bayesian probability, chaitin descriptions, and evolution

Sylas,If I don't respond this evening (which I probably won't) I definitely will tomorrow morning.Regards

We have a mixture of both, although the number of non-bible bashers far outweigh the number of bible bashers here.

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The Laminator

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Personally I am not aware of any credible objections to evolution as a natural process giving rise to all the many current diverse forms of life, from ancient and primitive common ancestors.... but that is another debate.Apart from that, the wonder you express at the natural world is echoed by many of our greatest scientists. While I certainly do not endorse all his metaphysical views, I expect you would be interested to read some of the writings of Paul Davies. I have enjoyed reading them. Davies is a significant modern cosmologist and physicist, who has become better known for his philosophical speculations about the order and apparent fine tuning of the universe. The apparent design he perceives is not in the particular complexities we seen in life on earth, but in the deepest levels of physics... themodynamics, particles physics, properties of space ... which in Davies' view are remarkably balanced to facilitate the spontaneous emergence of complexity, and life, and consciousness.Take all the time you need for any of the other stuff. There is no rush.Cheers -- Sylas

Well I wondered what error you thought you saw in my post and it seems that it was not the essential reasoning, but instead it is an assumption which I did not in fact make.Sid claimed that
"...the chance of the thing that generated B containing natural selection has to be factored in. When you do that, you have greater chance of getting intelligence in B through blind chance."Assuming that "contiing" means "produced by" (because that is relevant and because it is pretty silly otherwise - intelligence is a property of individuals and individuals don't "contain" natural selection) then Sid is wrong. And Sid is wrong for exactly the reasons I stated. He is looking at the wrong probability.Now you say But I didn't say HOW the origin was identified. It *is* necessary to identify the origin since we need to distinguish those produced by pure chance from those produced by natural selection to examine Sid's claim. How we do this is irrelevant to my point. And since there are ways to do this without relying purely on examination of the string (e.g. observe the origin and label the string) there is no need to assume that it is done solely from examination of the string.Sid on the other hand is saying "we'll take the probability of selecting an 'intelligent' string from a mixed population produced by chance and natural selection and call that the probability of chance producing "intelligence". Which is wrong, and obviously so.I'll add that, in my view, my post 17 which has yet to be addressed contains a point which is fatal to Sid's application of Chaitin.

Right. This appears right back in his first post. If you sort through the rather confused notation, it boils down the saying the following: The argument treats "naturally selected" and "intelligent" as predicates, and the argument is no more profound than which is of course true, and completely vacuous as a basis for skepticism about either predicate.Your response to the matter in Message 11 speaks of a collection of strings, some of which are produced by selection and some of which are produced by chance. For your model to work, you have to maintain labels on those strings to distinguish their origin, since you can't actually tell looking at the string alone how it was produced.However, assuming you do this, then your argument is correct. Sid is not really contrasting selection with chance since he mixes up distributions over the space of generating methods with distributions over the space of output results. I nearly commented on that post in my response to sid, except that I was not entirely sure what you were saying. I could not make sense of the phrase in parentheses: You can't have a single description method which can represent each of 2^n strings in n bits or fewer. Some of the strings must have descriptions that are longer than the strings themselves.Cheers -- Sylas

That's right - and that is the fatal flaw in Sid's argument. Yuu have to have a rule on interpreting the strings and you can't use "interpret as an algorithm for producing a sequence" AND "interpret as a pure sequence" at the same time.Consider a sequence of n bits that is highly compressible in Chaitin's sense such that it can be generated by an algorithm of m bits where m << n.We can either attempt to generate the sequence through pure chance - represented by the procedure listed in my earlier post, or use the algorithm.Suppose then that we have to generate the algorithm by pure chance, then we have to generate a sequence of m bits. Fortunately if we successfully generate the algorithm then the probability of getting from there to the sequence is 1.So the probability of generating the sequence by pure chance is 2^-n
The probability of generating the algorithm by pure chance is 2^-m,Therefore the probability of generating the sequence using the algorithm - *including* chance production of the algorithm is 2^-m.as m << n, 2^-m >> 2^-n directly contradicting Sid's argument.The probability of obtaining the algorithm and producing the sequnce using that IS higher than the probability of simply producing the sequence by pure chance.Sid's argument relies on including the probability of generating the algorithm *in* the probability of generating the sequence by pure chance. But as I point out above these are different methods relying on different interpretations of the string.

Agree with this and with the rest of your post. In fact, I nearly did point out your Message 17 to sid earlier as a good response, but I removed that mention during editing of the post because of the ambiguity in the parenthesized phrase. You clarify here nicely, and I have no objections.You express very clearly the problem with confusing random probabilities on the output strings themselves, and random probabilities on the method of generating output strings. This is a serious defect in sid's argument, which I did not address.Cheers -- Sylas

(I'm just responding in sequence to your last few posts, because its easier. I have no idea whether the following is directly applicable to anything you've posted since.) "The probability of set Pc given set Pn?"What is the probability of a set of numbers given another set of numbers?Also, your version drops B out of the picture completely which I wouldn't have, because it obscures the issue of the size of E relative to B (which you haven't addressed yet). Only dealing with sets obscures the same issue which of course is not something explicitly relevant to Bayesian probability, but I address it afterwards.Given some B chosen at random, and given some E chosen at random that outputs B. Those are the relevant starting assumptions I do not want to obscure, because I want to deal with properties of these two entities relative to each other.I understand that if B is some specific value, then Pi(B) evaluates to either true or false. and prob(false) for example is meaningless. I guess my meaning was, "assume B is a specific value, we just don't know what it is." The Bayesian argument has more relevanve if considered simultaneously with the size of E relative to B, that is the amount of information in E relative to B. You mentioned the thread in talk.origins I started, where I deal with this size issue more thoroughly.I didn't mean to imply that, "Evolutionists are wrong to consider natural selection the mechanism, because there's a greater probability of intelligence by blind chance." Rather the implication is that the burden of explanation is just shifted to Pn(E).Someone who says "goddidit" will not be disturbed that his explanation contains more information than what is explained.If B has the property of intelligence it is because its cause E had the property of outputting intelligence. These two properties contain the same amount of algorithmic information. As such the probability of either property occuring by blind chance is the same.To use your notation, prob(Pi) = prob(InvComp(Pi)).Or in terms of algorithmic information, i.e. 'H' (Chaitin's notation),H(Pi) = H(InvComp(Pi)).(This post is sort of rambling, I admit.)[This message has been edited by sid, 04-27-2004]

Its clear that both PaulK and Sylas are happy to treat "blind chance" as "anything that is not natural selection".I define the blind chance of X as prob(X), i.e. *the unconditional probability of X*.In essence, PaulK and Sylas define the blind chance of X as
prob(X|X is not generated by natural selection).Needless to say I think mine is the correct treatment.I also understand that Sylas thinks that
prob(A) >= P(A and B)
is vacuous.
I've tried to address that as best I can as well.(If there are any other points I should address, someone let me know.)But in any case I wanted to close with a summary of my second main point. (This is really redundant, but I feel compelled to sum it up again. I don't anticipate doing this again. If someone wants to discuss the following, I will.)To review, E outputs B.
B is a binary number representing the current state of our biological world. E is a program (i.e. a binary number), a portion of which (L) controls natural selection the other portion (M) controls mutations. E makes changes to its working memory, which contains the current state of the biological world, which goes from state to state e.g. (B'->B''->B'''->B''''...) as dictated by E.The final state of B when E halts is the output of E (w/ apologies to Sylas - I know this is *really* informal - I just want to get my point across.) To reiterate what I have said in a previous post, if B', B'', etc. are binary numbers representing the state of the biological world, then there *must* exist a program E that dictates how you get from B' to B'' and so on, as no intrinsic property of a binary number dictates that it should go to another binary number (aside from a program dictating such behavior.)In any case E, as coded, *always* outputs B, (its final state).If the question arises "Where did B come from?",
The answer is "E caused it, and E is a program that output B".If the question arises "Why does B have property Px?"
the answer is "E caused it and E has the property of outputting a number with property Px.Now if the question arises "Where did E come from?",
the answer is "No one knows. The mutations are random, the laws just exist."Assume that B is the works of Shakespeare.
Then the answer to "Where did the works of Shakespeare come from?",
is "E caused it and E is a program that outputs the works of Shakespeare."What is more complicated -
The Works of Shakespeare or a
Program that *outputs* The Works of Shakespeare?Isn't
The Works of Shakespeare almost the same thing as a
Program that *outputs* The Works of Shakespeare?Thus, E is virtually the same thing as B.Anytime you have a completely stated specification for a cause E that resulted in B, E and B are very nearly the same thing.So our answer to "Why does the biological world exist?" is,
"E is extremely similar to the biological world B. E predated B and caused B to exist".(w/ apologies for extreme redundancy.)

Paulk said: If I don't understand you here, I hope you or Sylas will condescend to correct me.However, the above is just wrong. Some sequences (e.g. a string of n 0's) will have a much higher probability than others. The probability of a particular string X is 2^-H(X) (not 2^-n). (Actually, its 2^-H(X) + C for some constant C, I believe.) An infinite number of programs will generate any particular string, but the probability of any string X is the above. The probability an arbitrary program will generate a string of n 0's is much greater than 1/2^n. (So obviously the probability of others sequences is lower than 1/2^n.)

Paul seemed to be talking specifically about unbiased sources.Wouldn't the above make it, by definition, a biased source? If not all outcomes are equally probable, it's a biased source.

No, Sid it isn't wrong. As I stated the procedure is to record the first n outputs from an unbiased source of binary digits. The only way to get a string of all zeroes from that procedure is if all n digits are 0.And simply recording the digits produced - the procedure described - WILL give a probability of 2^-n of getting ANY particular sequence. So it is not possible that any sequence has a lower probability. That really is basic probability theory. If you want to say that THAT is wrong then I'd like to see an argument which doesn't rely on your misapplication of Chaitin.

Well I won't speak for Sylas, although we seem to agree on most points here.The reason I am not considering possibilities other than chance or natural selection is that those are the two you are supposedly comparing and intorducing other options would just confuse the issue (for instance any options involving a pre-existing intelligence would need to explain the origins of that intelligence).The probability of getting an "intelligent" life form with the basic requirements for natural selection (notably reproduction) by pure chance is trivially lower than the probability of producing an intelligent life form by pure chance. But that *is* a complete irrelevance because it does not address the effects of natural selection at all - the whole point of invoking natural selection to explain the development of intelligence is to avoid resorting to pure chance.As for your summary nobody is saying that we don't know where "E" comes from. Strictly speaking "E" does not exist - it is a representation of various processes and contingent events and thiking of it in terms of a "program" - especially in terms of a Von Neumann architecture computer - is not helpful and may be actively misleading. Natural selection is an inevitable consequence of imperfect self-replication and limited resources. Mutations are typically the result of chemical interactions in the reproductive process (and there is a lot of research into how various mutations occur). There is also research into relevant historical events and their effects such as the meteor strike at the end of the Cretaceous (K/T boundary).
In short your "E" is an abstract representation of the history of life on this planet. Some of it we know. Some of it we are actively researching. Some of it may be beyond finding - the surviving evidence may be inadequate to work out what happened let alone why.

That is not a reasonable statement of my position. One of the many flaws in your treatment is that you conflate probability over results with probability over programs that produce results.You effectively speak of a space of result strings, some of which you identify as being "intelligent". (This is the predicate Pi)You also speak of strings as programs, or ways of generating result strings, and some of these programs you identify as "incorporating natural selection". (This is the predicate Pn)You introduce the notion that strings can be computed from other strings, though the mapping is not represented explicitly. I have made this explicit with the function "Comp" that maps a string to an output string, where the first is taken as an inputless program. Thus we have Comp(E) = B representing that string B is the result of computing string E.You propose one probability distribution function over strings that you identify with "blind chance". (This is the function prob)From this function, you speak of an unconditional probability of intelligence, being the probability that a string chosen at random by this distribution will have the intelligence property. (You write this as prob(Pi(B)) I'll refer back to this, so lets record sid's definition of "blind chance" of B being intelligent. This makes sense if we consider the free variable B to be the random variable that defines the event space.Then you speak of a probability conditional on selection being applied, written as prob(Pi(B)|Pn(E)).Now conditional probability for real mathematicians is defined by a restriction of some space of events. What are the events in question here? It is impossible to be sure, because of the awkward and ill-defined notation.If prob(Pi(B)) denotes the probability of a randomly chosen string being intelligent, then prob(Pi(B)|XXX) denotes the probability that the randomly chosen string is intelligent given that it satisfies some other property.But what other property? The condition is written as Pn(E) with an implicit assumption that B is Comp(E). But if E is the means by which B is chosen, then this is not actually a conditional probability at all, because it overrides the random choice of B altogether. We could make it a conditional probability, by saying that XXX is the probability that there exists some E which satisfies Pn, and which "computes" B. The following expression is a plausible expression of that notion. Note that this has only one free variable B, which is the random variable. E is bound by an existential quantifier. (The quantifier is given using a Unicode character. This should work on most browsers, but it is a hack and may fail for some readers. Sorry.)Or perhaps the intended concept is that that the randomly chosen string is actually E, and that this is how B is defined. In this case, what is written as "prob(Pi(B))" should actually be expressed The conditional probability becomes Note that the values of expressions 1 and 3 are different. One is the probability that a randomly chosen result B is intelligent. The other is the probability that a randomly chosen method E will give an intelligent result.My original way of resolving this confusion on variables, by the way, was to propose expressions 3 and 4, but expressed as sets rather than with an explicit random variable. Sid did not really understand that, so I have expressed the same thing again in terms of random variables. We could also get a consistent treatment using the expressions 1 and 2.Both are reasonable understandings of "pure random", but you can't have a meaningful argument which conflates these two ways of defining the random probability of intelligence.The whole discussion is riddled with errors, and the fundamental incoherence of the notation means that many valid criticisms have been given which are based on various ways of trying to resolve the inherent underlying confusions.In order to discuss this with any level of rigour, we need to back up and get the maths right. This is difficult, because sid is approaching this whole debate with the blas confidence of the M51 driver engaged in the following conversion on his mobile phone. Sid has been surprised that not a single person has acknowledged the force or validity of his argument. The reason is that the argument is not actually valid at all. Even the comments I give above only touch on a few of the flaws; but my hope is that with more mathematical rigour sid's argument can be made more precise. That won't make the argument correct, but it will give a solid basis for showing what is wrong with it. I'm still happy to attempt to work through a resolution of the problems, if this is likely to be any help. It is vacuous as a statement about B, because it is true no matter what B represents. B can be "natural selection", "intelligent design", "takes place in Africa", "occurred long ago", or anything else you like. The expression is true for ANY B, and hence does not tell us any useful ABOUT B. Snip rest. I think you mean that there must be a machine which defines how E is executed. E is already the program, by your own description. You can think of the machine as the natural world we live in, which is such that the "program" of evolution works just fine, and produces the marvellous complexity we see in the natural world.For Christians, God is the creator of this entire fecund and exuberant world; and many Christians see God's command for life to come forth expressed in Genesis 1:24 as implicit acknowledgement of the God-given capacity of the natural world to bring forth such complex phenomena as living things. I realize religion has not come up previously in this thread, but it seems to be invariably at the heart of an inability to accept the consequences of variation and selection over long periods of time to bring about living diversity on the scale we see around us.But this recognition of the creative potential inherent in the natural world is not confined to Christians, and it was this in mind that I propose you might like to read Paul Davies, back in Message 33Cheers -- Sylas

my original impression was that his post was word games dressed up in mathematical clothes. you have made that clear, and I can follow your arguments. thanks.