## Friday, August 5, 2011

### An unconfirmability argument

[A caution: This post is long, verbose, and overly technical. Read the dialogue I posted at the end. If it is comprehensible to you, you should have an adequate grasp of the contents of this and the previous post.]

In my last post, I interpreted Hume's argument against the confirmability of miracles and found it to be unsound. Now I want to affirmatively answer another question: given that Hume's argument is unconvincing, is there another unconfirmability argument which applies to paradigm miracles like the Resurrection?

If so, we are not doomed to Earman's (apparent) conclusion, that we must analyze the details of every miracle claim if we are to safely reject miracles (p.3). Rather, we can specify categories of claims, narrower than miracle', that cannot be confirmed by certain categories of evidence. Though we might not safely assert something like "no evidence you have should convince me of a miracle," we may be able to say something like "by itself, the evidence you present is by nature incapable of overcoming the prior probability of the type of miracle you assert." I will leave open the possibility that miracles like the Resurrection (R) can be confirmed; I am only closing a class of potential means of doing so. I do not think that I am providing or can provide "an everlasting check to all kinds of superstitious delusion" which "will be useful as long as the world endures" (EPHU, p.169); rather, I am giving something which, if successful, would obviate any non-pedagogical, rational need for any detailed Bayesian analysis of the Resurrection like this one, so long as we lack other significant arguments in favor of Christianity. One notices the number of qualifiers required to invest in such an argument, and there will be more. This form of argument does not constitute a license for ignorance, and it will require creative adaption to specific miracle claims.

A lot hinges on the problem of determining appropriate priors in subjective Bayesianism. In order to provide some convincing estimate of the prior odds on R, a trick - I think novel to me - must be applied.

To motivate this trick, I'll cite its precedent and inspiration: the method of reparation, originally due to Richard Jeffrey. This tool was invented for a specific problem, the problem of old evidence. Roughly, the problem is as follows: whenever a new theory is crafted, its ability to explain known phenomenon is considered to be of epistemic significance. For example, relativity theory's ability to predict' the perihelion precession of Mercury - a phenomenon which had long defied explanation in classical physics - is considered powerful evidence for that theory. But the hypothetico-deductive principle, which states that verification of an uncertain prediction of an uncertain hypothesis increases the probability of that hypothesis, is incapable of yielding this well-founded intuition; the prediction' is already known, i.e. not uncertain.

Reparation solves this problem by positing a hypothetical prior probability, an ur-distribution or ur-prior, in which the theoretical prediction itself in addition to the corresponding observation is treated as uncertain. The discovery' of that implication and the evidence then raises the probability of the hypothesis in question in a straightforward way. I want to extend this to a problem which plagues estimating the prior probability of miracles, i.e. the problem of old evidence and explanation, or if you prefer, the old everything problem. When priors are not in dispute, this method is unnecessary; this is not an actual case of confirmation. But on the very safe assumption that the prior probability of R, in the absence of other argument, is calibrated with respect to confirmation of natural principles, it is quite useful. We know' that the prior ratio is small, but we do not know how small. This is important when disputes depend on whether or not that prior is greater than 1/1000 or less than 10-44. We need something better than the vague intuitions' rightly deplored by the McGrews (CCRJ, p.50).

The idea is to posit an ur-prior u which is (partially) devoid of current background knowledge. The voided background knowledge is treated as uncertain in u, the recapturing of which yields through conditionalization a suitable prior p. In this context, the recaptured' knowledge is the bulk of the uniform evidence of sense' discussed by Hume, which is assumed to be wholly confirmatory of the law L. As I am working with the Resurrection, L is the principle that dead people remain dead. By definition and subset rule, we have for any probability pr that

$\{L\longrightarrow\sim R,\ R\longrightarrow\sim L\}\implies \{pr(L)\leq pr(\sim R),pr(R)\leq pr(\sim L)\}$.

By quick algebra, we yield the following inequality:

$\frac{pr(R)}{pr(\sim R)}\leq \frac{pr(\sim L)}{pr(L)}$.

So by calculating our confidence in L, we set a maximum confidence in R.

As I mentioned, we want to recapture' p using u by conditioning, that is,

$\frac{p(R)}{p(\sim R)}\leq \frac{p(\sim L)}{p(L)}=\frac{u(O|\sim L)}{u(O|L)}\times \frac{u(\sim L)}{u(L)}=u(O|\sim L)\times \frac{u(\sim L)}{u(L)}$,

where O is the set of observations confirming the putative law. But here I seem to have merely shifted the problem elsewhere, since we now need the u-odds on ~L. This regress is halted by the following abstract consideration: at some theoretical point of time, a hypothetical rational agent should have crossed the more likely than not' threshold in favor of the law. For this reason, we may assume that the u-odds on ~L is 1. From there we must estimate the raw confirmatory force of the remaining experience as captured by the u-Bayes factor. Cleaning up the previous mess, we want to analyze

$\frac{p(R)}{p(\sim R)}\leq u(O|\sim L)$.

Now here is the crucial question: what does ~L look like after this partial recapitulation of background knowledge? An anti-law' which states that all dead people resurrect will have probability zero, but this is an extreme; the preexisting information will select for those non-laws which ascribe higher probability to individual non-resurrections. These include, amongst other things, statements like "everybody who dies remains dead except under very rare conditions." However, even this will be made less probable by O so long as the further prediction of its elements are uncertain. Unless one claims to be able to specify in advance who should be exempted from death based on limited confirmation, the effect remains strong as regards statements like "everybody stays dead except for Jesus." (Lazarus and other proposed resurrections are not part of the background knowledge being recaptured here.)

To go further, I assume that O contains N instances of confirmation beyond those which lead our hypothetical agent to equivocate, each of which are of equal weight1, i.e.

$O=\{O_1,...,O_N\}\text{ and }u(O_i|\sim L)=u(O_j|\sim L)\forall i,j\in[N]$

where I use the combinatorial notation [N]={1,...,N} for the sake of cleanliness. Supposing that each element of O is independent modulo ~O, we have that

$u(O|\sim L)=u(O_1|\sim L)^N$.

So if N=106, even a quite large value for u(O1|~L) yields an absurdly small upper bound on the prior odds on R. If this assumption of independence is to fail in favor of the theists, there must be in the domain of u some significant set of propositions in ~L that correlate the elements of O. In other words, there must be plausible causal processes not implying L which probabilistically tend to keep dead people dead.

And here is where the remainder of our scientific knowledge and experience come into play. We do not consider L to be a fundamental feature of the universe but a consequence of other properties. Were a technology developed that allowed for the reanimation of corpses, we need not say that a putative law of nature had been overturned; instead, we would say that the ordinary course' had been altered by the introduction of a new yet naturalistic element. Without such technology, we expect other principles to produce L. These include thermodynamic principles. When Jesus died, there are three sorts of miraculous possibilities: either his vital tissues never fatally decayed by supernatural sustenance, his vital tissues decayed but were reconstituted gradually or suddenly prior to his reanimation, or he reanimated without the function of his vital organs. Each of these possibilities runs against other established regularities. Properly speaking, then, L is not merely the generalized rule that dead people remain dead but also the set of combinations of rules which produce that outcome. For the Resurrection to occur, every one of those combinations must fail to hold.

So a theory which advantageously counters the above independence assumption must probabilistically correlate the elements of O, have significant plausibility in its own right, and fail to assume or imply any element in L, be it the putative law itself or any combination of principles which produce that law. In this case and in the case of other miracles which violate mass/energy conservation and/or thermodynamic principles, we must toss out modern science before examining any evidence proposed against its principles. This is an absurd task, but that absurdity is consequent to proposing a suspension of the natural order.

In such a manner I tentatively maintain that the independent assumption as employed is valid in establishing an estimated upper bound on the prior odds on R. So long as we agree that it is disgustingly low, a precise value is unimportant for the strengthening of the argument. Since as you may have noticed this method of unconfirmability argument requires some detailed information concerning the relevant miracle, specifying any particular value is unimportant. For an approximate value's import to materialize, we must also delve into the category of evidence in play.

Hume appears to recognize the need for categorical bounds on the strength of evidence in his essay:
When any one tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that the fact, which he relates, should really have happened. I weigh the one miracle against the other; and according to the superiority, which I discover, I pronounce my decision, and always reject the greater miracle. If the falsehood of his testimony would be more miraculous, than the event which he relates; then, and not till then, can he pretend to command my belief or opinion. (EPHU, p.174)
I think that whenever Hume attempts to place such a bound on testimonial evidence', he overreaches by virtue of that category's broadness. "I would not believe such a story were it told to me by Cato" (p.172) does not a sufficiently general argument make, nor are vague allusions to particular failures of human reporting which remain in the background, as these do not necessarily apply to all claims that humans make. Instead, it is better to focus on more task-specific categories. It might be possible to do so, but there is no reason to argue for more than is required.

Now I must specify what I mean by a categorical bound in formal terms: roughly, a categorical bound B is a real number such whenever evidence E for an uncertain hypothesis H lies in a set C, any Bayes factor produced by E is less than or equal to B.2 That is,

$E\in C\implies \frac{p(E|H)}{p(E|\sim H)}\leq B$.

If H is a miracle claim violating a putative law L supported by recaptured observations O, then tautologically the argument is successful if

$B\leq\left[\frac{p(\sim L)}{p(L)}\right]^{-1}=\frac{p(L)}{p(\sim L)}=\frac{u(L|O)}{u(\sim L|O)}.$.

What remains is to show that the evidence proposed by the miracle-claimant lies in such a set C.

What would C look like? If we want to be trivial, we can use uncontroversial examples. For example, two gamblers who have vastly differing prior odds on the fairness of a coin should not be able to resolve their dispute by tossing the coin a small number of times to check the outcomes. With precise values available, one could derive the minimum number of tosses which could possibly result in agreement.

One may extend this notion by thinking of the epistemic limitations on experiments like "tossing a coin a certain number times" as instances of methodological bounds. In the medical sciences, anecdotal and testimonial evidence as applied to broad categories of claims is almost completely useless for confirmation; careful study is required. What this reflects is a well-founded confidence in systematic errors frequently present in anecdotal reports which cannot be sufficiently discounted in the absence of controlled study. Anecdotal evidence for a novel, popular medical hypothesis cannot by nature discount those established theories and credences, a task which is necessary in order to confirm the new remedy against a significant prior implausibility.

Similar considerations apply to many classes of potential miracles and the evidence most commonly presented for them. With the Resurrection, we need not attempt the task Hume appears to set for himself in arguing that any set of witness testimony is incapable of producing convincingly calibrated Bayes factors above something like 101000. In this case, we may instead suggest that the methods of historians in establishing the historicity of those reports and their background, while possibly very good, are not so good as to allow a confident assertion that the historical record produces a Bayes factor above 101000. Myth and utter fabrication, even if wildly improbable in this case, have precedents, and I do not think they are capable of being discarded with absolute or nearly absolute confidence given the difficulties in method.

Hoping my proposals have been convincing thus far, I conclude by giving the sort of conversation which two Bayesians could have.

Christian: Ah, skeptic! Just the person I was wanting to see! I have crafted a convincing case that the Resurrection did in fact occur, and I was wanting your feedback.

Skeptic: That sounds very interesting, but before we go into the details, would you consider the Resurrection, had it occurred, to have constituted a suspension of the natural order?

Christian: Certainly, as you well know. Otherwise, the Resurrection would be meaningless. If Jesus' Resurrection were, say, a mere product of absurdly unlikely but possible quantum fluctuations, then any argument to theism or Christianity from that event would be undermined. It would be an isolated physical anomaly; nothing more, nothing less, and surely not a communicative sign of divine endorsement of the validity of Jesus' teachings.

Skeptic: I'm glad that you and I agree. And within this possibly suspended natural order, would you admit that local, epistemic generalizations hold and that tremendous confidence in those generalizations is yielded by what Hume would call uniform sense data'?

Christian: While I obviously do not accept Hume's argument, I agree that incredulity prior to the examination of the evidence is wholly reasonable. That is why I have analyzed the evidence; one cannot say prior to analysis whether or not the evidence is sufficient or insufficient. You and I both know that this is merely a matter of probability.

Skeptic: I do not accept Hume's argument either, but one may with certain conditions be able to obviate the need to examine all of the details in advance by grounding a posteriori bounds on Bayes factors produced by certain types of evidence.

Christian: Part of this worries me, as it sounds like an excuse to avoid examining the evidence, which, as a good skeptic and Bayesian, you should be interested in doing. In any case, I suspect that such an approach would undermine important areas of scientific research, were it to be accepted.

Skeptic: I admit that this idea is a time-saver and has its ideological attractions, but allow me to specify some of those conditions. Hopefully, when you are satisfied with their stringency, your worries will vanish. But before I may do so, I nevertheless must know the nature of the argument that you are proposing. As I said before, the bounds to which I refer would be a posteriori, not some analytic consequence of Kolmogorov's axioms, uncontroversial metaphysical theses, or sound subjective Bayesian principles. I could not pretend to Hume's rhetoric and claim "an everlasting check to all kinds of superstitious delusion," or claim to have silenced any potential reasoned argument on your part or the part of your comrades in arms, be they future comrades or present confederates. In order to state exactly what I can say in advance of detailed analysis, I have to consider at least some of the details.

Christian: That at least sounds more interesting than another platitudinous regurgitation of Hume's breathless meanderings. Fine, I will play along. I am arguing, as against many prominent skeptics, to and from the historicity of the texts with respect to several key facts, especially those facts concerning the secular claims of witness testimony.

Skeptic: Have you accorded these facts certainty in your analysis as opposed to a more general analysis, for example using Jeffrey conditioning or classical conditioning on a partition of the historical possibilities?

Christian: For the facts concerning the witnesses, I strengthened the relevant arguments so as to make those facts not only plausible, but so overwhelmingly likely as to ensure that errors of omission do not seriously undermine the strength of the argument.

Skeptic: How overwhelmingly likely?

Christian: I think that I see roughly where you are heading with this. By your earlier hintings, it is clear that you are relying on some estimate of the prior odds of the Resurrection. Riddle me this: How do you propose to estimate prior odds on the Resurrection in any convincing way? You and I are both critics of equivocation and objective Bayesianism. You and I both acknowledge the limitations of current theories of calibration, especially as applied to claims like the Resurrection.

Skeptic: Properly speaking, I do no such thing.

Christian: Help me out here.

Skeptic: I rely on the notion that miracles, to occur, require a suspension of the natural order. As you have probably anticipated, I rely on the epistemic status of that natural order with respect to any potential exemption to gauge a suitable prior on the Resurrection...

Christian: Sorry to interrupt, but clearly you seem to be contradicting yourself.

Skeptic: Only if you assume that I need a specific range of prior odds. Instead, I use the deductive implications of putative laws to straightforwardly derive inequalities via the subset rule which by basic algebra translate into an upper bound on the prior odds of the Resurrection. I only need inequalities and bounds, not specific, well-defined ranges of reasonable discussion.

Christian: Ah, I see. You're assuming that the only relevant calibrating factor is the relation of a potential suspension of the natural order to the epistemic status of the natural order, of course.

Skeptic: That's right; hence why I do not claim that my approach, even if valid, would constitute the end of the discussion. One may still need to engage the evidence, but only if an adequate, well-established natural theology is formulated so as to calibrate the priors differently.

Christian: Which of course would present a serious difficulty, since the primary and standard means of evidentially filtering Christianity out of the more general category of theism is by arguing for the Resurrection. Now I am curious: supposing you could bound the prior odds on the Resurrection below 10-1000, what would you say to me if I claimed to have produced a Bayes factor based on a confidence some salient facts which is greater than 101000?

Skeptic: I would say that you have proposed the a posteriori equivalent of proving the rationality of the square root of two, Euler's number, or pi.

Christian: That's quite a strong statement; how do you mean it?

Skeptic: I might agree that your proposed facts are plausible, even extremely convincing. But I would nevertheless insist that they cannot be sufficiently plausible as to yield such a factor. Formally, I would put evidence like that you have proposed into a set of similar evidences and claim that Bayes factors in favor of the Resurrection produced by an element in that set are below 101000.

Christian: In which case, your argument would be tautological or trivial unless you can convincingly establish, before engaging all of the details, that my textual evidence cannot be stronger. Again, I do not see how you are avoiding the shortcomings of Hume.

Skeptic: Well, you have surely noted my insistence on your specifying the type of evidence in question. I doubt you fail to imagine how that might be relevant.

Christian: I have a rough idea: are you proposing theses, like those of Hume, against testimonial evidence? Just because testimonial evidence is always subject to some precedented, possible counter-thesis, that does not mean that one can say that testimonial evidence as a category must be at least this or that weak by that virtue. The details decide how significant those considerations need to be.

Skeptic: I agree, which is another reason why I do not claim to be vindicating Hume's essay. Testimonial evidence' is perhaps too broad a category to be subject to sufficiently small, convincing categorical boundaries. As I said before, some specifics are required. Allow me to motivate those which apply to the textual record on which you plan to rely: you can envision cases where an experiment, by its nature, cannot overcome discrepancies in prior odds so as to yield agreement between two rational agents, correct?

Christian: In highly idealized scenarios like fair dice rolls or well-understood machines and programs, sure, but I do not see the relevance to a scenario so complex and multivariate as eyewitness testimony.

Skeptic: You may at least be able to anticipate a generalization of simple and uncontroversial lessons to broader notions like historical methodology', correct?

Christian: Not exactly, as I see such a category as too vague to easily bound.

Skeptic: Again, it depends on specifics. What is the method which you used to arrive at your initial, secular factual claims? Presumably, you do not claim to have directly observed the events in question.

Christian: Of course not.

Skeptic: And so there is some significant uncertainty in the indirect inference methods, i.e. historical methods, which you employ?

Christian: At least in a trivial sense, but that need not translate into any boundary.

Skeptic: Actually, it does, unless you claim that there is no minimally significant alternative to your facts which your methods can not diminish to an arbitrary degree. For example, can you rule out as strongly as you like the possibilities of fraud and later myth-making with respect to these secular facts?

Christian: I wouldn't say that, but again, I see no reason why, in advance, I can not devalue such possibilities sufficiently as to overcome the prior implausibility of the Resurrection.

Skeptic: If by in advance' you mean in advance of all background knowledge, surely you are correct, but I mean the reliability of your methods with respect to our current knowledge about its reliability. If that reliability is such that the probabilities of hypotheses like frauds and myths cannot be convincingly grounded below 10-1000, I have established my case. For such extreme values, I would say that this can be said further in advance than I need to argue, but to firmly secure your methods into the category which I require, I will need to know a few more specifics.

Christian: I think I understand now, and I think that I see how you will be able to secure your conclusions were I to spell out more details. I suppose that I will have to qualify my paper with a placeholder for the time being and play with the formalisms to double-check your statements.

Skeptic: That sounds fine. In the meantime, I would be happy to read your paper. After all, you might be able to calibrate the relevant priors differently. It is still worth reading, for this and other reasons, if your conclusions are as strongly supported as you have suggested.

Christian: I look forward to your review. However, I hope you only resume technical blogging after all that wine you just drank leaves your system.

Skeptic: You're breaking the fourth wall.

1. The simplifying assumption of equal-weightedness is not generally true. If alternatives to the law include something like dead people remain dead unless you perform a certain magic ritual', then only failures of that ritual will contrast the hypothesis with L. We can recapture the plausibility of the assumption by stipulating that a theoretical agent at this theoretical threshold point has effectively ruled any particular such hypothesis.

2. I've been playing with this notion for some time, and I know of several generalizations if anyone is interested.