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From: [e-mail] (Simon Clippingdale)
Subject: Re: Americans and Evolution, now with free Ockham's Razor inside
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Date: Fri, 23 Apr 1993 18:49:02 GMT
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Sorry about the delay in responding, due to conference paper deadline panic.

In article <[e-mail]> [e-mail] (Robert Singleton) writes:
>In article <[e-mail]>  
>[e-mail] (Simon Clippingdale) writes:

[Alarming amounts of agreement deleted :-)]

> I made my statement about Ockums Razor from my experiences in physics. 
> Thanks for info in Baysian statistics - very interesting and I didn't
> know it before. I follow your proof, but I have one questions. We have
> two hypotheses H and HG - the latter is more "complicated", which by
> definition means P(H) > P(HG).

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.

> As you point out, it's a very simple matter to show P(x | H) = P(x | HG)
> ==> P(H | x) > P(HG | x), and thus H is to be preferd to HG. Now to say
> that H is as consistent with the data as HG is to say P(x | H) =  P(x | HG).
> Can you elaborate some on this.

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.

> Also, in the "real world" it isn't as clear cut and dry it seems 
> to me. We can't always determine whether the equality "P(x | H) =  
> P(x | HG)" is true. 

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.

> BTW, my beef with your Baysian argument was not a mathematical one - 
> I checked most of your work and didn't find an error and you seem very  
> careful so there probably isn't a "math mistake". I think the mistake
> is philosophical. But just to make sure I understand you, can please 
> rephrase it in non-technical terms? I think this is a reasonable 
> request - I always try to look for ways of  explaining physics to 
> non-physicist. I'm not a Baysian statistician (nor any type of 
> statistician), so this would be very helpful. 

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.

>> Constant observation of no evidence for gods, if evidence for them 
>                                                ^^^^^^^^^^^^^^^^^^^^
>> is at all possible under the respective theisms, constantly increases
>  ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>> the notional estimated probability that they don't exist, 

> It's important to draw a distinction between theism that could
> be supported or not supported by evidence and theism that can't.
> Given a theism for which evidence is in principle not possible,
> it doesn't make sense to say "lack of evidence" supports the contrary 
> view.

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.

> So it depends upon your conception of this god. If it's a conception 
> like Zeus, who happened to come down to earth to "play" quite 
> frequently, then I agree with you - lack of evidence for this conception 
> of god is evidence that it does not exist. But if your conception
> of God is one that does not make falsifiable predictions (see below
> on "falsifiable predictions"), then I disagree -- lack of evidence