Re: Thomas Bayes and Bayesian statistics

From: Iain Strachan (iain.strachan.asa@ntlworld.com)
Date: Sun Aug 31 2003 - 04:06:19 EDT

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In reply to: Steve Bishop: "Re: Thomas Bayes and Bayesian statistics"
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Steve,

Another good book that gives an easy(ish) introduction to Bayesian Theory is by a namesake of yours:

Neural Networks for Pattern Recognition - Christopher M. Bishop (OUP) 1995. Find this at:

http://www.amazon.com/exec/obidos/tg/detail/-/0198538642/qid=1062316097/sr=8-1/ref=sr_8_1/002-2611734-7901600?v=glance&s=books&n=507846

The book explains bayesian techniques for neural network training, and also how neural networks predict the a posteriori class probabilities, given that the training data reflects the a priori class probabilities. This is an oft-cited paper by Richard & Lippmann (1991), which unfortunately at the moment I can't track down on Citeseer. Furthermore, it is possible, if the training data didn't reflect the prior probabilities, that one can still compute the posterior probabilities from the neural network's outputs, by using Bayes' rule.

The book is one of the best around and is generally regarded as a classic text on the subject. I'm not just saying this because Chris Bishop was my PhD supervisor, either ;-)

The fact that the "training data" has to reflect the prior probabilities rather sinks Swinburne's hypothesis about Bayes being used to compute the probability of the resurrection. We only have a handful of examples of people being raised from the dead, and they are all in the Bible, which we take as true on faith. But if we take it on faith, why do we need Bayes' theorem? OTOH, if we don't take it on faith, then why should we trust the Bible, and regard the attendant circumstances described therein as anything more than myth?

In order to get the prior probability of someone being raised from the dead, you would have to have a large number of independently verified examples. Only then could you make a meaningful deduction of the posterior probability. And furthermore, the attendant "events", or bits of information (like in my gender example, where I told you it was a nurse, or a midwife who was 6ft 2), would have to be extra-biblical; if you just take the bible on faith as true, then the posterior probability is 1. Several independent witnesses saw him alive after the crucifixion; therefore he rose from the dead. But if you doubt those testimonies, you have to look outside the bible (perhaps one of the best arguments is that Christianity survived rather than fizzling out).
------------------------------------------------------------------
Iain .G.D. Strachan

There are 10 types of people in the world ...
those who understand binary and those who don't.

--------------------------------------------------------------------
  ----- Original Message -----
  From: Steve Bishop
  To: iain.strachan.asa@ntlworld.com ; asa@calvin.edu
  Sent: Saturday, August 30, 2003 10:30 AM
  Subject: Re: Thomas Bayes and Bayesian statistics

Hi Iain,

Many thanks for your comments - the link was particularly appreciated.

>From: "Iain Strachan"
>Date: Fri, 29 Aug 2003 20:49:53 +0100
>
>Steve Bishop:
>
>
> > Hi all,
> >
> > I have recently noticed how Swinburne and others have been using Bayes
> > theory to "prove" Christianity. Using Bayes' theorem [he] maintains that the
> > resurrection of Jesus is 97% probable.
>
>I find this hard to believe.
It's what Swinburne maintains!

"Mr. [sic] Swinburne, a commanding figure with snow-white hair and piercing blue eyes, proceeded to weigh evidence for and against the Resurrection, assigning values to factors like the probability that there is a God, the nature of Jesus' behavior during his lifetime and the quality of witness testimony after his death. Then, while his audience followed along on printed lecture notes, he plugged his numbers into a dense thicket of letters and symbols-using a probability formula known as Bayes's theorem-and did the math. "Given e and k, h is true if and only if c is true," he said. "The probability of h given e and k is .97"

"In plain English, this means that, by Mr. Swinburne's calculations, the probability of the Resurrection comes out to be a whopping 97 percent. "
http://www.selfknowledge.org/resources/press/nyt_eakin.htm