Sorry about the delay in responding, due to conference paper deadline panic.


[Alarming amounts of agreement deleted :-)]


That ("complicated") isn't in fact where P(H) > P(HG) comes from; it's more
the other way around. It's from

  P(H)  =  P(HG) +  P(HG')  where G' is the complement of G

and by axiom, P(anything) >= 0, so P(HG') >= 0, so P(H) >= P(HG).

In a sense, HG is necessarily more "complicated" than H for any H and G,
so I may be splitting hairs, but what I'm trying to say is that irrespective
of subjective impressions of how complicated something is, P(H) >= P(HG)
holds, with equality if and only if P(HG') = 0.


Well, "P(x | A) = P(x | B)" means that x is as likely to be observed if A is
operative as it is if B is operative. This implies that observing x does not
provide any useful information which might allow us to discriminate between
the respective possibilities that A and B are operative; the difference
reduces to the difference between the (unknown and unhelpful) prior
probabilities P(A) and P(B):

  P(x | A) = P(x | B)  ==>

    P(A | x)  =  k P(A),   and   P(B | x)  =  k P(B)

where k  =  P(x | A) / P(x)  =  P(x | B) / P(x).

So A and B are "equally consistent with the data" in that observing x
doesn't give any pointers as to which of A or B is operative.

In the particular case where A = H and B = HG, however, we know that their
prior probabilities are ordered by P(H) >= P(HG), although we don't know
the actual values, and it's this which allows us to deploy the Razor to
throw out any such HG.


That's certainly true, but the particular point here was whether or
not a `divine component' actually underlies the prevalence of religion
in addition to the memetic transmission component, which even the religious
implicitly acknowledge to be operative when they talk of `spreading the word'.

Now it seems to me, as I've said, that the observed variance in religious
belief is well accounted for by the memetic transmission model, but rather
*less* well if one proposes a `divine component' in addition, since I would
expect the latter to conspire *against* wide variance and even mutual
exclusion among beliefs. Thus my *personal* feeling is that P(x | HG) isn't
even equal to P(x | H) in this case, but is smaller (H is memetic transmission,
G is `divine component', x is the variance among beliefs). But I happily
acknowledge that this is a subjective impression.


Not that I'm a statistician as such either, but:

The idea is that both theism and atheism are compatible with all of
the (read `my') observations to date. However, theism (of the type with
which I am concerned) *also* suggests that, for instance, prayer may be
answered, people may be miraculously healed (both are in principle amenable
to statistical verification) and that god/s may generally intervene in
measurable ways.

This means that these regions of the space of possible observations, 
which I loosely termed "appearances of god/s", have some nonzero
probability under the theistic hypothesis and zero under the atheistic.

Since there is only so much probability available for each hypothesis to
scatter around over the observation space, the probability which theism
expends on making "appearances of god/s" possible must come from somewhere
else (i.e. other possible observations).

All else being equal, this means that an observation which *isn't* an
"appearance of god/s" must have a slightly higher probability under
atheism than under theism. The Bayesian stuff implies that such
observations must cause my running estimate for the probability of
the atheistic hypothesis to increase, with a corresponding decrease
in my running estimate for the probability of the theistic hypothesis.

Sorry if that's still a bit jargonesque, but it's rather difficult to
put it any other way, since it does depend intimately on the properties
of conditional probability densities, and particularly that the total
area under them is always unity.

An analogy may (or may not :-) be helpful. Say that hypothesis A is "the
coin is fair", and that B is "the coin is unfair (two-headed)". (I've
used A and B to avoid confusion with H[heads] and T[tails].)

Then

  P(H | A) = 0.5  }  total 1
  P(T | A) = 0.5  }

  P(H | B) = 1    }  total 1
  P(T | B) = 0    }

The observations are a string of heads, with no tails. This is compatible
with both a fair coin (A) and a two-headed coin (B). However, the probability
expended by A on making possible the appearance of tails (even though they
don't actually appear) must come from somewhere else, since the total must
be unity, and it comes in this case from the probability of the appearance
of heads.

Say our running estimates at time n-1 are e[n-1](A) and e[n-1](B). The
observation x[n] at time n is another head, x[n] = H. The estimates are
modified according to

                            P(H | A)
  e[n](A)   =   e[n-1](A) * --------   =   e[n-1](A) * m
                              P(H)

and

                            P(H | B)
  e[n](B)   =   e[n-1](B) * --------   =   e[n-1](B) * 2m
                              P(H)

Now we don't know P(H), the *actual* prior probability of a head, but
the multiplier for e(A) is half that for e(B). This is true every time
the coin is tossed and a head is observed.

Thus whatever the initial values of the estimates, after n heads, we have

                 n
  e[n](A)   =   m  e[0](A)

and
                    n
  e[n](B)   =   (2m)  e[0](B),

and since e[k](A) + e[k](B) = 1 at any time k, you can show that 0.5 < m < 1
and thus 1 < 2m < 2. Hence the estimate for the fair-coin hypothesis A must
decrease at each trial and that for the two-headed coin hypothesis B must
increase, even though both hypotheses are compatible with a string of heads.

The loose analogy is between "unfair coin" and atheism, and between "fair
coin" and theism, with observations consistent with both. A tail, which
would falsify "unfair coin", is analogous to an "appearance of god/s",
which would falsify atheism. I am *not* claiming that the analogy extends
to the numerical values of the various probabilities, just that the principle
is the same.



Quite so, but this type of theism is what I might call "the G in the HG",
in terms of our Ockham's Razor discussion, and I'd bin it on those grounds.


The hypotheses don't have to be falsifiable, and indeed in my `model',
the theism isn't falsifiable.



You don't have to. We don't need, in the above analogy, to know *any*
prior probabilities to deduce that the updating multiplier for the
fair-coin hypothesis is less than unity, and that the corresponding
multiplier for the two-headed coin hypothesis is greater than unity.
You don't need to know the initial values of the running estimates
either. It's clear that after a large number of observations, P(fair-coin)
approaches zero and P(two-headed-coin) approaches unity.

All you need to know is whether P(x | Ha) is larger than P(x | Ht) for
observed x, and this follows from the assumptions that there are certain
events rendered *possible* (not necessary) under Ht which are not possible
under Ha, and all else is equal.


Any observations you like; it really doesn't matter, nor affect the
reasoning, provided that there are some possible observations which
would count as "appearances of god/s". Examples of this might be
a demonstration of the efficacy of prayer, or of the veracity of
revelation.



OK, we'll downgrade "*does* interact" to "*may* interact", which would
actually be better since "does interact" implies a falsifiability which
we both agree is misplaced.


I'll explain, but bear in mind that this isn't central; all I require of
a theism is that it *not* make the prediction "Appearances of god/s will
never happen", as does atheism. (Before somebody points out that quantum
mechanics doesn't make this prediction either, the difference is that
QM and atheism do not form a partition.)

Predictions include such statements as "Prayer is efficacious" (implying
"If you do the stats, you will find that Prayer is efficacious"), or "Prayer
is *not* efficacious", or "Verily I say unto you, This generation shall not
pass, till all these things be fulfilled." I don't think we have any problems
of misunderstanding here.



That's fine; I don't claim that theism is false, merely that the [finite
number of] observations available to me so far suggest that it is, and
that as I continue to observe, the suggestion looks better and better.




I think you are; an "appearance of god/s" is sufficient to falsify
atheism, whereas in general the corresponding theism is unfalsifiable.


No: by way of a counterexample, let X = "the coin is fair", or more
accurately (so that not(X) makes sense) "the two sides of the coin are
different". This is unfalsifiable by tossing the coin; even a string of
heads is consistent with a fair coin, and you have to go to an infinite
number of tosses to falsify X in the limit. Its converse is falsifiable,
and is falsified when at least one head and at least one tail have appeared.




Oops. Sorry. Mea culpa.


We agree here.


"The Rapture will not happen on October 28 1992." Said Rapture would have
falsified atheism to my satisfaction had it happened, although its failure
to happen does not, of course, falsify any theisms other than those which
specifically predicted it.

"No phenomenon which requires the existence of one or more gods for its
explanation will ever be observed." That about sums the whole thing up.


Cheers

Simon