Can Bayes’Theorem, given the evidence of this universe, be used to support theism?

Can Bayes’ Theorem, given the evidence of this

universe, be used to support theism?

John Ostrowick

[email protected]

Abstract

Given ht as the hypothesis of theism, hm as the hypothesis of materialism, and e as the evidence of a complex

life-bearing universe, Swinburne presents these arguments in his "The Existence of God" (2004):

(1) That this ordered universe is a priori improbable (2004, p49, p150, 1991, p304 et seq.), given the stringent

requirements for life (cf. Leslie, 2000, p12), and the Second Law of Thermodynamics (Giancoli, p396);

(2) That this universe’s structure is evidence for theism, and that theism therefore explains this

universe. Swinburne argues that that because P(e| ht) > P(e| hm), it follows that P(ht |e) > P(hm |e).

(3) A theistic explanation for the universe is more probable because it is simpler.

Therefore it is more likely that God exists than not.

As I have addressed (3) in a prior paper, this paper will address the Bayesian argument that Swinburne

offers in (2) — i.e. that P(e| ht) > P(e). I draw a number of conclusions, most pertinently, that Hacking's Total

Probability Rule (TPR), for cases of mutually exclusive hypotheses [ht v hm] and evidence e entails that any h

can only be confirmed if P(e|~h) is low. I also conclude that if we follow TPR for Swinburne's argument, we

achieve the result that theism is at best slightly improbable, or equiprobable with materialism.

John [email protected]

Using Bayes’ Theorem with a Scientific Example

Suppose we start with the intuition that we are indifferent to the theory of relativity, and give it a prior

probability of 0.5.

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

If the probability of the evidence, e, is [1.0], given the hypothesis, h, then the hypothesis entails the evidence.

Hacking, 2008, pp69-70, shows that if P(e|h) = 1.0, in other words, if the hypothesis entails the evidence, then it

follows that the hypothesis is constrained by P(h v ¬h) — the law of excluded middle — that is, h is mutually

exclusive of ¬h. If this is so, then P(h v ¬h) must equal 1.0, irrespective of the actual value of P(e|h). It is a

general rule of the probability calculus that a disjunction of all the mutually exclusive possibilities has a

probability of 1.0 (Hacking, p59)

Hence


Think about it: If something is the case; e.g. the universe exists, then it is certain it exists (1.0). So if a range of possibilities predict this universe (ie if one of them is right), then again, that range of probabilities must add up to 1. Hence, if we consider the probabilities as exclusive, then that entire series of disjuncts, for say h1, h2, h3... must be a sum, or logical OR function, such that h1 v h2 v h3 = 1 or h1+h2+h3 = 1


(1) P(h) = P(h) x 1.0 [anything x 1.0 equals itself]

(2) P(e|h) = 1.0 [Hacking, 2008, pp69-70 and since e exists]

(3) P(h) = P(h) x P(e|h) [substituting (2) into (1)]

(4) P(¬h) = P(¬h) x P(e|¬h) [since P(h v ¬h) = 1.0, Hacking, 2008, p59]

Thus we arrive at the Total Probability Rule (TPR):

P(e) = P(h)P(e|h) + P(¬h)P(e|¬h) [sum of exhaustive hypotheses’ probabilities]

Now, let’s consider the Theory of Relativity. Using our prior probability, let us assume that P(h) = [0.5]; that

we’re indifferent to h. Let us further take it that the hypotheses of the falsity of h, and the truth of h are

mutually exclusive (h v ¬h).


Note the pattern: Probability of h and e given h, plus probability of not-h and e given not-h — opposites re h.


P(h|e) = (0.5)(1.0)/P(e) [by Bayes’ Theorem]

and

P(e) = P(h)P(e|h) + P(¬h)P(e|¬h) [by Hacking’s TPR]

Assuming that P(e|h) = [1.0] — i.e. that the evidence of the light bending, etc., is certain if General Relativity is

true, then

P(h|e) = (0.5)(1.0)/P(e) [by Bayes’ Theorem]

If we use Hacking's TPR and substitute it into Bayes’ Theorem where P(e) currently stands, we obtain:

P(h|e) = 0.5 / P(e)

= 0.5 / ( P(h)P(e|h) + P(¬h)P(e|¬h) )

= 0.5 / ( 0.5 x 1.0 + 0.5 x P(e|¬h) )



Let’s assume that P(e|¬h) = 0.1 - i.e. that the evidence would be unlikely given ¬h, so:

P(h|e) = 0.5 / ( 0.5 x 1.0 + 0.5 x 0.1 )

= 0.5 / ( 0.55 )

= 0.90909

This means that h — the Theory of Relativity — is highly probable, assuming that P(e|¬h) is improbable — say

[0.1].

There are three points that we can draw from this.

• Firstly, if the probability of the evidence being as it is, is low, on the assumption that the hypothesis is false,

and if the probability of the evidence being as it is, is high, on the assumption that the hypothesis is true, then

the hypothesis is a good one, because it corresponds to the evidence.

• Secondly, if the evidence supports the hypothesis better than others, then the hypothesis is likely true.

• Thirdly, we have to assume that P(e|¬h) has a definite value, and this assumption

could be subjective, and might be case-dependent.


ie we’re assuming General Relativity is 9/10 the most likely or correct explanation for light bending

Using Bayes’ Theorem with the case of Theism

Let’s see, then, how Swinburne uses Bayes’ Theorem to help make his case. Let’s start with the argument above

where we consider the probability of a universe such as ours arising (where e is the evidence of an ordered

universe containing intelligent life).

(1) P(e|ht&k) > 0.5 [Given God, the universe is likely]

(2) P(e|¬ht & k) < 0.5 [Without God, the universe is unlikely]

(3) P(e|ht&k) > P(e|¬ht&k) [(1) & (2) The universe is likely than not, given God]

Let’s now see if we can discover whether the evidence (e) confirms the hypothesis.

Bayes Theorem: P(h|e) = P(h) x P(e|h)/P(e)

P(e|ht) > P(e) [From (3) above]

P(ht|e) P(e) / P(ht) > P(e) [Substituting Bayes’ Theorem for P(e|ht)]

P(ht|e)/P(ht) > 1.0 [Dividing by P(e) on both sides]

P(ht|e) > P(ht) [Multiply throughout by P(ht)]


Using Bayes’ Theorem with the case of Theism

or, with the factor k reintroduced:

P(ht|e&k) > P(ht|k)

This result means that the probability of the hypothesis of theism, ht, given the existence of a complex

universe, e, is greater than the prior probability of theism alone. IE The existence of a complex universe

confirms or supports the view that God exists (Swinburne, 1991, p144, Holder, p296).

Swinburne’s explanations are probabilistic propositions (Brown, 2009) — God’s existence makes an ordered

universe more likely. The universe’s coming-to-exist is more likely given the truth of ht. But ht does not entail

an ordered universe’s existence. In other words, it’s not certain that (ht ⊃ e) or vice versa, it’s merely the case

that e renders ht more probable (i.e. P(ht |e) > 0.5)


Using Bayes’ Theorem with the case of Theism — Swinburne’s claim

Let’s assume that the prior probability of theism is high (Swinburne, 2004, p93), say [0.9].

Now, what is the probability of an ordered universe having come to exist — given theism?

God is omnibenevolent. So let’s assign a value of [0.5] for P(e|ht) (ibid. 2004, pp338–9).

And P(e)? Well, P(e|¬ht) could have the value of [0.1] — as probability ranges between [0;1] and, ht and ¬ht are

mutually exclusive, and thus their probabilities jointly add up to [1.0]. Thus to calculate P(e) we use TPR —

( P(ht)P(e|ht) + P(¬ht)P(e|¬ht) ) and find P(e) is 0.5:

P(ht|e) = P(ht) x P(e|ht)/P(e) [Bayes’ Theorem]

= (0.9 x 0.5) / ( P(ht)P(e|ht) + P(¬ht)P(e|¬ht) ) [TPR, Hacking, p70]

= 0.45 / (( 0.9 x 0.5 ) + ( 0.5 x 0.1 ))

= 0.45 / ( 0.45 + 0.05 )

= 0.45 / 0.50 = 0.9

Hence, the universe supports theism because P(e) > P(e|¬ht) & P(ht|e) = 0.9.


God would have reason to create a good universe, due to omnibenevolence, and God being infinitely simple

Swinburne himself supplies the amount 0.5

Using Bayes’ Theorem with the case of Theism — Swinburne’s claim

In short, Swinburne is right that the evidence of a life-supporting universe supports the hypothesis of theism

(ht) if the following are true:

— That P(ht), the prior probability of theism, is high

— That P(e|ht) is at least 0.5 and due to TPR or LEM, P(e|¬ht) is also 0.5

— And that P(e) is about 0.5

So Swinburne’s argument requires that we accept that theism has a high prior probability, and that the universe

is unlikely to have existed without God, or had about a 50-50 chance of not existing.

P(e|ht&k) > P(e|k), i.e., that theism explains this universe, or that this ordered universe is more likely, given

God, and, that taken with Bayes’ Theorem, this entails that:

P(ht|e&k) > P(ht|k), i.e., that theism is more likely given the evidence of the existence

of a complex physical universe, than it would be without that evidence.


Criticisms — 1. What if God doesn’t want to create?

If we grant Swinburne that P(e|ht&k) > P(e|k), then P(ht|e&k) > P(ht|k), since it follows mathematically.

But what if it is not true that the universe is more likely given God? i.e., What if we could make an argument

that God might not have incentive to create? Then P(e|ht&k) < P(e|k). If we apply that premise (God’s

uncreativity) to Bayes’ Theorem, we fail to obtain the result that P(ht|e&k) > P(ht|k) — the hypothesis is

supported by the evidence. In other words, this universe is not evidence of God, unless God would want to

create it more than he would not. Of course, there are arguments for this.

Suppose that God is perfect, and therefore has no needs or urges to fulfil, such as creativity. A perfect

omnipresent being would make the universe worse by adding to existence e, since e was already absolutely

perfect with just God in existence (can’t make something more perfect if it is perfect). So let’s assign a value of

[0.1] for P(e|ht). Then let’s assume that P(e|¬ht) has the value of [0.9], since the universe exists, and theism

might be false, and TPR; i.e. the sum of probabilities must be 1.0.

Take it now that the prior probability of theism alone, agreeing with Swinburne,

is neutral — [0.5].



P(ht|e) = P(ht) x P(e|ht)/P(e) [Bayes’ Theorem]

= (0.5 x 0.1) / ( P(ht)P(e|ht) + P(¬ht)P(e|¬ht) ) [TPR, Hacking, p70]

= 0.05 / (( 0.5 x 0.1 ) + ( 0.5 x 0.9 ))

= 0.05 / ( 0.05 + 0.45 ) = 0.01

This means that just assuming that God has no good reason to create the universe, with P(e|ht) = [0.1], is

enough to render theism ten times more improbable.

Even if we take it that P(e|¬ht) is low, say, 0.1, rather than 0.9, i.e. even if we agree with Swinburne that the

universe is improbable, we still only obtain P(ht|e) of [0.5]. IE Theism equiprobable to materialism since they

are mutually exclusive and jointly sufficient hypotheses for this universe existing.

This means that Swinburne has to show that God would create, AND that the universe is improbable without

God. The case for theism is worse if we recognise that there are an infinite number of possible universes

which would be good universes: lim P(e|ht) ➝ [1.0/∞] which is approx. zero. Even if half the possible universes

are good, lim P(e|ht) ➝ [2/∞] which is still approx. zero



Vilenkin argues that our universe is more likely mediocre

and therefore commonplace, rather than rare, statistically

speaking. Moderate evil suggests it’s mediocre.

Even if Swinburne is offering us a good C-inductive

argument, it can still be a bad argument. Good

C-inductive arguments are not necessarily good

arguments, since P(ht|e) could be accepted as

higher than P(ht) alone, and still be far below

[0.5] — i.e., P(ht|e) can still be very

improbable. E.g. P(ht|e) might be [0.1],

and P(ht) might be [0.01]; this is

a good C-inductive argument

since it makes ht 10 x more

probable, but it still doesn’t confirm h.


Criticisms — 2. Does ht entail e?

We expect good scientific theories to entail their observations in a deductive-nomological way. Theism does

not offer this. We cannot deduce from theism that this universe would exist, or be more likely, contrary to

what S. says. Swinburne seems convinced that the existence of this universe is staggeringly improbable without

God. But this is just his own opinion.

Stenger, for example, demonstrated the opposite using a simulation with the four fundamental forces, and

found that there are at least 1 x 10500 possible universes that would yield life-hospitality (Stenger, 2009, p89,

2011, p23).

So Swinburne has to first convince us that a life-capable universe is unlikely without God and that God would

create this universe.


Criticisms — 3. Probability base rates

Hacking, (2008, p73) argues that base probability rates are also highly relevant. If a witness, known to be

reliable, reports an accident caused by a green taxicab, we will object, because we know taxicabs are usually

yellow (or black, in the UK). Hence, the content of our background knowledge, namely, what is in k, influences

the plausibility of our hypotheses. So if k contained “there are no demonstrated supernatural phenomena”,

(which k does in fact contain), then P(ht|k) is very, very low. This affects the prior probability of theism.

4. Creatio ex nihilo and basic acts

Creationism depends on the view that creatio ex nihilo (CeN) is possible and that basic acts are possible. There

are problems with both. S. argues for the improbability of e|¬ht using the second law of thermodynamics, but

ignores the first law, which prohibits CeN. You can’t cherry-pick science. Some cases similar to CeN (vacuum

polarisation) are reported in quantum states near existing matter, but not in isolation (Stenger, pp146-7).

The problem with “basic acts” is that it depends on the doctrine of free-will libertarianism,

which would make God’s acts spurious, bereft of causal reasons, and hence, not

his acts, but random events. Does God even have free-will, e.g. to do evil?


Summary and Conclusion

1. Swinburne fails to demonstrate that this universe is evidence for the existence of God. He needs to show

that God would likely make this universe, and that this universe is highly improbable, prima facie. He has to

show why theism would REALLY make this universe more probable, since one can imagine an infinite number

of very good universes that are hence infinitesimally improbable due to being infinite in number. Of course, I

am aware of the “best kinds” argument (e.g. Hasker), but theism still leaves it up to chance/arbitrariness within

a smaller infinite. Vilenkin (in Stenger, 2011) argues that our universe is more likely mediocre and therefore

commonplace, rather than rare, statistically speaking. The problem of evil supports this view.

2. Even if S. has a good inductive argument for God, it is not demonstrative, and even may be not so taken with

his other arguments. P(ht|e) might still be very low, and S. doesn’t give us numbers.

3. There’s no D-N link (ala Hempel) from ht to e. S. tries to talk about omnibenevolence, but that entails angels.

4. Other probability issues affect the argument, e.g. the contents of k, which S. ignores, and which are prima

facie materialist and lend more support to hm.

5. CeN and basic acts have philosophical and scientific problems which might be insurmountable, and which do

not plague the scientific “big bang” model.

THANK YOU!


P-inductive and C-inductive

Swinburne introduces us to two kinds of inductive argument: P-inductive and C-inductive (2004, p6 et seq.). He

aims to show that by establishing an inductive argument which raises the probability of a hypothesis, it does

not, by itself, necessarily confirm that hypothesis. We need something more.

Swinburne’s goal is to produce a collection of good C-inductive (confirmatory) arguments for his overall P-

inductive (probably-true) argument that God exists.

“If P(h|e1&k) > 1/2, then the argument from e1 to h is a good P-inductive argument. If P(h|e1&k) > P(h|k), then the

argument is a good C-inductive argument.” (Swinburne, 2004, p15).

If P(h|e&k) > P(h|k), ∴ e ⊃ h (not logically entails, but makes highly probable)


“Let us call an argument in which the premisses make the conclusion probable a correct p-inductive argument. Let

us call an argument in which the premisses add to the probability of the conclusion (that is, make the conclusion

more likely or more probable than it would otherwise be) a correct c-inductive argument. In this latter case let us say

that the premisses ‘confirm’ the conclusion.” (p6, 2004)

This page is not required for understanding the argument

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Can Bayes’Theorem, given the evidence of this universe, be used to support theism?

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