P(h|e & k) = [P(h|k)P(e|h & k)] / P(e|k)

Where

P(h|e & k) is the probability of the existence of God, given all the various bits of evidence we will come to,

P(h|k) is what he calls the intrinsic probability of God (i.e. the probability before we consider the evidence),

P(e|h & k) is the probability that the evidence would be as it is, given that God exists,

P(e|k) is the intrinsic probability of the evidence.

He expands this last term (correctly) as follows

P(e|k) = P(e|h & k)P(h|k) + P(e|~h & k)P(~h|k)

The first term is the same as the top line of the equation. The second is the converse, i.e. the probability that the evidence would be as it is, given that God does not exist, multiplied by the intrinsic probability of God’s nonexistence.

So overall we get the following equation

P(h|e & k) = [P(h|k)P(e|h & k)] / [P(e|h & k)P(h|k) + P(e|~h & k)P(~h|k)]

Now, if it is a while since you last did any maths & algebra at school, all these letters and symbols might look a bit intimidating. Of course, they are intended to look that way. So let me show you how the theorem actually works on a more down-to-earth example.

Suppose that at a particular a school, 60% of the pupils are boys, and 40% are girls. There is a school uniform, and all the boys must wear trousers, but the girls have a choice between trousers and skirt. A quarter of all the girls choose to wear trousers. You see a child in the distance wearing the school uniform, and can see that the child is wearing trousers. What is that probability that the child is a girl?

We use precisely the same equation.

P(h|e & k) is the probability that the child is a girl, given the evidence of wearing trousers. This is what we are trying to calculate.

P(h|k) is the prior probability that the child is a girl. We know that number, it is the proportion of girls in the school, i.e. 40% or 0.4.

P(e|h & k) is the probability that any particular girl wears trousers. We know this is 0.25.

P(e|~h & k) is the probability that a boy wears trousers. This is 1, since all boys must wear trousers.

P(~h|k) is the prior probability that the child is not a girl. We know this is 0.6, because we know that 60% of the children in the school are boys.

So we have all the numbers we need, and can plug them into Bayes’ equation.

P(h|e & k) = ( 0.4 x 0.25 ) / [ ( 0.4 x 0.25 ) + (1 x 0.6 ) ]

Get the pocket calculator out (or multiply it out in your head), and it comes out at 1/7, or about 14%.

You can check this out by another method. Out of every 100 children, 60 will be boys, and 40 will be girls. Of the girls, 10 wear trousers, and 30 wear skirts. We aren’t interested in the skirt-wearing children – we can see that the child has trousers. Of the trouser-wearing children, 60 are boys, and 10 are girls. So 10 out of every 70 trouser-wearing children are girls, or 1/7.

You can extend this to cover multiple pieces of evidence. For instance, taking the above example, suppose that 80% of the girls in the school have long hair and only 10% of the boys, and you can see that the child in the distance has long hair and is wearing trousers, you can make the calculation simply by plugging the new numbers into Bayes’ theorem. First, you calculate for the trouser-wearing (as shown above). That gives you a new set of prior probabilities, for the probability that a trouser-wearer is a boy or a girl. Our new evidence is the long hair. Modify the definitions above, replacing trousers with long hair, and plug the numbers into the equation. We now get this

P(h|e & k) = ( 0.14 x 0.8 ) / [ ( 0.14 x 0.8 ) + ( 0.1 x 0.86 ) ]

0.14 is the proportion of girls among trouser-wearers, and similarly 0.86 is the proportion of boys (i.e. 1 – 0.14). 0.8 is the proportion of girls with long hair, 0.1 is the proportion of boys with long hair.

Calculate this out, and it turns out that the chance of a long-haired child wearing trousers being a girl is about 0.57, or 57%.

How can that be? The chance of any trouser-wearer being a girl was only 14%! Well, by being aware of this new piece of evidence, we can now eliminate from consideration a much larger proportion of the children i.e. all the short-haired ones.

We can check it out using the other method as well. We previously worked out that for every 100 children in the school, there were 70 trouser-wearers, of whom 10 were girls. 8 out of 10 girls have long hair, but only 6 out of the 60 boys have long hair. So among the long-haired trouser-wearers, 8 out of every 14 are girls, i.e. 57%.

And that in essence is what Bayes’ theorem is all about. There is a whole load of algebra which I shan't bother to go into here which shows why Bayes' theorem works. If you're interested in probability and statistics you shouldn't just take my word for it, I recommend that you look up the maths and understand it for yourself. But the fact is, Bayes' theorem does work.

What Swinburne hopes to do is to make estimates of the various probabilities regarding God, feed them into Bayes’ theorem, crank the handle and produce an overall probability for God’s existence.

In the previous chapter, Swinburne was trying to make an assessment of the prior probability of God’s existence, which he called the intrinsic probability of theism. In the example of the schoolchildren described above, we know the prior probability that any particular child is a girl. We know that because we know how many girls and how many boys are at the school, we can simply count them. Even when we can’t count an entire population, we can sample it – that is what opinion polls are all about. And with that sample we can get a fairly accurate estimate of probabilities which can be fed into Bayes’ theorem.

But we don’t have a population of gods, nor do we have a population of universes, some of which have been made by God and some of which haven’t. We have a hypothesis regarding the existence of just a single God, and evidence in the form of just one universe whose existence we know of. And yet, Swinburne wants to make use of Bayes’ theorem in order to assess the “balance of probability” of God’s existence. In order to do that, he will have to put a number on each of the following concepts.

- The prior probability of God’s existence
- The probability of God’s existence given each of the various “pieces of evidence” that he puts forward (the cosmological argument, the teleological argument etc)

In the previous chapter, Swinburne was working his way towards suggesting a figure for P(h|k), the prior probability of God’s existence. Even to describe it in those terms is to show the futility of the effort. Either God (by Swinburne’s definition) exists or he doesn’t, and scientific investigation should in principle be able to uncover evidence one way or the other. For instance, if God performs miracles, we might reasonably hope that one or two of them would happen while scientists have their instruments pointed in the right direction. Because Swinburne’s hypothesis is not looking for what proportion of all gods have a particular set of characteristics, there isn’t any way of expressing it as a numerical probability.

The same applies to the various conditional probabilities he is looking to assign numbers to. We know what proportion of girls at the school wear trousers, we can count them. In a larger population, we could sample. Either way, we would have a statistical method by which we could come up with a number. But how can we estimate the probability of whether God would make the universe the way it is, if he exists? There is only one way available, and that is quite simply to make the numbers up.

And that is what Swinburne does, though he disguises the fact under a huge torrent of words. By the end of this chapter, we have been through 132 pages of argument which describe Swinburne’s reasons for saying that it is justified to make up the numbers, and on what basis he will choose one number over another. In the latest chapter he describes various reasons for thinking it probable that God would want or need to create a universe containing humans. I’ll paraphrase it rather than directly quoting. He thinks that it is a good thing that humans exist, and since God is perfectly good by definition, creating humans is the sort of thing that we might reasonably expect God to do. God might also want to create animals as well, but there isn’t such a good reason for him to want to do that, so animals don’t tip the matter much one way or the other. Swinburne then for the first time in the book goes on to actually assign a number which will get plugged into Bayes’ equation. This really is worthy of direct quotation.

I have argued in this chapter that there is a modest probability intermediate between 1 and 0, to which I will give the artificially precise value of 1/2, that a God will create humanly free agents located in a beautiful physical universe, perhaps also containing animals.

If you understand probability and statistics it is possible to decode this. It means that by his own admission none of his arguments are sufficient to make any kind of estimate of probability in the matter (it's impossible by definition to have a probability outside the range 0 to 1, so he is in fact ruling nothing out at all), but because he needs a number in order to proceed to the next chapter, he's going completely arbitrarily to decide to use 1/2.

Yes, that really is how he proceeds.

Genius! I could be a religous apologist. I can make stuff up.

ReplyDeleteMuch more interesting than the 'arguments' themselves is the process of naked self-delusion behind them. How can an intelligent man with a background in philosophy publish this stuff without a deep sense of embarrassment? What contortions are actually going on in that mind?

ReplyDeleteI essentially agree. I don't think doing probability assessments without an established frequency is entirely useless, but it is subjective and and arbitrary.

ReplyDeleteThe benefit of such analysis is not in asserting final or prior probabilities, then, but rather in assessing the power of various arguments. The final verdict of probability is too dependent on the prior probabilities to be meaningful, but the way from the prior to the final probabilities can often be more or less agreed upon. For example, one can agree that evidence for a divine miracle will greatly increase the probability that the god hypothesis is correct, even if one doesn't evaluate god's prior probability, and hence the final probability, the same at all.

Calling P(h|k) "intrinsic" is greatly misleading. The intrinsic probability for h is P(h|{}), the probability of god given nothing. He can only put k there by insisting that k doesn't affect anything, but even then at best P(h|k)=P(h|{}). P(h|k) is not an intrinsic probability, it is prior probability.

I don't understand what he estimates as 1/2, though - is it the probability P(e|h & k), where e is "humanly free agents located in a beautiful physical universe, perhaps also containing animals"? I find this very odd. I can invent a god whose probability to create precisely this universe is 1. I wonder why Swinburne doesn't address this, or other god, hypothesis... oh, right, because then he'll have a far weaker case.

Hi Yair

ReplyDeleteI don't understand what he estimates as 1/2, though - is it the probability P(e|h & k)?Yes, that appears to be what he is estimating, in other words, the probability that a God in the form he has defined (should he exist) would create such "humanly free agents located in a beautiful physical universe, perhaps also containing animals".

"...one can agree that the evidence for a divine miracle will greatly increase the probability that the god hypothesis is correct..."

ReplyDeleteWell, that depends. If your god hypothesis is that 'there is a being out there who can do things we don't currently understand', then a miracle is evidence for that. But I don't see how anything could count as evidence for omnipotence, omnipresence or omniscience. How do you show that a god knows everything?

Jojnj - Well, clearly I overstated the case, but certain types of miracles would be good evidence for certain kinds of deities. If a guy prayed to Thor to strike his foes with lightning, and his foes got struck with lightning, repeatably and with selectivity, then that would make a line of evidence for Thor's existence and nature. It won't establish Thor's omnipotence, or even his existence (it could be some trick...), but it would be a line of evidence towards establishing his existence and nature.

ReplyDeleteI agree that nothing could count as evidence for an omni-x deity - if only for the simple reason that I think this is a self-contradicting concept.

Yair