Sorry about the delay in responding, due to conference paper deadline panic.  In article <1qsnqqINN1nr@senator-bedfellow.MIT.EDU> bobs@thnext.mit.edu (Robert Singleton) writes: >In article <1993Apr18.043207.27862@dcs.warwick.ac.uk>   >simon@dcs.warwick.ac.uk (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 > does not support a disbelief.   The hypotheses don't have to be falsifiable, and indeed in my `model', the theism isn't falsifiable.  > [...]  > I used the phrase "SHOULD obverse". Given any specific 'x' theism  > does not make the prediction "P(x | Ht) > 0". That's why I used the  > word "should" - theism makes no predictions about any specific event. > I can only say "I believe" that God did such and such after such > and such happens, or "I believe God will" do such and such. But > for any given 'x' I can never, a priori, say P(x | Ht) > 0. I can > not even say this for the set of all 'x' or some 'x'. This is what  > don't like about your use of probability. We also have no way of > assigning these probabilities - I hold science to positivistic > criteria - if someone cannot tell me how to measure, even in principle, > P(x | H), then probability is not applicable to hypothesis H. Such > is the case when H = Ht (theistic) and Ha (atheistic). For example, > P(x | Ha) = P(x & Ha)/P(Ha). What is P(Ha)?!? How do I measure it?   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.  > Baysian statistics relies upon a series of observations. But > what if the hypothesis isn't amenable to observation? And even for > statements that are amenable to observation, some observations are > not relevant -- a sequence of observations must be chosen with care. > I'm curious to know what types of observations x[1],x[2],... you have  > in mind concerning theism and atheism.  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.  >> But any statement about P(x | H) for general x still counts as a  >> prediction of H. If the theism in question, Ht, says that prayer may  >> be answered, or that miracles may happen (see my interpretation, quoted  >> again above, of what `God exists' means), then this is a prediction,  >> P(x | Ht) > 0 for such x. It's what distinguishes it from the atheist  >> hypothesis Ha, which predicts that this stuff does not happen, P(x | Ha) >> = 0 for such x.  > Theism does not make the claim that "P(x | Ht) > 0 for such x". > Or I should say that my "theism" doesn't. Maybe I was too quick to > say we had a common language. You said that by the existence of God  > you "mean the notion that the deity described by the Bible and by  > Christians *does* interact with the universe as claimed by those agents". > I agreed with this. However, I must be careful here. I BELIEVE > this - I'm not making any claims. Maybe I should have changed *does* > to *can* - there is an important shift of emphasis. But any way, > since I "only" have a belief, I cannot conclude "P(x | Ht) > 0 for  > such x".  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 don't think my theism makes "predictions". Maybe I'm not > understanding what you mean by "prediction" - could you explain what > you mean by this word?  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.  >> Persistent observation of this stuff not happening, *consistent* with >> Ht though it may be, is *more* consistent with Ha, as explained in the >> Bayesian stats post.  >> >> Even if Ht ("God exists") is unfalsifiable, that's >> no problem for my argument, other than that you have to let the number  >> of observations go to infinity to falsify it asymptotically.   > BTW, I do not consider an argument that requires an infinite number of  > observations as valid - or rather that part of the argument is not valid.  > We, as existing humans, can never make an infinite number of measurments  > and any conclusion that reilies on this I don't accept as valid.  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.  > [Renormalization stuff deleted]  >> In the Bayesian stats post, I assumed that theism was indeed unfalsifiable >> in a finite number of observations. Here's the relevant quote: >>  >> $ The important assumption is that there are *some* observations which  >> $ are compatible with the theist hypothesis and not with the atheist  >> $ hypothesis, and thus would falsify atheism; these are what I called  >> $`appearances of god/s', but this need not be taken too literally. Any  >> $ observation which requires for its explanation that one or more gods  >> $ exist will count. All other observations are assumed to be compatible  >> $ with both hypotheses. This leaves theism as unfalsifiable, and atheism  >> $ as falsifiable in a single observation only by such `appearances of  >> $ god/s'.  > Here is my problem with this. For something to be falsifiable it > must make the prediction that 'x' should not be seen. If 'x' is  > seen then the hypothesis has been falsified. Now, atheism is a word  > in oposition to something - theism. A theism aserts a  belief and an  > atheism aserts a disbelief. So there are certain atheisms that are  > certainly falsifiable - just as there are certain theisms that are  > falsifable (e.g. if my theism asserts the world is only 6,000 years  > old and that God does not decieve then this has been falsified). However,  > the atheism that is in oposition to an unfalsifiable theism is also  > unfalsifiable. I could be wrong on this statment - [...contd]  I think you are; an "appearance of god/s" is sufficient to falsify atheism, whereas in general the corresponding theism is unfalsifiable.  > I'll think more about it. Until then, here is a general question. > Suppse X were unfalsifiable. Is not(X) also 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.  >>> This is partly what's wrong with you Baysian argument - which  >>> requires observations x[1] ... x[n] to be made. There are simply  >>> no such observations that have a truth value in relation to the  >>> statement "God exists". Now, by use of your symmetry argument, I  >>> can understand why someone would say "Since the statement  >>> 'God does not exist' >>   ~~~~~~~~~~~~~~~~~~ >>> makes no predictions I will choose not to believe it." But none >>> the less this would be founded on a type of faith - or if you don't >>> like the word faith insert "belief for which there is no falsifiable >>> evidence" instead.   >> I'll assume you meant `God exists' up there at the highlight. But by our >> agreed definition of "exists", the statement makes predictions as I said >> above, although it isn't falsifiable in a finite number of observations.  > Actually, I mean 'God does not exist' makes no predictions.  Oops. Sorry. Mea culpa.  > The truth of this statment actually depends upon which god you are > refering to. But I can think of some conceptions of God for which  > it is true. But once again I'm open to the posibility that I could > be wrong. So give me some examples of predictions of the statment > "God does not exist". Here is one that I can think of. If true, then  > there would be no healing or miricles. But this can in principle never  > be determined one way or the other. There are cases in which people  > seem to recover and are healed without the help of a doctor and for no   > known reason. These situations do in fact happen. They are consistent > with a theistic hypothesis, but IN NO WAY support such a hypothesis.  We agree here.  > They are not inconsistent with an atheistic hypothesis. I can't > think of one "prediction" from 'God does not exist' that isn't of > this type. But I might be missing something.   "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.  > bob singleton > bobs@thnext.mit.edu  Cheers  Simon