Probabilifying probability

Jon Curry said:

“What evidentiary value? What do you mean? I’m simply saying that Gene’s is misguided to think that the contributors would agree on what actually happened with Jesus, or that the contributors should not be expected to offer a variety of opinions. The purpose of the book is not to show what actually happened with Jesus, but to show that in fact a miracle did not occur.”

How is Gene’s criticism misguided? If the contributors don’t know what really happened, then how are they in any position to say what didn’t happen?

They are assuming that the Resurrection didn’t occur, and then proposing alternative explanations. This assumes that we need an alternative explanation in the first place.

But if they best they can do is to offer a grab-bag of mutually exclusive alternatives, then they clearly have no idea what really happened, in which case they have no idea of what didn’t happen.

So the procedure is viciously circle. They assume the Resurrection didn’t occur. They justify their assumption by appeal to a variety of contradictory explanations. But their alternative explanations would only be needed on the prior assumption that the Resurrection did not occur.

“I suppose it is. Do you disagree with assigning a low initial value?”

Yes, I do.

1.To begin with, I don’t regard miracles as either probable or improbable. A miracle is not like a rare, but naturally occurring process (e.g. Halley’s Comet).

2.In addition, BT does not prove or prejudge the relative reliability of a historical witness, but, to the contrary, is simply a mathematical formulism into which we plug prior probability assignments that are independent of BT itself. As one philosopher explains:

“What this seems to show is that Bayesian theory can help us with the consistency of our commitments, and perhaps with clarification of what they are, but it is ill-placed to prove any sort of criterion for the acceptability of astonishing reports,” C. Coady, Testimony: A Philosophical Study (Clarendon (1994), 192.

“In the context of our present inquiry this conclusion is reinforced by the consideration that, even if we don’t know anything much about the witness…our broad philosophical assumptions about testimony and its epistemological significance will feed into the scales we use for the Bayesian operation or for any related probabilistic acceptability device. If we view testimony as a very weak evidential reed and our follow observers as by nature grossly credulous we will come to very different conclusions from those whose fundamental epistemic outlook is more trusting. There is a continuum of attitudes to reports possible here. Let us call those at the end of the continuum, respectively, the Sceptic and the gull. No one is consistently a Sceptic or a gull and I have argued that no o0ne could be consistently a Sceptic. For one thing, as I argued in earlier chapters, a consistent scepticism about testimony would deprive one of any content to the contrasting notion of experience—what Locke called ‘common observation in like cases.’ In the context of Bayesian considerations it is tantamount to depriving oneself of any secure grasp on the background facts,” ibid. 192-93.

Continuing with Curry:

“Do you agree that if someone claims they have ESP we should initially be skeptical, or should we assume that they probably do have it?”

1. Neither. My worldview is open to the paranormal. Therefore, I don’t come to the claim with any prior expectation one way or the other.

2.Rather, I’d judge it on a case-by-case basis. Is the claimant a credible witness? Is his claim verifiable? E.g. Does it involved privileged information which he would be in no ordinary position to know?

So I’d assess a claim like this based on a (i) general assumption of what’s possible along with a (ii) specific consideration of the particulars in this individual case.

“Very easily. If I claim that I flew around today in a space ship there are many explanations for my claim that do not require that I actually flew in a space ship. Perhaps I’m lying, or I’m delusional, or someone put me up to it. These are all contradictory, but as we come up with more and more alternative explanations acceptance of my claim becomes less and less rational. That’s the way Bayes’ Theorem works.”

1.This is deceptively simple. It has nothing to do with BT, per se. Rather, the irrationality of the claim and superior rationality of the alternative is a conjuring trick generated by the choice of one’s illustration.

Curry has done nothing whatsoever to prove that this claim is irrational. Rather, he has deliberately selected an example that he regards as ridiculous, and which he assumes his reader will regard as ridiculous as well. Given their shared assumption, then many alternative explanations are more plausible than the original claim.

So he’s assuming what he needs to prove. BT doesn’t show that ufology is improbable.

If he were to get into an argument with a savvy ufologist, an appeal to BT would beg the question of prior probabilities.

To make his case, Curry would have to bracket BT, and establish the prior improbability of alien abductions by raising a number of direct objections to ufology.

Only then would he be entitled to assign a prior probability value to the claim, and generate a posterior probability.

2.Curry is also glossing over a number of internal difficulties with BT. To take a few examples:

***QUOTE***

The assumption of logical omniscience.

The assumption that degrees of belief satisfy the probability laws implies omniscience about deductive logic, because the probability laws require that all deductive logical truths have probability one, all deductive inconsistencies have probability zero, and the probability of any conjunction of sentences be no greater than any of its deductive consequences. This seems to be an unrealistic standard for human beings. Hacking and Garber have made proposals to relax the assumption of logical omniscience. Because relaxing that assumption would block the derivation of almost all the important results in Bayesian epistemology, most Bayesians maintain the assumption of logical omniscience and treat it as an ideal to which human beings can only more or less approximate.

The problem of the priors.

Are there constraints on prior probabilities other than the probability laws? Consider Goodman's "new riddle of induction": In the past all observed emeralds have been green. Do those observations provide any more support for the generalization that all emeralds are green than they do for the generalization that all emeralds are grue (green if observed before now; blue if observed later); or do they provide any more support for the prediction that the next emerald observed will be green than for the prediction that the next emerald observed will be grue (i.e., blue)? This question divides Bayesians into two categories:

(a) Objective Bayesians (e.g., Rosenkrantz) hold that there are rational constraints on prior probabilities that require that observations support the green-generalization and the green-prediction much more strongly than the grue-generalization and the grue-prediction. Objective Bayesians are the intellectual heirs of the advocates of a Principle of Indifference for probability. Rosenkrantz builds his account on the maximum entropy rule proposed by E.T. Jaynes. The difficulties in formulating an acceptable Principle of Indifference have led most Bayesians to abandon Objective Bayesianism.

(b) Subjective Bayesians (e.g., de Finetti) do not believe that rationality alone places enough constraints on one's prior probabilities to make them objective. For Subjective Bayesians, it is up to our own free choice or to evolution or to socialization or some other non-rational process to determine one's prior probabilities. Rationality only requires that the prior probabilities satisfy relatively modest synchronic coherence conditions.

Subjective Bayesians believe that their position is not objectionably subjective, because of results (e.g., Doob or Gaifman and Snir) proving that even subjects beginning with very different prior probabilities will tend to converge in their final probabilities, given a suitably long series of shared observations. These convergence results are not completely reassuring, however, because they only apply to agents who already have significant agreement in their priors and they do not assure convergence in any reasonable amount of time. Also, they typically only guarantee convergence on the probability of predictions, not on the probability of theoretical hypotheses. For example, Carnap favored prior probabilities that would never raise above zero the probability of a generalization over a potentially infinite number of instances (e.g., that all crows are black), no matter how many observations of positive instances (e.g., black crows) one might make without finding any negative instances (i.e., non-black crows). In addition, the convergence results depend on the assumption that the only changes in probabilities that occur are those that are the non-inferential results of observation on evidential statements and those that result from conditionalization on such evidential statements.

Objective Bayesianism and Subjective Bayesianism are two opposite extremes. There is plenty of room for a compromise position that there are further rationality constraints on prior probabilities that can be added to the Bayesian framework, without supposing that the additional constraints will determine a uniquely rational prior. For some examples of some additional rationality constraints, see the next section. However, because there is no generally agreed upon solution to the Problem of the Priors, it is an open question whether Bayesian Confirmation Theory has inductive content, or whether it merely translates the framework for rational belief provided by deductive logic into a corresponding framework for rational degrees of belief.

The problem of rigid conditional probabilities.

When one conditionalizes, one applies the initial conditional probabilities to determine final unconditional probabilities. Throughout, the conditional probabilities themselves do not change; they remain rigid. Examples of the Problem of Old Evidence are but one of a variety of cases in which it seems that it can be rational to change one's initial conditional probabilities. Thus, many Bayesians reject the Simple Principle of Conditionalization in favor of a qualified principle, limited to situations in which one does not change one's initial conditional probabilities. There is no generally accepted account of when it is rational to maintain rigid initial conditional probabilities and when it is not.

http://plato.stanford.edu/entries/epistemology-bayesian/

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Continuing with Curry:

“So it becomes more and more likely that I flew in a spaceship if you come up with a variety of plausible alternatives? Generally speaking you would expect the opposite to be the case.”

1.Once again, the persuasiveness of this retort is an artifact of his chosen illustration, and not of BT, as such.

If we don’t believe in alien abductions, then many other explanations are more plausible than the prima facie claim.

What makes them at all plausible or more plausible than something else is entirely relative to the plausibility of the contrasting claim.

2.But there’s another fundamental flaw in his reasoning:

Why should we isolate one explanation out of many, then bundle the remaining explanations and oppose them to that particular explanation?

Take the following theories of the empty tomb:

a) Resurrection
b) Swoon theory
c) Stolen body (by friends or enemies?)
d) Reburial/wrong tomb
e) Mass hallucination
f) Jesus’ identical twin
g) Jesus as space alien

Why should (b)-(g) count against (a)?

By the same logic, (a) & (c-g) count against (b); (a)-(b) & (d-g) count against (c); (a-c) & (e-g) count against (d), and so on.

Each theory is inconsistent with the other theories; each theory has its own burden of proof; each theory has its own advocates.

So why single out one theory (Resurrection), and deploy all of the alternative explanations against that one theory, rather than bundle the competing theories in a rotating one-to-many estimate?

Since each theory is inconsistent with every other theory, considered separately or collectively, it’s quite arbitrary to group one particular set of explanations and limit their explanatory power to an isolated target instead of allowing them to target each other in various combinations.

3.Apropos 1-2, suppose we were apply Curry’s reasoning to a different illustration:

A rich man is murdered. The police indict the wife. She has no alibi. She recently found out that her hubby was having an affair. Traces of his blood were found in her bathroom sink.

Ah, but the defense attorney has a number of alternative explanations:

1.He was murdered by a business rival. The evidence was planted to frame the wife and thereby divert attention away from the real killer.

2.He was murdered by vampires.

3.He was murdered by space aliens.

4.He was murdered by the Feds because he discovered the truth about Roswell.

5.He was murdered by members of the International Jewish Conspiracy.

6.He was murdered by members of Opus Dei.

7.He was murdered by members of the Illuminati.

8. He was murdered by Rosicrucians.

9.He was murdered by house-burglars, who washed their hands in the sink.

10.He was murdered by O.J.

11.He was murdered by jihadis.

12.He was murdered by…(fill in the blank).

Does the multiplication of imaginative scenarios really count as cumulative exculpatory evidence?