Thank you. But you are no longer talking about anything relevant to my comment on Rosson. On why you are not entirely correct that “a frequency is a real property of some system, not some hypothetical construct,” because we often have to use hypothetical and not actual frequencies to establish a sound logical argument in the absence of omniscience, see PH, pp. 257-65. Which actually addresses the very example you are using (albeit with dice instead of coins).

If we were omniscient, and thus knew the actual frequencies of things, then your entire case before would be even more undermined–because then when I said “likely x,” rather than x is impossible, it would always literally mean sometimes ~x. It can only be allowed to ever include “never ~x” when we aren’t speaking about the actual frequency (because it can never be known to us). Otherwise, if we know the actual f, then we would know that f = 0. Unless it doesn’t.

But even when we don’t know, and therefore “never ~x” is possible, my comment about Rosson was simply that it is incorrect to conclude that when I say “likely x” I mean “never ~x.” That was my original point, and it remains correct.

Hi Richard,

Glad we got one thing cleared up, at least.

I have been assuming you knew the original sentences in PH that we were talking about.

As it happens, I have read those sentences, but they are irrelevant to my point. Your assertion was

[that x is] unlikely entails sometimes x

This statement is abstract and general, and is thus independent of any context. Perhaps you feel I’m nitpicking, but it started as a minor point, and then escalated when you defended it. Note that the wrongness of your statement absolutely does not depend on any probability or rate being vanishingly small.

I don’t think then that we actually disagree about anything.

I’m very glad we understand each other a little better, but I’m afraid there still are some issues on which we disagree. I’m not sure, however, how much miscommunication remains to be corrected, as a result of your disorientation, regarding the drive of my initial comment.

I very much doubt that any of your probability calculations can have been adversely affected, but the problem I perceive is with the way the theory is understood. If you are interested to understand better my objections, I’m happy continue the discussion here, or offline, if you prefer a more relaxed environment – either by email, or Skype, or what not (I believe you can find my email address). I’m pretty busy at the moment, though, as I expect to travel to Europe within a few days later.

For now, however, I’d like to express why I feel these issues I perceive are important. I appreciate your indulgence.

When people start to learn probability theory, if they have enquiring minds, the first niggling doubt usually comes when they ask themselves, “how do I know my priors are correct?” In fact, as you know, this problem is trivial – the priors are entailed by the pre-existing data and the probability model – but many never get past it (it’s a standard frequentist objection, that I still hear often, and apparently (weirdly) one that some Bayesians also subscribe to). Those that do get past it face another source of doubt and confusion: “how do I know that my probability model is correct?”

Whatever objections people might be stuck on, their confusion undermines their faith in probability theory, and many have struggled to find alternative theories of inference, several of which remain quite popular. As none of these alternatives is probability theory, however, they are all necessarily bullshit, which leads people, including many active scientists, into all kinds of trouble regarding the relevance and meaning of their results, and sometimes causes people to reject useful and valid methodologies, or to directly employ inappropriate mathematical procedures.

Now, the great worry, “how do I know that my probability model is correct?” has been the particular anguish of many. The failure of probability theory to answer this question has led many to abandon it in favor of low-grade alternatives. But this question is not something probability theory claims to answer, nor can it. Nor, in fact, can any possible alternative. What probability theory does provide is a mechanism to proceed if we have particular concerns about some model: we apply the theory recursively, using the method of model comparison. We assign our current model to a point in some higher-level parameter space, and crank the Bayesian handle again. This can be continued, as I mentioned earlier, to any arbitrary degree of sophistication. This is the best that any theory of inference can deliver, and is not a bug, but a marvelous feature.

The problem with your statement that probabilities are frequencies is that it demands that probability theory can guarantee the correctness of one’s probability model.

This objection survives whether the frequency you are identifying relates directly to the phenomenon under investigation (e.g. how biased is that coin?), or if it’s the frequency with which one will be right about similar problems of inference. This, I’ve tried to explain in the preceding comments. I understand, though, that my ability to explain is imperfect, and often assumes the triviality of things others may not have spent time thinking about.

Please note that a frequency (as opposed to “a frequency,” somebody’s idea of a frequency) is a real property of some system, not some hypothetical construct. If I toss a newly minted coin 1000 times, getting 522 heads, then destroy the coin, the frequency of heads for that coin was, and remains forever exactly 0.522. (If a person lives and dies, having committed not a single murder, then their murder rate is and forever remains exactly 0.) Note also that had I not destroyed the coin, but kept it and declared its frequency of showing heads to be 0.522, or 0.5 (different numbers, corresponding to two possible probability models), for all future tosses, I would have been notably wrong, had somebody remotely activated a mechanism inside the coin, making it show heads on every subsequent occasion.

Once again, if you’d like to discuss this further, I’ll be happy to oblige.

With genuine appreciation for your work promoting probability theory and rationality,
Tom

(PS: every time I see “OHJ,” my mischievous brain sends it to my inner voice as “OMG”)

Ah, then I mistook what you were trying to argue. I don’t think then that we actually disagree about anything.

Note that as I have explained, there is in fact a relationship between your rate of error claiming x and the rate of ~x. It is only when the rate of ~x is vanishingly small (or relevantly small, e.g. smaller than can be expected to occur even once in a human lifespan in the murder case) that it becomes inaccurate to say ~x sometimes occurs when you say x is likely. Otherwise, the entailment does obtain. That’s what I’ve been saying.

But, you are using the wrong analogy. Rosson is talking about my statement “it’s inherently unlikely that any Christian author would include anything embarrassing in his Gospel account, since he could choose to include or omit whatever he wanted,” p. 134 PH, which is not a statement of my rate of error, but an actual statement about the frequency of x occurring (my confidence in that statement would then be close to 100%; your Big Bang example was using the confidence level and mistakenly equating that with the frequency we are claiming to have confidence in). Indeed, in the Rosson case we are talking about my even being explicit about this, e.g. “The exceptions are very few and hard to establish in particular cases,” p. 136 PH, could hardly be clearer. I even discuss the mathematical rate in detail, with equations (pp. 163-66).

I have been assuming you knew the original sentences in PH that we were talking about. I see now that you must not have. Which is why we’ve been talking past each other.

Then you lept to the defense of Rosson.

I did not. The very first thing I said about this was that you were correct in identifying Rosson’s argument as an error.

That you don’t see the difference between what I said and a defense of Rosson’s argument is ample evidence that you don’t comprehend what I am talking about.

Thus what is duly entailed by “it is unlikely that Richard Carrier commits murders” is “sometimes I am wrong about things like that,” not “sometimes Richard Carrier commits murders.”

This is my exact point, and contradicts your original statement, “[x is] unlikely entails sometimes x,” along with several you have made since.

This is a side issue from the above, but this also suggests you are not listening.

If you say “It is unlikely that Richard Carrier commits murders” you actually mean “it is unlikely that I am wrong that Richard commits no murders.”

The first thing being frequency-counted here is not my acts of murder, but your acts of being right or wrong in your epistemic judgments.

Thus what is duly entailed by “it is unlikely that Richard Carrier commits murders” is “sometimes I am wrong about things like that,” not “sometimes Richard Carrier commits murders.”

Nevertheless, if you genuinely thought that 10% of my time I am committing murders (if that is what you actually meant by “It is unlikely that Richard Carrier commits murders”), then indeed “sometimes Richard Carrier commits murders.”

You are thus confusing an epistemic statement about my having ever done x, with a statement about the frequency with which I do x.

Certainly, if you thought that latter frequency was so high that odds are I have murdered at least one person by now, then you would indeed be thinking “sometimes Richard Carrier commits murder” (to whatever same probability of being correct you assign to your estimate of that frequency). But usually when you say it is unlikely that I have ever committed murder, you are saying the frequency of it is so low that odds are I have not murdered at least one person by now, and that you are (let’s say) 90% certain the frequency is that or lower. The “unlikely” is then referring to how likely you are to be wrong, not to how often I commit murder (the latter frequency would be something far lower than that, in order for it to be unlikely that I have ever committed murder).

It is true that this then entails “sometimes Richard Carrier commits murder” if “Richard Carrier” lived to infinity and thus your estimate of the odds of my ever doing so approach 100% as t approaches infinity, but t doesn’t approach infinity for human lives. Indeed, this can be used to calibrate your estimation of how likely it is that I will commit murder: what frequency (in murders/year) do you think has a better than 50% chance of being true given that I will live for 1,000,000 years? You could say it would even then be below 1 (and since you can’t have half a murder, it would then be false to say “sometimes” I commit murder–but only because now we are dealing with vanishingly small probabilities, where I am expected to die before even one event occurs). But if you’d say it would be 1 or above, then you are indeed saying that sometimes, per million years, I will commit murder (more probably than not). It’s just that I’m not likely to live that long, so this is kind of a moot point.

Hence my point about vanishingly small probabilities not being relevant to what I said about Rosson.

In reality you would not say it’s 10% likely I have committed murder. You would know that the actual prior probability that any random American will ever commit murder is well below 1 in 300, or 0.3%. And if you allowed any evidence to reduce that in my case (e.g. I am not an active member of a drug gang, nor have any other markers indicative of being in the murderer class), it would drop even further. But let’s stick with that, as per 85 years (average lifespan). So it would be incorrect to say “it is unlikely [= 0.3%] RC will commit murder” therefore “sometimes [less than 1 per 85 years] RC will commit murder” only because in that condition odds are I will die before even 1 murder occurs. If I lived a million years, you can legitimately expect me to commit murder sometime, unless you can adduce evidence that reduces its frequency in my case to below once in a million years. And so on, as life is extended.

Your confidence level would then correspond to a confidence interval (the maximum and minimum frequency with which RC commits murder, at that confidence level, let’s say 90%), which could include zero at the low end. But that would not mean you believed it was zero, much less that you were saying it was zero. That’s my point.

Moreover, that your confidence in that interval is 90% does not mean you think there is a 10% chance I have committed murder by now. It only means there is a 10% chance the interval is larger than you estimated. What that entails regarding what you would be compelled to believe about the actual probability I’ve murdered someone would depend on how much larger the interval is widened, and how much of that interval crosses into at least 1 murder per 50 or so years, and what the probability is of the true frequency falling in that section of the resulting bell curve (formed by each frequency being assigned a probability of being the true frequency, all of which probabilities summing to 1).

Can you please point out where I made those errors?

I did not claim you did. I claim Rosson did. Then you lept to the defense of Rosson. For some reason I can’t fathom.

If that is not what you thought you were doing, then you clearly didn’t understand what I was saying about Rosson. I was being charitable in assuming you did not go off the rails that badly right from the start, and that you did understand which errors I was calling Rosson out for (or theorizing he was falling victim to, since I am not sure that’s what happened, his statements only imply it), and therefore you were defending Rosson.

Otherwise, what on earth were you arguing?

Rosson acts like when I said “likely x” I was saying “never ~x.” But “likely x” does not mean “never ~x.” And the converse of never is sometimes. So if I did not mean “never ~x” I must mean “sometimes ~x.” QED.

It is not a legitimate objection that “I could have thought never ~x,” because had I thought so, I would have said so. That “never ~x” is logically possible is wholly irrelevant to what I was saying about Rosson’s erroneous inference. Indeed, we only get “never ~x” when we reframe the discussion with margins of error. But Rosson wasn’t confused about there being margins of error. His mistake was not in reading me having said “the probability of x is between 67% and 100%” and then interpreting me as having said “the probability of ~x is 0%.”

You have severely mischaracterized my statements, so I feel compelled to continue.

If your current workload prevents you from responding properly, then just say so.

Here is where you have gotten confused. We were not talking about vanishingly small probabilities.

I never mentioned vanishingly small probabilities, nor relied upon them for any of my arguments.

… in which there is no dispute that the thing claimed to be frequent sometimes nevertheless doesn’t happen.

The claim was actually about something being unlikely, not frequent, or even possessing a small non-zero frequency. Contrary to your claim, there is nothing about the word unlikely that guarantees (a) that a thing sometimes happens, or even, acknowledging the difference between frequency and expected frequency, (b) that I believe that a thing sometimes happens.

Example: It is unlikely that Richard Carrier commits murders. Is the truth of this statement sufficient to have you locked up? Or for you to sue me for libel?

(In the above, (a) and (b) are different things, despite you denial. See for example comment 5.1 : “Bayes’ Theorem gets you to an answer. But that is still just a frequency: the frequency with which a conclusion reached that way will be wrong. ” There is nothing to guarantee this frequency, as I have explained multiple times. And no, this does not render probabilities meaningless, as you claimed in the same comment – probability is the amount of belief a modeled rational agent would possess if the prior information, and the probability model were guaranteed correct, a situation that cannot ever arise, but an approximation that allows rational decision making to any arbitrary degree of sophistication.)

You can’t claim that when I say the frequency of x is 99 in 100, that therefore I am saying ~x never occurs. Nor can you adduce a single example of ~x and thus argue that when I said the frequency of x is 99 in 100, I was wrong.

Those were the errors I was calling out.

Can you please point out where I made those errors?

(1) A thing is not the same as a person’s concept of the thing.

That’s irrelevant here. If you think a frequency is x, you are not saying it is x in concept, you are saying it is x in fact but that you might be wrong. So the reality-concept distinction is not at issue here. Saying a frequency is x is still making a claim about an actual frequency. Thus claiming the frequency of y is x, still entails claiming that maybe sometimes ~y occurs, and indeed by definition that it occurs at a frequency of 1 – x with the exact same confidence that you have that y occurs at frequency x, as the one statement entails the other (unless you are saying x is zero, and not only zero, but zero with no margin for error, which neither you nor I are talking about).

Moreover, there will also be a frequency with which you are wrong when you say the frequency is x. That is a different frequency than x. It is measuring a different thing–not the frequency of the thing you are talking about, but the frequency of your being wrong about things like that.

But that is still a frequency. And it is an actual frequency. It is not a mere concept of a frequency, but a guess at an actual one.

And if you keep wide enough margins of error, your guess (that the frequency falls “somewhere” within your declared margins) will be correct to a very high frequency. Which is the object of all human knowledge.

I don’t see why you are having trouble understanding this. But maybe you are misunderstanding that this is even what we’ve been talking about…

(2) An estimate of a frequency is not the same as the frequency with which I expect such an estimate to be true.

I have repeatedly been saying exactly the same thing.

I don’t understand why you haven’t noticed this.

When you say “I may not be very confident of this” you are saying “the frequency with which I am right about claims like this might not be very high.” Ergo, this is a claim about a frequency. It’s frequencies all the way down. It also entails the converse frequency of being wrong (“I am highly confident that x entails I will sometimes be wrong about x,” which does not entail you are wrong on this specific occasion, only that you could be, to the converse of the stated frequency of your confidence).

See my discussion of your own example (I just use dice instead of coins) in Proving History, pp. 265-80.

(i) the statement that event x occurs with low frequency does not entail that x sometimes happens (contrary to what was suggested by your wording)

Here is where you have gotten confused. We were not talking about vanishingly small probabilities. We were talking about merely low ones, and in which there is no dispute that the thing claimed to be frequent sometimes nevertheless doesn’t happen.

What you are trying to argue is that such claims amount to something like “if the frequency of x is no greater than 1 in 100, then the frequency of ~x is somewhere between 0 and 1 in 100,” therefore it is possible x never occurs.

But we weren’t talking about whether it was logically possible for x never to occur. We were talking about everything we are allowing to be possible. In other words, we were talking about exactly the opposite thing: that agreeing that x only occurs at a certain rate, we are agreeing it is possible for x not to occur.

We can agree that the frequency of x might indeed be zero. That makes no difference to the fact that we are also admitting that it might not be zero, and indeed might even be respectably high–as in my case, explicitly allowing a frequency of historicity for high-scoring Rank-Raglan heroes that isn’t zero…nor even vanishingly small. You can’t claim that when I say the frequency of x is 99 in 100, that therefore I am saying ~x never occurs. Nor can you adduce a single example of ~x and thus argue that when I said the frequency of x is 99 in 100, I was wrong.

Those were the errors I was calling out.

You tried defending the errors I was calling out. For some reason.

Now you have abandoned that (rightly) and descended into defending claims wholly irrelevant to anything I was talking about in the first place.

(1) A thing is not the same as a person’s concept of the thing.

That’s irrelevant here. If you think a frequency is x, you are not saying it is x in concept, you are saying it is x in fact but that you might be wrong. So the reality-concept distinction is not at issue here. Saying a frequency is x is still making a claim about an actual frequency. Thus claiming the frequency of y is x, still entails claiming that sometimes ~y occurs (unless you are saying x is zero, and not only zero, but zero with no margin for error, which neither you nor I are talking about).

Moreover, there will also be a frequency with which you are wrong when you say the frequency is x. That is a different frequency than x. It is measuring a different thing–not the frequency of the thing you are talking about, but the frequency of your being wrong about things like that.

But that is still a frequency. And it is an actual frequency. It is not a mere concept of a frequency, but a guess at an actual one.

And if you keep wide enough margins of error, your guess (that the frequency falls “somewhere” within your declared margins) will be correct to a very high frequency. Which is the object of all human knowledge.

I don’t see why you are having trouble understanding this.

(2) An estimate of a frequency is not the same as the frequency with which I expect such an estimate to be true.

I have repeatedly been saying exactly the same thing.

I don’t understand why you haven’t noticed this.

When you say “I may not be very confident of this” you are saying “the frequency with which I am right about claims like this might not be very high.” Ergo, this is a claim about a frequency. It’s frequencies all the way down.

See my discussion of your own example (I just use dice instead of coins) in Proving History, pp. 265-80.

(i) the statement that event x occurs with low frequency does not entail that x sometimes happens (contrary to what was suggested by your wording)

Here is where you have gotten confused. We were not talking about vanishingly small probabilities. We were talking about merely low ones, and in which there is no dispute that the thing claimed to be frequent sometimes nevertheless doesn’t happen.

What you are trying to argue is that such claims amount to something like “if the frequency of x is no greater than 1 in 100, then the frequency of ~x is somewhere between 0 and 1 in 100,” therefore it is possible x never occurs.

But we weren’t talking about whether it was logically possible for x never to occur. We were talking about what we are allowing to be possible. In other words, we were talking about exactly the opposite thing: that agreeing that x only occurs at a certain rate, we are agreeing it is possible for x not to occur.

We can agree that the frequency of x might indeed be zero. That makes no difference to the fact that we are also admitting that it might not be zero, and indeed might even be respectably high (as in my case, explicitly allowing a frequency of historicity for high-scoring Rank-Raglan heroes that isn’t zero…nor even vanishingly small).

One more time, there are a few subtleties you seem not to grasp. I appreciate your patience.

(1) A thing is not the same as a person’s concept of the thing. A statement of pobability guarantees nothing about any physical frequency. I believe you get this, though your wording isn’t clear enough to demonstrate it, and in fact indicates the opposite.

(2) An estimate of a frequency is not the same as the frequency with which I expect such an estimate to be true. My best evidence might indicate that a coin is heavily biased, though I may not be very confident of this. (A coin might be randomly selected from a box containing 51 coins that always show heads, and 49 coins that show heads and tails with equal frequency.)

(3) A statement like “x is unlikely” serves to summarize the properties of some probability distribution. If x is a statement about a unique event, it says: “The probability mass associated with the frequency that I expect to be wrong about problems such as that relating to x, if I assert that x is false, is somehow predominantly concentrated in the region of low frequencies.” It does not say that I am confident that x occurs with non-zero freuqency – as illustrated by my earlier proposition about pixies. Also, and very importantly, it does not say that I am confident that the error frequency associated statements such as “x is false” is non-zero. It is not a definitive statement about that frequency, but a summary of the main properties of the probability distribution that I associate with this error frequency.

Taking all of this into account,
(i) the statement that event x occurs with low frequency does not entail that x sometimes happens (contrary to what was suggested by your wording)

(ii) the statement that event x occurs with low frequency does not imply that I believe that x sometimes happens (contrary to what you perhaps meant). If my probability distribution over f overlaps zero, then I still believe f = 0 is possible.