cnearing's argument, subjective probability and the whole Reverse Monty debacle

Over at Craig-Land, in his thread on what constitutes a "good argument", a forum member called cnearing wrote this:

I would point out that I take a strictly subjectivist approach to probability.  Probability is not an objective feature of reality (probably--set aside potential quantum weirdness for the moment) but rather a representation of the uncertainty one has about the way things behave.

Probability comes from models.  Models are built through inference from experience.  Both are subjective.

This reminded me of the terrible trouble I got myself into with respect to the "Reverse Monty Hall Problem".  I went into it with an assumption which, for one reason or another, had become implicit rather than explicit and which then became forgotten, hidden and/or overlooked.

There are a lot of articles connected with the one I've provided a link to, so I'll try to distil it down to essentials.

If we don't know, and have no way to know, the probability of a particular claim or premise in terms of it being true or false, we are required by the principle of indifference to assign it a notional probability of 1/2.  If we know more, for example that there are more options, say:

A is true, B and C are false.

B is true, A and C are false.

C is true, A and B are false.

One of A, B and C is true.

Then, knowing nothing else, we are required to assign a notional probability of 1/3 to A, B and C.  We don't know anything that makes A more or less likely than the others.

Say then, that we have an urn in which there is an arbitrarily large number of balls that are identical in size and that is all we know.  We draw out all but one of them (say 999 of them), and they are all white.

What is the likelihood that the last one is also white?  If we know nothing else, then the answer is one in a thousand, the same likelihood of a single non-white ball being placed in any specific position in a sequence of one thousand extractions.

What was the likelihood, after having drawn 99 balls, that the 100th was non-white?  The same logic applies and it was one in one hundred.  As more white balls were drawn, the likelihood of a non-white ball being drawn went down.

Now, compare this to the likelihood of drawing a non-white ball as assigned by someone who has one more piece of information, the knowledge that the balls were initially drawn from an enormous barrel in which there were 900,000 white balls and 100,000 black balls.

Unlike us, who had no additional information and could only work on the basis of what balls we had drawn out, this other person will know that there is an increasing likelihood of drawing a black ball after each white ball is removed.  In fact the likelihood of the 1000th ball being non-white after a sequence of 999 white balls is very slightly higher than one in ten.  The likelihood of drawing 999 non-white balls in a row is extremely low, but that is immaterial, since we are only looking at the likelihood associated with the next draw once this extremely unlikely scenario has already played out.

We can fiddle with the figures to make it more explicit.  Say we only know about 999,999 balls that we've drawn, over a period of a couple of boring days.  All of them are white.  We have to say that the likelihood of the next ball being non-white is one in a million.

But if our more knowledgeable friend knows also that there is one black ball in the barrel, then she will have to say that the likelihood of the next ball being non-white is one in one, 100%.

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My point here is that we have evidence, and we might also have assumptions.  The assumptions that we make about the distribution of balls in the urn will change our assessment of the likelihood of a non-white ball being drawn.  If we characterise our (potentially false) assumptions as "knowledge" - as theists are often wont to do - we will consistently misjudge the likelihood of our premises (and subsequent conclusions) being true.

Add to this the possibilities that we don't consider (i.e. thinking only of A being true or false, rather than factoring in other possibilities like B and C), then we end up with very little likelihood of reaching reliable conclusions.

For Your Information

There are arguments for the existence of god that rely on the notion that the only source of ((brand) new) information is a mind.  There are, of course, problems with these arguments, foremost of which is equivocation.  Information means different things to different people and in different contexts.  Pretending that the term means one thing only and gliding from one meaning to another permits all sorts of self-deception on the part of the apologist.

I want to try to explain (and dismiss) one example of this apologetic trickery, but to do that I have to make it quite clear what information is and isn't.

When an apologist, like Max Andrews, provides this equation:

I(m)=-log_x(P(m))

or one like it, it is clear that he is talking about a different sort of information than he is when using a variation of this equation:

P(A|B)=P(B|A).P(A)/P(B)

and he talks about the evidence B in terms of "background information".  (He actually splits the "evidence" into two parts, one that he calls "evidence" (call it E) and the other that he calls "background information" (call that B).  This is something we've come across before.  The effect is, as I have argued, to make P(A|(B&E)=P(A).  In other words, you get nothing from the faffing about with impressive-looking probability equations other than some mild confusion.)

The Shannon entropy equation refers to the information content of m, which could be a message, and is basically a measure of how unlikely m is - which means you need to know what values m may take.

I'll try to make this concept a little more concrete.

Say that a father is away on a business trip and sends a message back to his teenage son: "send socks".  Even before we try to quantify the amount of information in this message, we could be safe in saying that it isn't enough.  How to send them (post, express mail, courier pigeon?) isn't detailed.  When to send them isn't detailed.  Quantity is not detailed.  And the teenager might be quite justified in asking "Which socks?" - unless of course the father only owns one type and colour of sock, all of which are of equal quality (either all with holes in them or none with holes in them).

There's a subtle linkage between how much information is in the message and how much information is required.  For example, this message could contain one single bit of information for the son, a request to transition to the "send state" after having previously been in a "no-send state".  There could have been an existing agreement as to how, when and what to send based on the receipt or non-receipt of this short message.  So the knowledge state of the recipient has an impact on the amount of information being conveyed.

If the father had written "Send one new, unused pair of ankle-high blue socks with yellow stripes, located to the left in the top-drawer of dresser, via express post, immediately" and this course of action had already been agreed, then no more information would have been conveyed than in the message "send socks".

But imagine that everything had been agreed apart from precisely which socks had to be sent.  Imagine further that the father had two types of socks, black sock and brown socks.  Adding the word "brown" to the message adds one bit of information to the message because there is a range of two possible valid messages "send black socks" and "send brown socks".  The more possible valid messages that could be sent, the more information is contained in the message "send brown socks".  We can think of that in this way, the message "send brown socks" also contains the information "don't send black socks".  If the father has a wide range of socks, then the message "send brown socks" also contains the information "don't send black socks", "don't send pink socks", "don't send orange socks", etc, etc.

And it's here that information can be thought about outside the context of sending a message.  Say we were in the father's bedroom standing in front of his dresser with the top drawer open.  We reach in and take out a pair of socks, we identify them as black.  Given that we know that he has only black and brown socks (and that we have assumed, due to the principle of indifference, that he has equal numbers of black and brown socks), then we know that the probability of having selected a pair of black socks was P(Black)=0.5 and so the information associated with that selection was I(Black)=-log₂(p(Black))=1bit.  If there were 20 different types of socks in the drawer, the probability of having selected a pair of black socks would have been P(Black)=0.05 and we'd have I(Black)=-log₂(p(Black))=4.3bits of information.

A standard CD can hold 847MB of data, with 110MB of that being set aside for error correction.  This is close to about 6.8 billion bits which means that the information on a CD is equivalent to uniquely identifying one pair of socks out of 10^2000000000 pairs (847MB is equivalent to about 2GBan).  Or one star out of 10^2000000000 stars (although there are only about 10²¹ stars in the observable universe).  Or one grain of sand out of 10^2000000000 grains of sand (but there are only about 7.5x10¹⁸ grains of sand on Earth, meaning that if every star had two temperate rocky planets as sandy as the Earth (I'm saying that Mars is sandy and ignoring that it's smaller than the Earth, so probably has less sand), then there'd be about 1.5x10⁴⁰ grains of sand in the universe).

(This is probably why Max Tegmark feels confident enough to suggest that the initial data in the universe could have been so simple as to be contained on a single CD-ROM.  Although he immediately suggests that even the CD-ROM might not be necessary - in accordance with his suggestion that the universe (or multiverse) overall may contain hardly any information at all.)

A complicating factor here is the one that probably trips up overly excited apologists such as Max Andrews.  Selecting one specific grain of sand from the universe, using the assumptions above, represents a rather unimpressive 16.558 bytes of data (and I'm overselling it slightly since 256^16.558=7.509955x10³⁹, meaning that I am being cavalier and suggesting that it doesn't matter if I add about 10³⁷ additional grains of sand to universe … about half the number of grains of sand that would exist on as many Earths as there are grains of sand on Earth).  But if we didn't care about the specific grain of sand and just wanted a grain of sand, the information in that grain of sand would be much, much less.  The less specified the selection is, the less information there is in that selection.  There's still some information in the selection of "grain of sand" from the range of things that could be subject to this process, but the game "20 questions" demonstrates just how many different things can be uniquely identified with a sequence of 20 carefully framed "slice questions" - a little over a million.  Double the number of questions and you can uniquely identify 10¹² different items … with precisely 5 bytes of information.

Going back to the sock drawer, if the father just wanted any old pair of socks, then the probability of drawing them from the sock drawer would be Pr(socks)=1.  (Note that I am assuming here that there are socks in the sock drawer, which may not always be the case, but the point at the moment is about the probability of selecting a pair of socks from a set of existent socks, not about the probability of the socks being existent.)  The information in this is precisely zero because log_x(1)=0, irrespective of the value of x.

In other words, the more you don't care about the result, the less information there is in that result.  This, hopefully, makes intuitive sense - think about the difference between the news that Khloe Kardashian has been on a new diet and the news that Prince has died (Purpleness Be Upon Him), both reported in April 2016.  One event led to widespread dismay and spontaneous demonstrations of grief on the streets of Minneapolis and the other one I only noticed because I was momentarily delayed in a supermarket checkout aisle and it was easy 2 see (it) on the cover of a magazine.  Reports of Prince (PBUH) dying contained information.  Reports of Khloe getting marginally less fat is so lacking in information that I lack the care factor necessary to

And this is where the argument of people like Max Andrews becomes circular.

We, as part of the universe, do care about the fact that we exist.  Therefore, for us, this is interesting information.  We are also part of a selection event, in so much as the universe does appear to be fine-tuned, in that were certain constants and laws slightly different then life like ours would not be possible.  We can imagine situations, or alternate universes, in which we would not exist (counterfactuals), and from this glean "information".  For example, if the strong nuclear force were stronger by more than one part in 50 (and all other constants were held at existing values), then diprotons would be stable leading to the failure of stars to form (due to the rapid consumption of all hydrogen in the first minutes after the Big Bang).  So we can posit a universe in which the strong nuclear force was stronger by one part in 20, or twice as strong, or four times as strong, or seven times as strong.  And we could keep going, infinitely, providing different multiples and then claim that just this one single fact contains infinite information.

(A problem here, of course, is that we don't know that all of these values are valid, let alone equally valid.  And there was a caveat, "all other constants were held at existing values".  There are solutions in which the strong nuclear force may be higher and other constants being at different values still allows for unstable diprotons.  The bottom line is that we don't really know how unlikely it is that the strong nuclear force lies in the convenient range that it does, and "we don't know" is equivalent to "no information".)

But when it comes to the existence of the universe, it doesn't really matter how interested the constituent elements are in it.  It's a question of how interested the universe is and, unless that is a mind, it's not a huge leap to suggest that the universe isn't interested at all.  In which case, there is no information in the universe as a whole.

What appears to be a novel argument on the part of Max Andrews is that because there is so much information in the universe, and minds are the only originators of "brand new information" (this part is a standard creationist claim, usually centred on genetic speciation), then there must have been an antecedent mind to have initiated the universe (or multiverse).  But this is circular, the universe as a whole only has information if the universe itself is a mind or there is a mind external to the universe that cares about what is inside it.  (The universe (or multiverse) would also have to be contingent, meaning only the universe/multiverse could have been different.  There's no information in a necessary thing being as it necessarily must be.)

So, unless he is suggesting that the universe itself is a mind, Andrews is presupposing that there must be a god of some sort, that therefore the universe contains a huge amount of information (presumably due to extreme contingency of it) and therefore that there must be a god as the source of that huge amount of information.

This is completely circular and therefore totally useless as an argument for what he so desperately wants to argue.  It's amazing that this got accepted as a thesis, but I guess that when both of your supervisors are dedicated god-botherers then so long as your answer is "god did it", you can get away with any old nonsense.

The King of Spades and the Strong Anthropic Principle

I was introduced to the works and thoughts of Max Andrews by Skydive Phil.  In this, and at least one other article, I'll be addressing some issues that I have with Max' published take on the multiverse and fine-tuning.  Note that unlike some others that we won't mention, Max is upfront about being theist (and Christian) but, as is the case with Jeff Zweerink, his support of fine-tuning as an argument for god has faded of recent times.

It should be noted that what I address below was written quite some time in the past and does not necessarily represent Andrews' current position.  Therefore, I am not arguing against Andrews per se, but rather against a position that he no longer appears to hold (at least not in entirety).  Keep in mind that it's quite likely that there are apologists out there who would be more than happy to call upon the 2013 version of Max Andrews in support of their argument for god, while totally ignoring the fact that the 2016 version of the same person no longer agrees with everything that the 2013 version had to say.

For an insight into Andrews' most recent position (at time of writing), listen to his appearance with Luke Barnes (yes, that Luke Barnes) on Justin Brierley's Unbelievable show.

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Max writes at his blog, in the article Do Multiverse Scenarios Solve the Problem of Fine-Tuning?:

The role probability serves in this argument does not favor the non-fine-tuning hypothesis (either chance, necessity, or a combination of the sort) in multiverse scenarios.  If the objector to a fine-tuner argues that the odds of having a finely tuned universe, which harbors life, increases given the vast number of universes there is bound to be one with the values we have.  This is an abuse of probability and commits the gambler’s fallacy.  This claim assumes a general disjunction rule of probability.  For example, this rule of probability suggests that the probability of drawing a king from a deck of cards increases when each card drawn is not replaced.  If the deck has all the cards it is supposed to have the probability of eventually drawing a king is 1.  The multiverse is like the restricted or general conjunction rule of probability. By simply increasing the number of possibilities, one does not increase the probability of selection.  For example, say you randomly draw a card from the deck and you want the king of spades.  The odds of you drawing a king of spades are 1/52.  Say you draw the three of hearts.  When you replace the card and draw another card from random the odds of you getting the king of spades is not increased by the first selection.

What Max writes about cards is true, but it misses the point.  A multiverse is more like multiple decks of cards.  With each deck there is a 1/52 chance of drawing the king of spades from the top - assuming that they have all been randomised by thorough shuffling such that there is an equal likelihood of any particular card being on top.  This is equivalent to using the one deck of cards, selecting a card at random, replacing it, and then selecting a card at random again - so long as we use a probability measure such that each of the 52 cards has an equal likelihood of being selected (this comment is intended to pacify mathematicians, just in case they are reading).

But while each random draw from a single deck and each draw from the top of a randomised deck has a 1/52 chance of resulting in a king of spades, across all the draws, the chance of one of them resulting in a king of spades is higher.  By the time you have drawn 1066 cards, using either method, there is one about a 1 in a billion chance that you have not drawn a king of spades (by the way, I am not implying that William the Bastard [aka William the Conqueror] was a dastardly king of spades, the 1066 figure is just a happy coincidence).  Draw 10000 cards and that becomes about 2 in 10⁸⁴.

In other words, as the number of universes in the multiverse increases, the likelihood of any unlikely event happening in at least one of them also increases.  If there is an infinite number of universes, then anything that is at the very least possible must happen.

Interestingly, this lends credence to a formulation of the Strong Anthropic Principle that I don’t normally favour.  The argument goes a little like this (per Carter):

(at least one) universe (and hence the fundamental parameters on which it depends) must be such as to admit the creation of observers within it at some stage

We know that life is possible (if you doubt this check your pulse, if you don't have one, try looking out your window and you'll probably see either a tree or some grass and possibly both).  If there is an infinite number of universes then we know that life must therefore be instantiated in at least one of them.  And it would appear that we are in one of them.  Therefore, life becomes less miraculous, it's just something that we'd expect.  (This is the case even if there is a finite but very large number of universes.  Andrei Linde and Vitaly Vanchurin calculate that there are in the order of ten to the power of ten to power of ten to the power of 7 (10^{10^7)}) universes.  While this might not make very very very very very unlikely (yet possible) things certain, the odds of possible things that are merely very very unlikely happening are pretty high.

This seems a little circular, which is why I don’t give it too much credence, but it does seem to speak rather strongly against many probability-based (ie appeal to ignorance) arguments for the existence of god.

If there is an infinite or extremely high number of universes …

WLC Does an About Face

Who would have guessed it?  William Lane Craig has, despite his appearance of overweening arrogance, admitted to being wrong.  You might have to read the article carefully because otherwise you might not see it, but he has come to the conclusion that his use of statistics has been inadequate.

In many of his arguments, Craig works from the principle that while we might not know with total certainty that each of his premises are not true, they are individually "more likely than not" and that, as a consequence, the conclusions reached as also "more likely than not" (this to me appears to be borrowed from Plantinga, but maybe it's an idea that infests apologetics as a whole).  [I address related issues in Planting a Demigod and Planting a Tiger.]

In his own words:

… that raises the further question of what qualifies as a “good” deductive argument. I take it that a good argument is one whose conclusion is shown to be more plausible than not. So under what conditions is an argument good? As you note, I have long said that in order for a valid deductive argument to be a good one, it suffices that each individual premise of the argument be more probable (or plausible) than its contradictory.

What Craig has written here should be parsed carefully.  Note that he uses the phrase "more plausible than not" and then implies that "probable" and "plausible" are interchangeable.  The problem is that these terms are not interchangeable, especially not in the context in which they are used and Craig appears to equivocate between use of the term as meaning "probable" and use as meaning "able to be believed" (that is, not totally impossible).

Then there is a question regarding precisely what Craig means by "probable" when he uses that term (or "plausible" in its stead).  I would suggest that when he talks about a premise having a probability of X, he means that there is a probability X that the premise is true.  Say we are tossing a fair coin.  The probability that we will get a head in any individual toss is Pr=0.5 - and the probability that it is true that we have tossed a head (assuming that I can't see the result) is Pr=0.5.  The premise "we have tossed a head" doesn't really have a probability, but the statement "it is true that we have tossed a head" does.

(We get around this in logic by assuming that any statement is also a statement to the effect that the statement itself is true.)

I think that it might be useful to introduce a new term: "assuredness".  Assuredness is the probability that what you believe to be true is actually true.  This might seem to be a nugatory term, but I hope to demonstrate that it is actually useful when considering probabilistic syllogisms.

Say we have a scenario in which Ted, our research assistant, draws a ball from an urn.  There are two balls in the urn, a black ball and a white ball.  If he draws the black ball, he picks up a fair coin and tosses it fairly.  If he draws the white ball, he does something completely different, so long as it doesn't result in a coin being tossed (he can toss a die, or slap his own face or sing the national anthem in his underpants, we don't know and to some extent we don't care as long as he doesn't damage our laboratory).  From this we can draw the following probabilistic syllogism:

(It is true that) if Ted tosses a coin, a head will result (Pr=0.5)

(It is true that) Ted tosses a coin (Pr=0.5)

Therefore, (it is true that) a head will result (Pr=0.25)

Note that I've constructed this scenario so that the fairness of the coin (making it 50-50 that a head will result from a fair toss) is totally independent of the process of selecting a ball from an urn (which makes it 50-50 that Ted will toss the coin).

A question that can be raised about the syllogism above is … what does the remaining 0.75 represent?  Keep in mind that we are primarily interested in whether a head results.  A table might help (note it has to be reordered so as to be chronological):

Coin tossed (Pr=0.5)		No coin tossed (Pr=0.5)
Head (Pr=0.5)	Not Head (Pr=0.5)	Not head (Pr=1.0)
Head (Pr=0.25)	Not Head (Pr=0.25)	Not Head (Pr=0.5)

Quite obviously then, Pr=0.75 is the probability that a head does not result.

This Pr=0.75 figure results because the experimental protocol prevented Ted from getting a head unless he drew a black ball.  But we can change this.

Let's say instead that if Ted draws a white ball, takes up and tosses a biased coin, which he then tosses as if it were a fair coin.  Say further, that we don’t know what the bias is on the coin.  The results change significantly:

Coin tossed (Pr=0.5)		Random biased coin tossed (Pr=0.5)
Head (Pr=0.5)	Not Head (Pr=0.5)	Probability of result unknown (Pr=1.0)
Head (Pr=0.25)	Not Head (Pr=0.25)	Probability of result unknown (Pr=0.5)

This is where "assuredness" comes into its own.  The probability of a head resulting is no longer Pr=0.25, but rather Pr=0.25+0.5xPr(Head|Biased Coin) - and we don't know what Pr(Head|Biased Coin) is.  This means we only know the interval in which the probability of a head lies: [0.25,0.75].  There is, therefore, a lower bound on the probability of a head, namely 0.25 and it is this that I want to associate with "assuredness".

We can be 25% assured that the coin tossed will show a head.  We can be 25% assured that the coin tossed will not show a head.  The remaining 50% is an ignorance interval - we simply don't know what the associated probability is.

For Craig's purpose, I suggest that this lower bound, this assuredness, is all that he can use to support his argument because it is not reasonable to base his argument (even if in part) on ignorance.  If he can only state that his premises are more probable (or plausible) than not, then he is saying little more than each premise has Pr(true)=0.51.  This means in turn that his assuredness in a simple syllogism is 26.01%.

It is certainly true that, if he were able to raise the "plausibility" of his premises such that Pr(true)>0.71414, then his assuredness would rise to 51%.  But he would have to provide good argumentation for that increased "plausibility".  I do note that he makes the claim that the probability of some of his premises approach unity.  I note the claim, but I note also that it's a bald claim with little to back it up other than Craig's confidence that he is right.

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Anyway, as mentioned, Craig has now accepted the error of his ways.  Although, he probably hasn't.  Now he just claims that the conjunction of his premises has a probability of at least 51%.  (It's unclear how much ignorance is supposed to contribute to this figure.)

There's a problem with this approach though.  Isn't one of the most important, in fact the only, relevant conjunction of premises in a syllogism a little something that we like to call the conclusion?  This is the happy conjunction when both (or all) the premises are true.  (There might be lesser conjunctions, but those would likely be subsets of the conclusion anyway.)

It seems to me here that the first thing Craig has done after doing his about face is to fall on it.

There Probably are Bayesians in Foxholes

Many years ago, I almost fell into bad company.  I frequented a philosophy channel on IRC that included a small, but very vocal band of what I like to call Randians – primarily because they didn’t like being called Randians.  These Randians preferred to be called objectivists and apparently their greatest love in life was sticking it to subjectivists.  Boy, did they hate subjectivism!

(Why Randians?  Because they were devotees of Ayn Rand and her philosophy of "objectivism".  I have, since that time, had an abiding distaste for Ayn Rand and anyone who adores her.  I do understand that in some instances a thoroughly reviled person like Ayn Rand may be misunderstood, or misjudged, and there is the possibility that, despite appearances, such a person does in fact have redeeming features.  For example, they might like puppies and kittens.  Ayn Rand, however, delighted in ripping the legs off puppies and kittens and throwing them at small children that she had cast into a deep well at the bottom of her garden.

If I were given the choice of spending eternity with Ayn Rand or William Lane Craig, I would choose laughing boy, despite being aware that the notion of spending eternity with anyone would indicate that he was right about the whole god thing and he would undoubtably spend some excruciating proportion of that eternity crowing about how he was right and I was wrong.  But at least we would both be in hell.)

The odd thing was that, at the time, when I searched for these subjectivists, they didn't seem to exist.  The Randians appear to have constructed an army of straw men to attack in their objectivistic fervour.  Sure, anyone who crossed them would get labelled as a subjectivist, but this term appeared to be more pejorative than accurate.

A similar sort of one-sided battle appears to be underway between Bayesians and frequentists.  Now perhaps the baying of the Bayesians is a little more accurate than that of those objectivists, perhaps there are people out there carrying the torch for frequentism, but I've not seen any evidence of it.  It seems to me that in some instances a frequentist interpretation of probability is appropriate and in other instances a Bayesian interpretation is appropriate.  See the second last page of this – note however, that the author is a statistician, the sort of person who uses probability all the time.  In other instances, such as the on-going cat fight between Luke Barnes and Richard Carrier, the participants of a recent probability-centred spat are not statisticians – they are a cosmologist and a historian.

What truly boggles the mind is the fact that Barnes has recently devoted an entire post to lambasting Carrier on what started out as a response to short statement from Jeff Lowder in support of a criticism from Barnes, all predicated on a single word.  Carrier wrote (in his essay in "The End of Christianity"), my emphasis:

Bayes’ theorem is an argument in formal logic that derives the probability that a claim is true from certain other probabilities about that theory and the evidence.

To say that Bayes' theorem is an argument is possibly a bit of an awkward way of putting it.  It could, possibly, be written as an argument in formal logic, in much the same way as 1+1=2 was by Russell and Whitehead, whose work Carrier references, but that's not really how Bayes' theorem is thought of.  So, Carrier's claim is not worded particularly well.  No big deal.  Barnes however leapt gleefully onto that fact, spending some considerable time in savaging it and Lowder later concurred that Carrier's wording wasn't completely accurate.

Sadly, rather than 'fess up to having (at least) one sentence in his essay that was a tiny bit stilted and moving on, Carrier gracefully conceded that Bayes' theorem might not be an argument per se, but then went on to claim it is the form of an argument (after having claimed, apparently off the cuff, that one simple derivation of Bayes' theorem is the derivation of Bayes' theorem rather than a derivation).  He subsequently went on to point out that the issue is not so much the validity of Bayes' theorem (which no-one appears to be contesting) or the formulation of Bayes' theorem (which, again, no-one appears to be contesting), but rather the issue is what may be input to the equation that is the expression of Bayes' theorem.

In effect this was saying "any argument about Bayes' theorem, the derivation of Bayes' theorem or the description of Bayes' theorem is moot, because the argument isn't about Bayes' theorem itself but rather about what we bring to Bayes' theorem".

So, did Barnes pick up on this implied appeal to stick to what is relevant and not get bogged down in irrelevant detail?  Of course not.  His latest (and perhaps last) attack on Carrier is focussed, laser sharp, on Carrier's use of "the" rather than "a".  In the conclusion, Barnes challenges Carrier to release a new variant of probability or, in effect, choose a side: "Bayesian" or "frequentist".

What I find even more absurd is Barnes' statement in the comments:

(Carrier)’s not a frequentist. Frequentists don’t believe that prior probabilities exist, but Carrier does.

What?  Are alternate interpretations of probability now to be considered as competing ideologies?  Here Barnes appears to either have forgotten what he wrote in his piece 10 Nice things about Bayes' Theorem or he is accusing these mythical frequentists of being complete morons.

Prior probability is a defined term.  It's not something like climate change, free will or the Loch Ness Monster – it's not something that you can really question the existence of (complete morons aside).  Perhaps it's a term that you can, under certain circumstances, question the utility of – like the term "European" (do you mean people who have some combination of the SLC24A5, SLC45A2 and HERC2/OCA2 genes or people who currently reside on the continent of Europe, or people who define themselves as members of the European Union, or something else?) – but you'd be crazy to deny that prior probabilities ever exist.

It seems completely bizarre to me that there are these mad keen Bayesians, apparently snug in their foxholes, taking the occasional pot-shots at their enemies, the dastardly frequentists, totally oblivious to the fact that – in the right circumstances – these "frequentists" would be more than happy to utilise a Bayesian interpretation.  It reminds me somewhat of those theists who become defensive at the mere mention of even the mildest expressions of atheism, as if the lack of belief on the part of one were necessarily a full-scale, frontal attack on the other.  (Note that this is a position which only encourages some of the more mildly oriented atheists to move towards the militant position.  It basically becomes a question of self-defence.)

Oh, hang on.  These people, the manic Bayesian defenders and the aggressively defensive apologists, they seem to be one and the same.  What was the likelihood of that!

Anyway, Barnes has stated that he might no longer be attacking Carrier.  I don't know if that will be the case though.  I think I identified a misplaced comma in one of Carrier's latest sentences and that could be strung out into a couple of a rants.  Don't you think so, Dr Barnes?

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(Yes, I do appreciate the irony in that final comment, since I have written numerous articles sniping at Barnes, spinning dross from essentially nothing.  But I am small fry, a mere gnat, a bit-part player while Barnes and Carrier are more substantial actors.  That's not to say that my inconsequentiality renders my criticisms invalid though, even if Barnes may well choose to ignore them.

And, again, yes, I do realise that I have a sample size of one here, regarding the correlation between Bayesian defenders [or at least attackers of frequentism] and apologists.  I just provided an example, that was not intended to be conclusive evidence in support of my case.  It happens to be a nice coincidence that the anti-frequentism argument is provided here in the context of a defence of fine-tuning as an argument for god.)

Lucky Luke and the Twenty Royal Flushes (Another Parable)

Little Luke Junior (also referred to as "Cool Hand" Luke after that incident with the ice cream) loved nothing more than to sit at the knee of his father, the legendary "Lucky" Luke, and have him recount the story of the day he had dealt twenty royal flushes in a row.

Lucky Luke would start with rambling stories about his past adventures, how smart his horse at that time was, how stupid his dog was, how it would still be years before he and Luke Junior's mother would meet and how, even to this day, he sometimes misses the first cigarette of the day (usually accompanied by an attack of coughing and spitting blood).  Then, once warmed to his topic, he'd talk about how, all those years ago, he met a gambling man whose penchant for evil eclipsed that of Black Bart (who was actually a nice gentleman who just happened to commit a stage-coach robbery here and there).

This character, "No-Eggs" Oliver, had little going for him except his reputation of meaningless violence and OCD.  His final obsession, the one that brought Lucky and No-Eggs together, was to see 20 royal flushes dealt in a row.  He rode up and down the country, visiting saloons, bordellos and gambling dens, challenging people to deal 20 poker hands from 20 individual packs of cards.  He rode quickly and, quite frankly, had nothing else to do with his time, so he extended this challenge to millions upon millions of people.

The challenge went a little like this: "Shuffle these 20 packs of cards and deal a royal flush from each.  If you fail, I shoot you in the head.  If you don't try, I shoot you in the head.  If you cheat, I shoot you in the head."

So, naturally, they all tried, and they all failed.  Until "Lucky" Luke (who until that time was only known as Luke).  He tried and he succeeded.  No-Eggs, who was getting rather old by this time, having killed millions upon millions of unsuccessful croupiers, was so astonished that, despite a lifetime spurning eggs due to the possibility of cholesterol and the suspicion that even "free range" eggs might not really be free range, suffered an enormous cardiac arrest and died on the spot.

At this point, little Luke would pipe up, claiming that the odds of dealing 20 royal flushes in a row are so incredibly unlikely that his father must have cheated – and No-Eggs simply hadn't noticed.

At this point, Lucky Luke would say: Ah, my son, it might look to you as if I had cheated, but the whole point of this story is that if I had not dealt those 20 royal flushes, you would not be here to accuse me of cheating.  An alternative way of looking at it is that many a man (and perhaps lady) died at the hands of No-Eggs before I succeeded in dealing 20 royal flushes.  For all we know, No-Eggs, with his uncontrollable compulsion and amazing fortune in evading the law with regards to his card-related homicides, could have kept going until someone else dealt 20 royal flushes.  It's possible that someone else had already succeeded, and his son or daughter also accuses him (or her) of having cheated.  It's even possible that No-Eggs hadn't really challenged anyone before, that I was the first, and that I am merely phenomenally lucky, as my nickname suggests.  Either way, cheating or not, the result is that I survived my encounter with No-Eggs and I was thus able to meet your mother some years later.

It would appear, my son, that it is only because you are already so totally committed to the idea of my having cheated that you fixate on cheating as being the singular reason for my success in dealing the royal flushes.  But consider this:

The meeting between myself and your mother was unlikely.  There are millions of people in this country that we could have met up with, but it transpired that, one day, we just happened to be in the same place at the same time and we noticed each other and we liked what we saw and we had the time and inclination to act on that mutual attraction.  That attraction lead eventually to, ahem, shall we say "intimacy" and one sperm of the approximately 525 billion that a human male produces over a lifetime managed to make its way to one 400 (out of the 400 thousand potential) eggs produced by your mother over her lifetime – which necessitated, ahem, intimacy at precisely the right time for that particular sperm and that particular egg to meet.  Then environmental conditions had to be just right to switch on and off the genes that were needed to make you.  And this incredible good fortune stretches back over thousands of generations, hundreds of millions of generations in fact, if you include the species from which humans evolved.  So, Junior, you are an incredibly unlikely result, even if I may have cheated at cards.  Surely you aren't suggesting that these unlikely events wer all orchestrated merely so that you, "Cool Hand" Luke, would be born – or are you suggesting that we, your parents, somehow cheated to bring you about?

To this little Luke would reply that he doesn't really like the back story involving multiple attempts at dealing twenty royal flushes (although he would strenuously deny that this objection is largely due to the conflict with his preferred narrative), and that the disturbing discussion involving parental intimacy and sperm was even worse.  So, therefore, he would simply ignore it all.

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What the hell is all this about?  Good question!

Look here.  Now, admittedly, Barnes is talking about sitting down and dealing 20 royal flushes in a row, the likelihood of which is staggeringly unlikely.  Little Luke is indeed justified for thinking that his father probably cheated, it's the by far most likely option in the scenario as described – even if "No-Eggs" Oliver had had the time and inclination to challenge billions of people, rather than just millions.  But the point is that the fact that Luke Junior exists at all is evidence that 20 royal flushes were dealt – even if it is true that Luke Senior didn't cheat (and so long as Luke Senior didn't just create the story out of whole cloth).

However, in this scenario, cheating is most definitely an option.  Another option is magic, if we are inclined to distinguish that from more mundane forms of cheating such as palming or counting cards.

What would be interesting to know is, if we could somehow eliminate illusionist-like cheating (say that "Lucky" Luke's feat of dealing 20 royal flushes was recorded in high definition video from multiple angles and could be scrutinised by Penn and Teller and however many other professional illusionists that are thought to be necessary), would Barnes plump for magic as being the most likely explanation?

Because, after all, this is the real equivalent to the NID option that Barnes appears to favour.  (NID means "non-terrestrial intelligent design" for those who haven't followed the Carrier-Barnes squabbles over the past couple of years or so.)

A cosmologist who puts his money on magic?  Hm, now that is a guy I'd like to play poker with.

Fine-Tuning a Return to Sweet Probability

In A Return to Sweet Probability, I argued that the resurrection does not increase the likelihood of William Lane Craig's god, largely because the credibility that you give reports of the resurrection depends very heavily on your predisposition to believe that his god (or some very similar god) exists.

I want to do something similar with the fine-tuning argument.  I'll be using a variant of it implied by my old buddy, Luke Barnes (also referring to this earlier comment):

Let

o = intelligent observers exist

f = a finely tuned universe exists

b = background information.

NID = a non-terrestrial intelligent designer exists.

…

I can admit that “p(finely tuned universe | observers exist) = 1” and still conclude that

p(NID | f.o.b) >> p(~NID | f.o.b)

Now this argument between Richard Carrier and Luke Barnes was about Bayesian probability, so we know that:

P(NID | f & o & b) = P(NID & f & o & b) / P(f & o & b)

Holding f and o together (for reasons which should become apparent shortly) this eventually becomes:

If you want to see the working for this, look at A Return to Sweet Probability.

Now we need to define our terms.  I think it’s reasonably clear what the term o means, the intelligent observers in question are us.  By “intelligent”, we are not implying that the universe in question must be crammed with Albert Einsteins.  We merely need to be intelligent enough to observe the universe and note that the universe contains observers who are intelligent enough to observe their own existence.  Tick.

The term f is a little more contentious.  I’m going to use the definition that Carrier uses (see Part Deux), in part because it’s something that Barnes appears to have consistently overlooked:

If fine tuning is necessary for life, and there is no God, then necessarily life will only ever exist in correlation with fine-tuning. This is because all universes without fine tuning (sic) will thus by definition not contain life.

Note that this does not say anything about the hypothetical case in which there is a god.  A god may choose to fine tune the universe for life, or it may choose to magically create and sustain life even in universes which are not at all tuned for life, or it may choose to create a universe in which the range in which life is possible is very wide indeed.  Carrier appears to be conceding a point to apologists (such as Barnes) for the sake of the argument.  The fine-tuning argument will only work in a universe which is apparently fine-tuned for life, if there are observers in a hypothetical universe which is not fine-tuned for life then the fine-tuning argument fails (and, hypothetically, the pseudo-scientific, apologetic denizens of that universe will deploy the “non-fine-tuning argument” for whatever god they have imagined into existence).

So, in the instances in which there are observers, the universe will either be fine-tuned (either naturally or as the result of the intervention of a god) or non-fine-tuned (as the result of the invention of a god).  There is the possibility that some universes will be fine-tuned for life but, for some reason, life doesn’t manifest.  I don’t think we are particularly interested in those universes but in any event, those universes won’t have observers.  I think we can get around this scenario by strengthening the concept of fine-tuning – if a universe is sufficiently fine-tuned for intelligent life, then there will be, at some point, intelligent life.  Any tuned universe which doesn’t manifest intelligent life simply isn’t sufficiently tuned to consider it fine-tuned.  It certainly could be excluded from the set of universes that our universe fits into (also known as the reference class – that is: what is it about our universe that is pertinent to our consideration in this instance?  It would appear to be the fact that it contains not only life, but also intelligent observers).

So fine-tuning is defined as the sort of fine-tuning in our universe, that may or may not have been due to the intervention of a god, and resulted in intelligent life, life that was intelligent enough to observe that the universe is sufficiently fine-tuned to produce intelligent observers.

This is an extremely long and convoluted way to say what Carrier had been saying, P(o|f)=1 and, therefore, P(f&o)=P(f).  Or, in other words, once you have fine-tuning (as defined) you necessarily get observers, so the probability of fine-tuning and observers resolves down to the probability of fine-tuning.  This allows us to change all the (f&o) terms to f:

Perhaps you might want to weaken the definition of fine-tuning, but I don’t think that apologists want to.  If they do, I think they merely weaken their fine-tuning argument as a consequence but that can be an argument for another day if any apologist wishes to take that route.

We know that NID is the “non-terrestrial intelligent designer”.  I’m not as coy as some others, so I am just going to call this the god of people like William Lane Craig, Plantinga and (almost certainly) Barnes (but certainly many of Barnes’ adoring fans).  For the sake of the argument, let’s change NID to R as in “cReator” or “Representing god” or “what the apologists are Really arguing for”.

Fine-tuning (f) is, of course, the evidence in this argument, so let’s call it E.  If you are reluctant to roll o into f per the argument above, we could instead say that E=(f&o).

Then there’s our background, b (which we will capitalise and call B).  Background is a little vague in its definition.  What exactly constitutes background?  Fortunately, we can rely on Barnes again, who has written a piece on precisely this topic.

Background is everything we know, with the exception of the evidence that we are currently looking at, so in this case everything we know with the exception of fine-tuning (and thus observers).  However, this in effect resolves down to everything we know that is relevant.  I suspect that we have to be very careful about doing this step manually – because that which is relevant might not be immediately obvious and eliminating relevant data will skew the result.

I think we have enough to be getting on with.  We’ve renamed NID, b and f (or (f&o)) to R, B and E, respectively, so we have:

For anyone who has read A Return to Sweet Probability recently, this equation should be familiar.  On the right hand side of this equation we have four terms to consider:

P(E|R&B) – this is asking us: Given the hypothesis that there is a cReator, and our background information (everything we know, bar fine-tuning), what is the likelihood of our evidence (fine-tuning)?

P(R|B) – this is asking us: Given our background information (everything we know, bar fine-tuning), what is the likelihood of there being a cReator?

P(E|^R&B) – this is asking us: Given the hypothesis that there is not a cReator, and our background information (everything we know, bar fine-tuning), what is the likelihood of our evidence (fine-tuning)?

P(^R|B) – this is asking us: Given our background information (everything we know, bar fine-tuning), what is the likelihood of there not being a cReator?

On the left of the equation is P(R|B&E), which asks us: Given our evidence (fine-tuning) and our background information (which is, cumulatively, everything we know), what is the likelihood of a cReator?

The value of this equation relies very heavily on two questions having already been answered, namely the likelihoods of there being a cReator and not being a cReator, given what we know, apart from fine-tuning.  It also relies, rather heavily, on the assumption that fine-tuning doesn’t follow necessarily from what we already know.  If we were, in the future, to discover that B→E (if B then (necessarily) E), then B&E would equal B, so the left had side of the equation would become P(R|B).  This would naturally follow because P(E|R&B) = P(E|^R&B) = 1 and P(R|B)+P(^R|B)=1.

Therefore, there is an assumption that fine-tuning is consistent with our background (everything we know, bar fine-tuning) but our background is insufficient.

Now, we should make clear something about “background”.  Barnes’ says, of background, “tell me everything” and warns that information should not be arbitrarily ignored.  So, my question is this: is it arbitrary, given that no time related caveat is placed on the cReator hypothesis, to limit “background” to that which we know now?  How extensive is this “everything” of which you speak?  Is it possible that, in the future, we will discover that the fine-tuning we observe is a necessary consequence of relatively simple, yet entirely natural laws?  It would seem that the answer to that question, according to an apologist, should lie somewhere between “maybe” and “I don’t know” – perhaps somewhat closer to “maybe” given that “I don’t know” is saying that it isn’t necessarily impossible, and is therefore possible, making the answer an unambiguous “yes, it is possible”.

So, it seems to be unreasonable to assume that our background is always going to be insufficient to explain fine-tuning.  The assumption might be valid if we presupposed an interventionist god, or if there is some natural limit on the information that we can glean about the universe.  In the case of the latter, the argument by apologists is unacceptably stacked in favour of their god (assuming the existence of god, what is the likelihood of god?)  In the case of the latter, such a limit would imply that the question of the existence of a god may never be resolved, a god may never be proved but it may also never be disproved.

This is, I guess, a wonderful conclusion for apologists.  Potential job security until the end of time!

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I think that I am now ready to assign tentative probabilities to each of the terms in the equation.  I’m going to be as charitable as I can to the apologetic cause.

P(E|R&B) – we don’t know

P(R|B) – we don’t know

P(E|^R&B) – we don’t know

P(^R|B) – we don’t know

Plugging those values in, we arrive at P(R|B&E) = P(R|B) = we don’t know.

So, we could argue endlessly about how to manipulate the identities in the Bayesian calculations, and how likely or unlikely fine-tuning itself is, but at the end of the day, we merely arrive at “we don’t know”.  Sure, an apologist may immediately transmute that “we don’t know” into “god exists” via the subtle alchemy of an appeal to ignorance.  The more intellectually honest among us, however, are left with a mystery to investigate – why does the universe appear to be fine-tuned? – without any real need to worry about the superstitious wretches who continue to dog our investigative heels.

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It’s vaguely possible that someone is going to be upset that there were no jellybeans in this article.  I’m sorry, but the probabilities are so nebulous that I didn’t see any point in assigning any, even tentatively.

If someone else is keen to do an analysis using the jellybean model, feel free.