Bayes, a subjective prior is nothing.

Bill Curry said...

Joel,

I would phrase my criticism of Swinburne’s approach differently. I think that you are implying that Swinburne has ignored relevant background information in his assessments. I certainly agree with you there. However, most Bayesians would say that the use of subjective assessments of data may be necessary for the evaluation of some hypotheses.

I do think that the problem of evil has a great deal of force against the Christian God. However strict Bayesians don’t assign probabilities of either 1 or 0 to contingent assessments. The reason is that once either of those values is assigned, the assessment can no longer be changed no matter what additional evidence is given.

Also note that just because someone thinks the P(Evil | God) is 0.9 does not in itself imply they think evil is evidence for the existence of God. The key quantity is the ratio:
P(Evil | God)/P(Evil | no God)

For example if someone not bothered by the problem of evil assigns P(Evil | God) = 0.9, but admits that evil is likely if there is no God, P(Evil | no God) = 1, then they would still acknowledge that evil is weak evidence against the existence of God.

franky said...

I'm going to have to agree with Joe. Swinburne's "data" was just plucked from thin air. He starts off with a completely nonsensical probability assigning a 50% probability that God exists or doesn't exists.
Huh? Really. How does that make any sort of sense? How can that number possibly be defined correctly?
If the numbers used are nonsensical, then the answer is likewise nonsensical.

Anonymous said...

Bill, you are right that it is the ratio that matters. I was oversimplifying in pursuit of my main point, that all the data numbers - the "contingent assessments" - are plucked out of the air.

Bayesianism can be forgiven for being subjective in its use of priors, but the data should be non-subjective and found by "trial" - experiment or survey.