When to Bayes

Richard Swinburne

The Following essay review of Richard Swinburne’s The Resurrection of God Incarnate appreared originally in Ars Disputandi (Utrecht) and is reprinted here without editorial changes.

The Resurrection of God Incarnate
By Richard Swinburne (Oxford: Oxford University Press, 2003; 232 pp.; hb. £ 45.00, pb. £ 16.99; ISBN: 0-19-925745-0/0-19-925746-9.)

Reviewed by Andrew Wohlgemuth
University of Maine, USA

1 Introduction
[1] Swinburne states, ‘New Testament scholars sometimes boast that they inquire into their subject matter without introducing any theological claims. If they really do this, I can only regard this as a sign of deep irrationality on their part. It is highly irrational to reach some conclusion without taking into account 95 per cent of the relevant evidence…But of course they couldn’t really do this if they are to reach conclusions about whether the Resurrection occurred…For you couldn’t decide whether the detailed historical evidence was strong enough to show that such an event as the Resurrection occurred without having a view whether there was prior reason for supposing that such an event could or could not occur. What tends to happen is that background theological considerations—whether for or against the Resurrection—play an unacknowledged role in determining whether the evidence is strong enough. These considerations need to be put on the table if the evidence is to be weighed properly.’ (p.3) Swinburne’s book has this ambitious and worthy aim.

[2] The book is in three parts, and has an appendix in which he uses the probability calculus to formalize his arguments. He concludes that the probability of Jesus being God Incarnate and being raised from the dead is very high. His assignment of what he feels to be conservative probabilities to the relevant data leads, via the probability calculus, to a probability of 97% that God Incarnate in the person of Jesus was raised from the dead.

2 The Probability Calculus
[3] Swinburne describes three types of probability: physical probability, statistical probability, and inductive (or logical) probability. (Some people identify physical probability with statistical probability.) Statistical probability is the most widely known. It rests on events—technically, subsets of a probability space. A typical probability space might be the set of all possible outcomes in some game of chance. Actuarial science and the physical sciences make use of statistical probability—which has been well developed mathematically. Probabilities in statistical probability can be assigned with precision.

[4] Logical probability is an extension of the propositional and predicate calculus—the formal logical structure of mathematical argument itself. It was developed by J.M. Keynes (A Treatise on Probability, MacMillan, 1921), because in many real-life situations one proposition, say q, does not follow another, say p, with the complete certainty of ‘p implies q’ in a mathematical argument. Instead, we might only be able to say that we are fairly sure that q would follow, if we knew p. Thus, a probability, a number from 0 to 1, might be assigned to the expectation that q would be true, if we knew that p was true. This probability is denoted by P(q/p) (the ‘probability of q given p’). Thus p implies q in the logical, or mathematical, sense provided that P(q/p) = 1.

[5] I think we all feel that it is reasonable and meaningful to ask if something is likely to happen, or likely has happened. To give an easy example, consider the forecast that the chance of rain today is 80%. We base this on experience. The forecasters notice that it actually did rain on 80% of the days that had the same early-morning conditions as today. This is an example of statistical probability. The underlying probability space is the set of days with the same initial conditions as today. The event we’re concerned with is rain.

[6] Suppose, however, that our neighbor Tom is accused of knocking his wife unconscious while in a rage. Although there may be no way to form a meaningful probability space here, we can nevertheless feel strongly that Tom is likely to have done it—or very unlikely. We do this by considerations that run deeper than the merely statistical. Of course, if Tom habitually knocks people about while in a rage, then we may not need to go any deeper than the statistical. But if the accusation is unexpected and unique, then we begin to rely on things such as Tom’s character, as it is known to us, in order to support our feelings of the likelihood of his having done the deed.

[7] This is what Swinburne is doing in his book. He is asking whether God is likely to have done certain things, and he is adding that in with the smaller world of history. Christians, of course, do believe that some things can be known about the character of God. I’ll look first at the formal treatment in the appendix, and then go to the material in Chapter 1.

[8] Swinburne lists 5 axioms of the probability calculus. (The axioms of the predicate calculus are implicitly also needed.) Axiom 4, which will play a prominent role, follows.

(4) P(p&q/r) = P(p/q&r)P(q/r)

[9] Substituting h, e, and k (letters Swinburne will use later) for p, q, and r gives

P(h&e/k) = P(h/e&k)P(e/k)

[10] Dividing both sides by P(e/k) gives

P(h&e/k)∕P(e/k) = P(h/e&k)

[11] Since h&e is logically equivalent to e&h, we can substitute

P(e&h/k)∕P(e/k) = P(h/e&k)

[12] Now by Axiom 4, P(e&h/k) = P(e/h&k)P(h/k), so we can substitute for P(e&h/k) to get

P(e/h&k)P(h/k)∕P(e/k) = P(h/e&k)

[13] Interchanging left and right sides of the equation gives

(4′) P(h/e&k) = P(e/h&k)P(h/k)∕P(e/k)

[14] Swinburne states, ‘Among the theorems that follow from the axioms is a crucial theorem known as Bayes’s Theorem. I express it using letters ‘e’, ‘h’, and ‘k’ which can represent any propositions at all; but we shall be concerned with it for the case where e represents observed evidence (data), k represents ‘background evidence’, and h is a hypothesis under investigation’ (p. 206) Equation 4′ above is Bayes’s Theorem as Swinburne expresses it. I have derived it to show that it follows from the axioms by the two simple algebraic operations of substitution and dividing both sides of an equation by the same thing. It is customary, when talking about formal languages (like the propositional, predicate, and probability calculus) to refer to anything that follows from the axioms as a ‘theorem’. In other mathematical branches with which the reader may be more familiar (like geometry or calculus, for example), the use of the word ‘theorem’ is reserved for deeper results. The foregoing should take away any mystery from the use of ‘Bayes’s Theorem’. It is really just a rephrasing of an axiom.

[15] As to the axioms, Swinburne states, ‘It is very easy to see intuitively the correctness of these axioms.’ (p. 206) At which point he explains them in words. When he gets to axiom 4 however, he appeals to successive tosses of a coin—which doesn’t model the situation accurately. We don’t know what p, q, and r are. In order to see why axiom 4 is true, we can relate the logical probabilities to conditional (statistical) probabilities. Thus let p, q, and r be events with probabilities P(p), P(q), and P(r). Let p be the proposition ‘p occurs’, and similarly for q and r. The conditional probability P(a/b) (the ‘probability of ‘a’ given ‘b’)’ for events a and b is defined to be P(a&b)∕P(b). In this case

Axiom (4) P(p&q/r) = P(p/q&r)P(q/r)

[16] in terms of conditional probabilities is

P(p&q/r) = P(p/q&r)P(q/r)

[17] which by definition is

P(p&q&r)∕P(r) = [P(p&q&r)∕P(q&r)][P(q&r)∕P(r)]

[18] which is an identity, since the factors P(q&r) cancel.

[19] It should be noted here that while any conditional probabilities (of statistical probability) can be seen as propositions of logical probability (as we have done), the reverse is not so—simply because there may not be any well-defined probability space. It is crucial for the case Swinburne makes that meaningful probabilities can be assigned to the factors on the right-hand side of equation 4′. Once that is granted, the probability on the left side must be accepted as calculated. I have shown the ‘intuitive correctness’ of axiom 4, since it follows from definition in the realm of statistical probability, which can be viewed as a restricted case of logical probability—the case in which we would find illustrative examples.

[20] Specifying the factors in equation 4′, Swinburne states, ‘Let k now be…the evidence of natural theology (including the sinning and suffering of humans). Let e be the detailed historical evidence, consisting of a conjunction of three pieces of evidence (e1 &e2 &e3 ). e1 is the evidence of the life of Jesus set out in Part II. e2 is the detailed historical evidence relating to the Resurrection set out in Part III. e3 is the evidence (summarized in Chapter 3) that neither the prior nor the posterior requirements for God being incarnate were satisfied in any prophet in human history in any way comparable with the way in which they were satisfied in Jesus.’ (p. 210) ‘Let h1 be the hypothesis that God became incarnate in Jesus, and h2 the hypothesis that Jesus rose from the dead. h is the conjunction (h1 &h2 ). Now at the end of the day this book is interested in P(h∕e&k)—the probability that Jesus was God Incarnate who rose from the dead (h), on the evidence both of natural theology (k) and of the detailed history of Jesus and of other human prophets (e).’ (p. 211)

[21] Assigning probabilities to the factors of equation 4′ is done by building up from other factors: ‘Let us represent by t theism, the claim that there is a God of the traditional kind. P(t/k) is the probability that there is such a God on the evidence of natural theology. I suggested in Chapter 1 that we give this the modest value 1∕2.’ (p. 211) Swinburne backs up this value only in the last paragraph of Chapter 1: ‘This evidence, the evidence of natural theology, provides general background evidence crucially relevant to our topic. I have argued elsewhere the case for this evidence giving substantial probability to the existence of God. (See esp. my The Existence of God and the shorter Is there a God? (Oxford University Press, 1996)). I cannot, for reasons of space, argue that case again here. But to get my argument going here, I will make only the moderate assumption that the evidence…makes it as probable as not that there is a God…’ (p. 30) I’ll return to more in Chapter 1, Principles for Weighing Evidence, after another illustration of assigning probabilities.

[22] ‘Then let us represent by c the claim that God became incarnate among humans at some time with a divided [’…he could act and react in his human life in partial ignorance of, and with only partial access to his divine powers.’ (p. 52)] incarnation, a more precise form of the way described by the Council of Chalcedon…and set out in Chapter 2. I suggested there that if there is a God (and there are humans who sin and suffer), it is quite probable that he would become incarnate…I suggested that it was ‘as probable as not’ that he would do this, and so in numerical terms the probability of his doing it is 1/2. The probability of 1/2 is clearly unaffected if we add to [should read ‘t’] all the data of natural theology, and so P(c/t&k) = 1∕2.’ (p. 211) Since P(c/k) = P(c&t/k) = P(c/t&k)P(t/k) by Axiom 4 and the logical equivalence of c and c&t, P(c/k) = 1∕4.

3 The Grand Philosophical Principle
[23] The two paragraphs above suffice to illustrate the completely subjective nature of assigning probabilities to the factors involved in the calculations. I don’t mean to imply that being subjective is necessarily bad, although I would not want to be involved personally with arguing the case for certain subjective probabilities. In the main body of the book, there are arguments for why Swinburne believes these probabilities to be reasonable—even conservative.

[24] The most problematical assertion in Chapter 1 is the following: ‘It is a further fundamental epistemological principle additional to the principle that other things being equal we should trust our memories, that we should believe what others tell us that they have done or perceived—in the absence of counter-evidence. I call this the principle of testimony. It must be extended so as to require us to believe that—in the absence of counter-evidence—when someone tell us that so-and-so is the case…they have perceived or received testimony from others that it is the case. Without this principle we would have very little knowledge of the world.’ (p. 13) There is no doubt that we get almost all of our information about the world in this way—but we also get a very good amount of misinformation too. For example, in a letter to John Norvell in 1807, Thomas Jefferson wrote, ‘Nothing can now be believed which is seen in a newspaper. Truth itself becomes suspicious by being put into that polluted vehicle. The real extent of this state of misinformation is known only to those who are in situations to confront facts within their knowledge with the lies of the day.’ If we change the word ‘lies’ to the word ‘fancies,’ we get a fair account of my own experience. I also have a great skepticism of grand philosophical principles that are used to draw inferences in special cases in arguments. If the special cases are not seen to be true themselves, how can the generalization be seen to be true?

[25] Swinburne would be on much sounder ground to take, as his ‘principle of testimony’, something in his next paragraph: ‘Testimony by more than one witness to the occurrence of the same event makes it very probable indeed that that to which they testify is true—to the extent to which it is probable that they are independent witnesses.’ (p. 13)

[26] A discussion of the probability of a miracle must, I suppose, bring up David Hume. Swinburne says, ‘Hume’s discussion suffers from one minor deficiency, one medium-sized deficiency, and one major one.’ (p.24)…‘But Hume’s worst mistake was to suppose that the only relevant background theory to be established from wider evidence was a scientific theory about what are the laws of nature. But any theory showing whether laws of nature are ultimate or whether they depend on something higher for their operation is crucially relevant. If there is no God, then the laws of nature are the ultimate determinants of what happens. But if there is a God, then whether and for how long and under what circumstances laws of nature operate depends on God. And evidence that there is a God, and in particular evidence that there is a God of a kind who might be expected to intervene occasionally in the natural order, will be evidence leading us to expect occasional violations of laws of nature.’ (p. 25)

4 Proof by Lack of Imagination
[27] Since I faulted Swinburne on using grand generalizations in a logical argument, I feel the need to fault Hume on the same account. Hume states, ‘It being a general maxim, that no objects have any discoverable connexion together, and that all the inferences, which we can draw from one to another are founded merely on our experience of their constant and regular conjunction; it is evident, that we ought not to make an exception to this maxim in favor of human testimony, whose connexion with any event seems, in itself, as little necessary as any other.’ (An Inquiry Concerning Human Understanding, edition by The Liberal Arts Press, 1955, p.119) This ‘maxim’ is Hume’s own grand philosophical principle. It elevates mere correlation, and pronounces the discovery of causation as hopeless. The most obvious counter-example is modern medical science, where correlation most often prompts the question—to which the discovery of causation constitutes the answer. One may not think it fair to fault Hume for not being familiar with modern medical science, but that gets us to an important point. Hume’s assertion that ‘all the inferences, which we can draw from one (object) to another are founded merely on our experience of their constant and regular conjunction’ is a mere proof by lack of imagination—which, in general, would run something like this: ‘I can imagine it being like this. I can’t imagine it being any other way. Therefore, it must be like this.’ Logical possibilities cannot be ruled out simply because they do not present themselves to even the best human imagination. For a statement or argument to be truly logical, it must exclude the possibility of a counter-example. That’s what makes it logical (instead of empirical). If a counter-example is ever found, it shows that the statement or argument was not logical in the first place.

[28] There is no doubt but that Hume intended his maxim to be part of a logical argument. He begins, ‘I flatter myself, that I have discovered an argument of a like nature, which, if just, will, with the wise and learned, be an everlasting check to all kinds of superstitious delusion, and consequently, will be useful as long as the world endures. For so long, I presume, will the accounts of miracles and prodigies be found in all history, sacred and profane.’ (ibid. p. 118) And concludes, ‘The plain consequence is (and it is a general maxim worthy of our attention), ’That no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous…’ (ibid. p. 123) And right in the middle of this argument is his maxim—which appears ridiculous to scientific eyes.

[29] It is interesting, to me, that Hume thinks his argument will be effective with the ‘wise and learned’. When looking through Swinburne’s references and related material, I noted numerous statements of Hume’s brilliance. Hume ponders his own ‘genius’, and is concerned with the ‘admiration of mankind’. (ibid. p. xi) I am uncomfortable in a field where people feel it appropriate to attest to the brilliance of anyone. It smacks of whistling in the dark—and I suspect the praise is lavished on those with a philosophy close to one’s own.

[30] Except for one place in which his ‘principle of testimony’ creeps in, Swinburne’s five-and-one-half page introduction states his case well. At the end of the introduction, he states, ‘Although there are, I believe, a number of original detailed historical arguments in this book, its main task is to put arguments developed by others into a wider frame so as to form an overall picture.’ (p. 6) In the body of the book, he addresses the program of the introduction, and motivates the assignment of probabilities assigned in the appendix.

[31] Swinburne’s main thesis, that one should make decisions about the likelihood of things only in the broadest context available, is very well taken. For example, consider suffering. Swinburne says, ‘I argued in The Existence of God that it is “more probable than not” that there is a God. However, my subsequent more satisfactory argument in Providence and the Problem of Evil to show that suffering does not count against the existence of God relied in part on the supposition that God would become incarnate to share our suffering and to make atonement for our sins.’ (p. 31 note)

[32] Suffering has been felt to be inconsistent with an omnipotent, good, and omniscient God. The only way I can see to reconcile these is to observe that the evidence is not all in yet—except in one case. Who could say that anyone suffered more than Jesus—with sweating blood (hemathidrosis) in Gethsemane, even before the physical abuse began. Yet who would want to say that Jesus himself would be better off, now, without the suffering. Jesus is the only one of us for whom we have enough information to decide that ‘suffering does not count against the existence of God’. And St. Paul says, ‘Christ has been raised from the dead, as the first-fruits of all who have fallen asleep.’ (1 Cor. 15:20) And, ‘Just as all die in Adam, so in Christ all will be brought to life; but all of them in their proper order: Christ the first-fruits, and next, at his coming, those who belong to him.’ (1 Cor. 15:22,3) And St. James says that ‘we should be a sort of frirst-fruits of all his creation.’ (James 1:18, italics mine, of course) So where the results are in, we see that God is justified, and we have promises that when all the results are in, God will be justified.

2 thoughts on “When to Bayes

  1. Karl Popper (1990) has recounted that when he learned that Carnap treated the probability of a hypothesis as a mathematical probability, he “felt as a father must feel whose son has joined the Moonies” (p. 5). I share the worries of those statisticians, philosophers, and psychologists who caution that the laws of probability do no apply to all kinds of statements about singular events, but apply only in well-defined circumstances (Gigerenzer at al,. 1989).

    Gigerenzer, G. (1996). On narrow norms and vague heuristics: A reply to Kahneman and Tversky. Psychological Review, 103, 592-596. [pdf]

    • Well, this really should have been submitted as a comment for the prior post (What to Prove?), rather than here. My apologies.

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