Following my three critiques (one, two, three) of Richard Carrier’s view on the fine-tuning of the universe for intelligent life, we had a back-and-forth in the comments section of his blog. Just as things were getting interesting, Carrier took his ball and went home, saying that any further conversation would be “a waste of anyone’s time”. Sorry, anyone.

I still have questions. Before I forget, I’ll post them here. (I posted them as a comment on his blog but they’re still “awaiting moderation”. I guess he’ll delete them.)

## The Main Attraction

What is Carrier’s main argument in response to fine-tuning, in his article “Neither Life nor the Universe Appear Intelligently Designed”? He kept accusing me of misrepresenting him, but never clarified his argument. I’ll have another go. Let,

o = intelligent observers exist

f = a finely-tuned universe exists

b = background information.

NID = a Non-terrestrial Intelligent Designer caused the universe.

We want to calculate the posterior: the probability of NID given what we know. From Carrier’s footnote 29, introduced as the “probability that NID caused the universe”, we can derive (using the odds form of Bayes’ theorem),

(1)

Carrier argues in footnotes 22 and 23 that,

implies , (2)

because o is part of “established background knowledge” and so part of b. Thus,

(3)

Conclusion: the posterior is be equal to the prior (as seen in footnote 29). Learning f has not changed the probability that NID is true. Fine-tuning is irrelevant to the existence of God.

**Question 1: **Is the above a correct formalisation of Carrier’s argument? (If anyone has read his essay, comment!)

The closest he came to clarifying his argument in our exchange was to say “P(e) = 1 for all observers. You have no valid argument against that”. I’ll give my response, but first I need to clarify a few things. Carrier’s approach to probability is unfamiliar, to say the least.

**Question 2: **If o is part of b, am I allowed to separate b into the part that contains o and the rest? In other words, can I write

,

where b’ doesn’t contain or imply o?

**Question 3: **Is Bayes’ theorem an identity? That is, can I apply the formula

(4)

to ** any** propositions x, y, and z? Or are there certain types of propositions to which Bayes’ theorem does not apply?

**My reply**

I agree with (1), (2) and (3). I disagree with the interpretation of (3).

I’ll work backwards. The answer to question 3 is yes. Bayes’ theorem follows from Cox’s theorem, which assumes only some reasonable desiderata of reasoning. So it is an identity – given any propositions x, y, and z, I can use Equation (4). This is not to guarantee that I can actually calculate any of the terms on the right hand side, of course. But breaking up a probability into pieces is sometimes helpful and never wrong.

The answer to question 2 is yes. Conjoining or separating propositions is a mere relabelling. Thus, if x = yz, then it doesn’t matter whether I write it one way or the other. It is one of the desiderata of probability theory that identical states of knowledge should result in identical assigned probabilities.

[Aside: But wait ... The statement o says that intelligent observers exist. Surely, if I know anything, I know that I exist. Thus, any statement that I know implies o. So there is no such statement b', right?

Nope. I know o, and I know that "2+2=4". This is true:

"I know that 2+2 = 4" implies that "I exist".

However, this is false:

"2 + 2 = 4" implies that "I exist".

I can distinguish the statement "2 + 2 = 4" from the statement "I know that 2+2 = 4". While my existence follows from the fact that I know something, it does not necessarily follow from every fact that I know. If b' contained only the statement "2 + 2 = 4", o would not follow from b'.]

So my first worry is with placing o in b. Of course, o is true. But the decision to place o within b is a mere labelling. It does not change our state of knowledge. If Carrier’s argument depends crucially on placing o in b, then something is wrong. I’ve explained this in more detail here. The stuff we label “background” is not special, not the stuff we *really* know. As Jaynes’ notes, the background “denotes simply whatever additional information the [probability calculating] robot has beyond what we have chosen to call the data” (pg. 87).

The distinction between E and B dissolves when we know more than two things. Suppose I know facts . Then the posterior of the hypothesis H is,

A given proposition can play the role of “background” or “evidence”, depending on the term. No proposition is locked into the background. There is no need to divide the into two and decide which are evidence and which background. Certainly, the posterior doesn’t depend on that decision. The order of can be shuffled at your convenience: they’re just labels. Use whatever order allows you to estimate the terms on the left. Talking about “*the* prior” or “*the* likelihood” in such a context is ambiguous. Better to use notation.

Now, we substitute b = o.b’ into (3), giving

(6).

Bayes’ theorem is an identity, so I can expand this expression as follows,

(7).

I got to this point using the basic machinery of probability theory, nothing more.

Look closely 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 exactly the probability that Carrier is afraid of, the one that could equal an “** ungodly percentage**” (pg. 293). It is the probability that “the universe we observe would exist by chance” (pg. 293). Carrier argues that this term is irrelevant because ignores o. It does, but

*. The posterior does not ignore o. Look at Bayes’ theorem: p(H|EB) = p(E|HB) p(H|B) /p(E|B). Both E and B are known, and yet the likelihood p(E|HB) just ignores the fact that we know E! Rightly so! This is the whole point of Bayes’ theorem.*

**rightly so**Here’s the problem with the argument above. What (3) shows is that, since f follows from o, I need not condition the posterior on f. There is a redundancy in our description of what we know. But that does not mean that the posterior p(NID|f.b) is independent of the “ungodly percentage” p(o | ~NID.b’). The surprising fact on ~NID, that a life-permitting universe universe exists ** at all**, cannot hide in the background. We can draw it out. It’s right there in equation (7).

**Carrier’s Prior**

What is Carrier’s estimate of p(NID | b)? He says, “we’re really asking how frequently are things we point to (in all our background knowledge) the product of NID?”.

Recall that, for the frequentist, the probability p(A|B) asks: how frequently, amongst all the known B-like cases, do we find known (A and B)-like cases? Formally, p(A|B) = p (A and B) / p(B). We need to be clear about what makes something an NID-like case. There a couple of different versions of NID floating around Carrier’s essay.

**NID1 = a non-terrestrial intelligence (NID) exists**

This version (on page 280) is in real trouble. A single case of an NID-made thing proves that they exist. So the probability of NID1, that an NID *exists*, is not the *frequency* of NID-made things.

Suppose I want to know if a car factory exists. Even if there were only one factory that made only one car, and a billion things not made by a car factory, the probability that a car factory *exists* equals one, not one in a billion. The prior (and hence the posterior) is one. You can’t assign a probability for an existential claim by counting examples.

Conversely, if there are no known cases in b, then the frequentist must conclude that the prior is exactly zero. Since it follows that the posterior is zero, it is impossible for anything that is unknown to exist. So, another massive frequentist face-palm there.

**NID2 = this randomly-chosen thing is a product of an NID**

The frequency at which things we point to (in all our background knowledge) are the product of NID is the prior probability that some randomly chosen thing is the product of NID. In the car example, the probability of a randomly-chosen thing being car-factory-made is one in a billion. If I’m interested in the existence of car factories, or whether a car factory made this particular ** car**, such a probability is irrelevant.

**NID3 = the universe (totality of physical existence) is a product of an NID**

Carrier considers this version on page 296, when he calculates the “probability that NID caused the universe”. The prior p(NID3 | b) doesn’t ask for the frequency of ** things **that are the product of NID. It asks for the frequency of

**that are the product of NID. “Things [exhibiting apparent design] made by people (trillions upon trillions of things)” (pg. 282) are not in the relevant reference class. If my grandma knits another tea cosy, does it become less probable that the universe was created by a designer?**

*universes*Putting aside our universe as a contested case, we have no known cases of NID3, and also none of ~NID3. This is the “zero-over-zero” problem for frequentism again. The universe is unique; where are your frequencies now?

### Carrier’s Generous Prior

Carrier’s maximum prior of 25% only seems generous because Carrier’s thinks that “trillions upon trillions” of human made things are cases of ~NID. This is only true of NID2. A tea cosy is not a case of a reality that does not contain an NID, and not an example of a universe not produced by NID. Carrier has the wrong prior, or at best, the right prior for the wrong hypothesis.

We saw above that p(NID | b) (specifically NID3) depends on the “ungodly” probability of a life-permitting universe by chance p(o | ~NID.b’). Thus, unless one of the other terms in the equation is similarly small, the probability p(NID | b) will be very close to one.

## A Few More Questions

**Question 4:** Carrier mentions a “formal demonstration online” that “it is effectively 100 percent certain an infinite multiverse exists” (footnote 20). In Footnotes 3 and 21, when Carrier’s article refers to “a formal demonstration”, he means Ikeda and Jeffrey’s, and Sober’s articles. Neither of these argue for an infinite multiverse. What “formal demonstration online” should I be reading in support of footnote 20?

**Question 5:** What mathematician should I read to learn about reference classes and why probabilities measure frequencies? Is Carrier a frequentist or a Bayesian?

**Question 6:** “When b and e are swapped, so a prior is swapped for a posterior … “. That’s a mistake, surely. You get from a prior to a posterior by conditioning on e. Did he mean to say “when e is added to what we know already b, we update the prior p(h|b) to become the posterior p(h|e.b)”?

**Question 7:** What is the general, formal version of the statement “if the evidence looks exactly the same on either hypothesis, there is no logical sense in which we can say the evidence is more likely on either hypothesis”? Is it

Carrier hinted that the prior was involved but I’m still not clear …

**Question 8:** “When b and e are swapped, so a prior is swapped for a posterior, the effect of changing the event (and thus the reference class) is effected at the prior probability, not the likelihood.” So, when we get new information e about an event, we have to change the prior? Is that what he’s saying? That’s wrong. The prior is independent of e. (Also, I can swap e and b? So, the answer to question 3 is yes?)

We were discussing a poker player who deals himself 20 royal flushes in a row; call it . Carrier argues that one royal flush does not tell us whether the player is cheating or fair .

“If the evidence looks the same on either hypothesis, there is no logical sense in which we can say the the evidence is more likely on either hypothesis. Think of getting an amazing hand at poker: whether the hand was rigged or if you just got lucky, the evidence is identical. So the mere fact that an amazing hand at poker is extremely improbable is not evidence of cheating.” (pg. 293).

Carrier follows the quote in my question 8 with this,

Thus, when you query the posterior probability that twenty royal flushes is by design and not chance, the fact that such an event has a vastly lower prior probability than a single royal flush produces the conclusion favoring design.

These passages seem contradictory. The first says that the low probability of the amazing hand is not evidence of cheating. The second says that the low probability of 20 royal flushes is evidence of cheating. This could be true in practice – the second probability will be much lower than the first – but not in principle. Either low probabilities count against hypotheses or they don’t.

More confusion. Carrier talks about the *prior* probability of the *event. *But prior probabilities are of hypotheses, not events. If Carrier means the probability that will happen, given a fair dealer, then that probability is a likelihood not a prior. If he means the hypothesis “20 royal flushes by chance”, then he’s talking about the conjunction . The “prior probability” of this composite hypothesis is , and again it’s the smallness of the likelihood that matters. Either way, sometimes, extreme improbability * is* evidence of cheating.

(It’s asking for trouble to discuss probabilities in words. Prior, posterior, likelihood, background, outcome, hypothesis – these are equivocal terms. The confusion could be cleared up in an instant by using notation.)

Here’s the problem. The first passage, from page 293, is illustrating a general principle. We are invited to apply the principle to fine-tuning. I should be able to apply this principle more generally. Take that passage, and replace “an amazing hand” with “20 amazing hands”. Two possibilities. If this replacement is invalid, then there is some unstated restriction on the principle. We must be told of this restriction before we can properly apply the principle to fine-tuning. On the other hand, if the replacement is valid, then the principle implies that 20 royal flushes in a row is not evidence of cheating. *Reductio ad absurdum*, I’d suggest.

Carrier goes on to say: “how often does that kind of event [20 royal flushes] happen in family games of poker as a result of fair draws relative to cheats? The ratio is very different than it is for a single royal flush, why, golly, isn’t it?”. This does nothing to answer the point. Of course 20 royal flushes occur less often. But why does the rareness of 20 royal flushes establish cheating but the rareness of 1 royal flush fail to do so? If the “extremely improbable is not evidence of cheating”, then the lower ratio for shouldn’t matter. (Also, almost certainly, no one in the family has seen 20 royal flushes dealt at all, fairly or by design. What now? No frequencies, no probability, no conclusion about cheating?).

**Question 9: **Moving on to Carrier’s scientific claims, there’s some explaining to do.

- What are the “six constants of nature“? Why is it that no physicist thinks that?
- Where are the peer-reviewed scientific publications that show how fine-tuning has been “refuted by scientists again and again”?
- Where are the peer-reviewed scientific publications that “only get [a] “narrow range” by varying one single constant“?
- What about all the non-theist scientists who who think that the universe is fine-tuned, and that this needs to be explained by a multiverse?
- Would you cite a calculation by Hugh Ross, if a physicist told you that three quarters of his equations were incorrect? Why, then, cite Stenger’s MonkeyGod?
- Why think that inflation itself is not fine-tuned?
- If the universe is fundamentally law-less, why do we observe such a large region of order, and more order than life needs?

### Postscript: Show, Don’t Tell

“Show, don’t tell” is a basic principle of writing. Don’t just tell us of Romeo’s love for Juliet. Show him scalinh the orchard wall, risking life and limb, and declaring that “With love’s light wings did I o’erperch these walls”.

Carrier wastes countless words * telling* the reader that I am wrong: “The Barnes pieces don’t even respond to my argument … Barnes is something of a kook … a series of completely irrelevant points … stock fallacies … selective quotations … ignores what I did say … wasting tons of time on a footnote [130 words, FYI] … the same scam you keep pulling … bogus claims of inconsistency … [you] do the math wrong … your handwaving … is just desperate … completely strange and irrelevant remarks … you seem to have some sort of fictional argument in your head … You seem pathologically incapable of understanding any argument you rebut … you can’t even understand what my arguments are … you still, after all this time, egregiously fail to acknowledge or grasp … like arguing with a box that spouts random sentences”.

This time could have been spent * showing* that I am wrong. More time is spent attacking me than defending, or even explaining, his case. Take the comment on January 7, 2014 at 8:43 am. Of 14 sentences: 1 clarification of a previous comment, 2 repetitions of points from his article that I agreed with, 2 claims contrary to mine (hurray! interaction!), and 9 that merely accuse of error and incompetence.

Take Carrier’s main argument. I explained what I thought his argument was, multiple times. The best way to *show* that I don’t understand it is to summarise the argument, so that the reader can see the difference. Instead, Carrier says “Huh? No one argues that. This is the kind of nonsense I am talking about. You don’t even understand my argument, and aren’t even replying to it” and moves on.

Another easy opportunity is a challenge set by the critic. Carrier says “probabilities are frequencies”. Frequentism faces well-known problems (I didn’t make them up). I raised those problems. His entire article is one long probabilistic argument, so it’s no small matter for the assumed interpretation of probability theory to be flawed. *Show* I’m wrong by answering the challenge. Give a defence of frequentism, or at least a reference to a mathematician. Likewise, if you claim that something has been refuted by scientists again and again, a request for examples isn’t unreasonable. It should be easy.

Too much telling looks evasive. A reasonable challenge ignored is suspicious. If the criticisms are so easily defeated, then why not just defeat them? Whinging is not replying.

on January 22, 2014 at 9:00 pm |thermodQuestion 4 seems to be missing a question.

on January 22, 2014 at 9:33 pm |lukebarnesWell spotted. I’ve added the missing sentence.

on January 22, 2014 at 10:12 pm |GGDFan777Spot on Barnes, I think a lot of confusion by Carrier could be prevented if he’d put his reasoning in mathematical notation. Btw Luke, what do you think of his argument:

“the fact in conditional probability that P(INTELLIGENT OBSERVERS EXIST) = 1 (the probability that we are mistaken about intelligent observers existing is zero, a la Descartes, therefore the probability that they exist is 100 percent)”

He then continues his calculation using P(B) = 1, but isn’t he really talking about P(B|B)? which trivially equals to 1?

on January 23, 2014 at 12:21 pm |lukebarnesHe’s back! More conversation here: http://freethoughtblogs.com/carrier/archives/4973

on January 27, 2014 at 8:02 pm |ldslogicdirectscienceAre you going to reply to Carrier?

on January 27, 2014 at 10:16 pm |lukebarnesI have replied. My comment is “awaiting moderation”. I’ll post it here in the meantime. Carrier says:

“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.)

on January 27, 2014 at 10:20 pm |lukebarnesMy reply:

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: http://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 (http://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).