“Bayesian tools and conditional independence applied to selection effects and fine-tuning: no help for a multiverse”

Link: http://beam.acclab.helsinki.fi/~vpalonen/bayantrPalonen.pdf

He argues among other things that:

“In the case of the fine tuning of the universe, the observation is not conditionally independent of the hypothesis and it follows that information about the observation should be used as data and not merely as background information.”

And…:

“In fact, using O as background information is equivalent to assuming that all hypotheses produce only observable universes.”

Any thoughts on his article?

]]>Since this is certainly a daunting task, I’d be delighted to discuss with you on this topic.

Cheers.

]]>Ok … let’s start with probability theory and frequencies.

* I am an objective Bayesian. Probabilities describe states of knowledge, not frequencies of actual events. For a full exposition, read “Probability Theory” by Jaynes (Cambridge University Press). HIGHLY RECOMMENDED. The article you cited gives a reasonable exposition, and also cites Jaynes. Here is my own introduction: https://letterstonature.wordpress.com/2013/10/26/10-nice-things-about-bayes-theorem/

* There is a deep problem with frequentism that you need to address. Frequentism cannot calculate the probability of a hypothesis. This is not a criticism by the opponents of frequentism. It is emphatically defended by frequentists as a *feature* of frequentism. Ronald Fisher, the patron saint of frequentism: “We can know nothing of the probability of hypotheses or hypothetical quantities”. (quoted here: http://ba.stat.cmu.edu/journal/2008/vol03/issue01/aldrich.pdf)

The reasoning is simple. Consider: “the probability that general relativity is true is 0.8”. If probabilities measure frequencies, what frequency? General relativity applies to the whole universe, so to the frequentist it could only mean that 8 out of 10 universes that we have observed obey general relativity. We cannot make such a claim, and so frequentism can know nothing of the probability of general relativity.

“NID exists” is a hypothesis. It’s either true or false – there cannot be more than one actual case. If you believe that “probabilities measure frequencies”, then you can know nothing of the probability of (NID exists), prior or posterior. Your own frequentist teachers forbid it!

This shows in your discussion of the prior probability that (NID exists). p(NID | b) cannot be the *frequency* with which things in our background knowledge are NID-made (pg 282). One such thing proves NID exists. Adding a second NID-made thing changes the frequency but not the probability. “Things made by people (trillions of things)” are not cases of ~(NID exists). For example, show me all the cows you like – it doesn’t make (horses exist) any less probable. A million cows are not a million cases of ~(horses exist). [They are a million cases of ~(this thing is a horse) – a different hypothesis] Thus your statement that “we have trillions of cases of ~NID and no cases of NID” (page 283) is false. We have no known cases of either. The frequency is zero over zero. Frequentism can know nothing of the probability of hypotheses.

This is why I spent my first post (https://letterstonature.wordpress.com/2013/12/13/probably-not-a-fine-tuned-critique-of-richard-carrier-part-1/) discussing your interpretation of probability. Frequentists explicitly, proudly reject the idea that prior and posterior probabilities are meaningful. The statement “The prior probability of NID … [is] really asking how frequently are things .. the product of NID?” (page 282) gets you expelled from both schools. Neither interpretation can make sense of that claim. Your interpretation of probability theory sinks your argument.

* You must abandon the idea that “probabilities measure frequencies” if you want to calculate the posterior probability of (NID exists). Here’s how a Bayesian does it. [Note: this section reproduces some of this blog post]

“There can be universes in which [2 + 2 = 4] is true and ~o.” Exactly. So I can write

b = o.b’ (1)

where b’ lists all the statements that we know and which could be true in a world where ~o. So, b’ contains mathematical truths, but also statements like “Jupiter goes round the sun” and “like charges repel”. By construction, b’ does not imply o.

Now, consider the ratio of posterior probabilities (call it R)

R =def. p(NID | f.b) / p(~NID | f.b)

Because p(f | o) = 1, and b contains o, p(f | b) = 1. This shows that the posterior equals the prior (as in footnote 29),

R = p(NID | b) / p(~NID | b)

So, we need to consider about the prior. First, I can substitute from (1), since conjoining statements is mere labelling. It doesn’t change our state of knowledge.

R = p(NID | o.b’) / p(~NID | o.b’) (2)

Now, Bayes’ theorem is an identity. It follows from whatever probability axioms you prefer (Kolmogorov, or Cox’s desiderata http://en.wikipedia.org/wiki/Cox's_theorem) without any constraint on the propositions to which it applies. So I can apply it to (2),

R = p(o | NID.b’) / p(o | ~NID.b’) x p(NID | b’) / p(~NID | b’)

Now, take a close look at p(o | ~NID.b’). This is the probability that a universe with intelligent observers exists, given that there is no intelligent cause of their universe, and given background information b’ that does not imply o. This is the probability that a life-permitting universe “would exist by chance” (page 293). It is exactly the probability that you say “doesn’t matter”, even if it equals some “ungodly percentage” (page 292-3). Well, there it is. Using only the identities of probability theory, we can draw it out.

Note well: we aren’t ignoring the fact that we know o. It’s right there, known, in the posterior. The fact that we know something doesn’t mean it must be taken as given in every probability we calculate. Look at Bayes’ theorem: p(H|EB) = p(E|HB) p(H|B) /p(E|B). Both E and B are known (and so, given in the posterior), and yet the likelihood p(E|HB) just ignores the fact that we know E! Rightly so! That’s the whole point of likelihoods – to calculate the probability of something we know, given the theory. That’s what p(o | ~NID.b’) does.

If it is true that “in the set of possible universes, the subset that permits the existence of life is extremely small”, then p(o | ~NID.b’) will be very small. The smaller that probability, the larger the posterior p(NID | f.b) is. The surprising fact on ~NID, that a life-permitting universe universe exists *at all*, cannot hide in the background. It makes NID more likely, and by an “ungodly” factor.

If I’m wrong, show me. Don’t just tell me. Where, and why?

_______________

A few minor points.

* “I am not going to answer irrelevant questions.”

The question was “what follows about the posterior probability of NID from the fact that p(f | o) = 1?”. A few sentences later, you say “[p(f|o) = 1] proves that f cannot be evidence of NID”. Which answers my question. All you had to say was: b). Was that so hard?

* “their discussion of reference classes is self-evidently a discussion of frequencies”. It is. I don’t deny the necessary link between reference classes and frequencies. Frequencies involve counting some group of things. Nor do Bayesians deny that frequencies can inform probabilities. Frequencies are data. The problem is the claim that probabilities *are* frequencies. The Bayesian can use reference classes and frequencies without becoming a frequentist.

* What probability textbooks are on your shelf? The article you cited turned out to be on my side. The wikipedia page explained the problem of reference classes, but didn’t offer a solution. This isn’t a trick question. You must have a favourite article or book on frequentism.

* If you’ve got a proof that an infinite multiverse exists, why not submit it to “Physical Review D” or “Monthly Notices of the Royal Astronomical Society”? It’s very relevant to cosmology. Articles on the mulitverse have appeared in those journals. If you can do probability in your infinite multiverse, then you’ve solved the measure problem (e.g. http://arxiv.org/abs/1301.0121).

]]>“You still haven’t answered the question. Which of the options is it?”

I am not going to answer irrelevant questions.

“I agree with you that Dembski’s threshold doesn’t apply to the universe itself. Not having any problem with that claim, I haven’t discussed it.”

That’s cheeky. Are you now talking in a circle? You keep ignoring the role of prior probability in this discussion. And that note contains a crucial point about priors that explains why your “twenty royal flushes” argument is a straw man. And still you don’t see it. Astonishing. This is like talking to a wall.

“At no point have I questioned that p(f|o) = 1. I accept that that is true. You do not need to convince me of that. I’m not trying to get around it. I just want to know what it proves about NID.”

It proves that f cannot be evidence of NID. Because you can’t get a likelihood higher than 1, and the only way for any e to be evidence for any h is for the likelihood of e on h to be higher than the likelihood of e on ~h.

That alone does not refute NID. It just means you can never use f to argue for NID. Exactly as explained in my chapter.

* Consider these two statements:

A: 2 + 2 = 4

B: I know that 2 + 2 = 4

o = an intelligent observer exists.

Obviously, o follows from B. Does it follow from A?”

There are no observers in A. There can be universes in which A is true and ~o. But no observer can ever observe that outcome. Because those universes lack observers. By definition. That’s the point. So you are asking pointless questions here.

Meanwhile, B entails o. You cannot have B and ~o. Therefore you cannot separate B from o. Any set that contains B contains o by logical necessity. Get it?

“But, surely, if there is one thing in our background knowledge that is the product of NID, then NID exists.”

To the same probability of that one thing, certainly. Can you point to one? If not, then as I said, “so far, that frequency is zero” (p. 282). We then have to evaluate what its highest prior can be (since we cannot presume it is actually zero). I then do that. If there were a confirmed case of NID, I wouldn’t have to do any of that and my chapter would be written very differently indeed. But we have to attend to things as they actually are. Not as you would wish them to be.

If we had any confirmed cases of NID, then the analysis would depend on reference classes exhibiting NID, e.g. if we were visited by aliens (not secretly but publicly) then the reference class in which NID has a high prior would be things those aliens could plausibly do (and whether that included originating and evolving earth life and creating the universe would depend on what facts are known about those aliens, and/or evidence of coincidences of correlation, hence p. 283, a page you seem to be ignoring). We would then not be talking about whether NID exists, but whether that agent created the universe or originated earth life or meddled in its evolution. Hence my chapter would be completely different were that the case. But it’s not the case. So it’s moot.

This is an example, BTW, of you not understanding my argument.

“I’ve clicked every link on this page, and searched the HTML source – I can’t find a link to the post “Ex Nihilo Onus Merdae Fit”. Did I miss it?”

I see this is my error. It’s in the God Impossible link (so you should have found it had you read that; so you just confirmed to me you didn’t). But the link after that was supposed to go to it directly, but I evidently duplicated the previous URL instead of put in the correct one. So I’ve corrected the error. Apologies.

“In case you missed it, the paper you cited argues that, according to (objective) Bayesianism, which it explains and defends, probabilities do not measure frequencies.”

Do you agree with them? That’s what I’m asking.

(If you want to know why they are wrong, see Proving History, pp. 265-80. But already their discussion of reference classes is self-evidently a discussion of frequencies. You should have noticed that.)

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