She doesn't particularly suspect that there's a Grimdark Conspiracy out to sabotage everyone else's ability to do science.  But only because Thellim has concluded that the Earth's moon is sabotaging fully everyone's ability to do science.

(In Thellim's defense, it is an extremely suspicious moon.  It grants magical powers during lunar eclipses.  Few dath ilani would not be suspicious that this moon had something to do with the problem.)

The problem with Bayesianism* is that it's too subjective.

(*)  A four-or-five-syllable Earth term that refers, if anything in their language does, to the forms for reasoning lawfully about probability as spotlighted by coherence theorems; though even among the few Earthlings who've heard of the word, far fewer have heard of coherence theorems.

Too subjective.

Too subjective.

Too subjective.

Would the Earthling care to explain what is less subjective about any of the deranged pseudo-mathematical constructs used in Earth 'science'?

Well, for example, let's say you flip a possibly biased coin 10 times.  It's an objective fact that if the coin is not biased, the long-run frequency of occasions where 10 coinflips have 9 or more heads, or 1 or fewer heads, is 22/1024.  So if we adopt the rule 'say the coin is biased if there's at most one head or at most one tail after 10 flips', we're adopting a rule which only rejects the null hypothesis, on occasions where the null hypothesis is in fact true, with 2% frequency in the long run.

This is a fact.  It is an objective fact.  Right?

It's a valid derivation from the stated assumptions, given the laws of probability, yes.  If you write a simulation program to flip fair coins ten times, and beep when the number of heads is 0, 1, 9, or 10, the program will beep with 2% probability each time it's run -

The more correct way to say it would be that the long-run frequency of occasions where the program beeps will tend towards 22/1024 as a limit.  That's an objective statement that is definitely true, where it's more controversial to say that it's meaningful to talk of there being a 2% probability of the program beeping on the next occasion it runs.  That either happens or it doesn't, so in what sense could the number 2% be true about it?

The way that truth-functionals on propositions generalize to truth-functionals on probability assignments, is that you say 2% and then you lose, like, 5 or 6 bits or so, if you're wrong, and lose a small fraction of a bit, if you're right.

It's slightly more complicated than mentally comparing the proposition 'snow is white' to a handful of snow and thinking that it's meaningful to talk of this proposition being 'true'.  Very slightly.  If you can do one you're probably smart enough to do the other, really.

If one denies that there is such a thing as a 2% probability that the program beeps on the next occasion, why couldn't you just imagine that you repeatedly run the program and it never beeps on any occasion forever?  That's a possible thing that happens when you keep flipping coins.  It's improbable but it's possible.

The probability tends to zero as a limit.

So this quantity that is supposedly meaningless tends to zero as a limit, which is, itself, apparently a very meaningful thing to say?  Yes, that sounds like a very consistent philosophy to have about the world.

After any finite number of steps, it will still, apparently, on his view, be objectively meaningless to talk about the probability that a program beeps at least once.

Therefore, the meta-proposition 'is it meaningful to talk about the probability of this program beeping at least once' will tend towards 'false' as a limit, on his philosophy -

Okay, look, it's not like that, it's more like - that the long-run frequency of programs beeping is 22/1024, is a valid consequence of the premise that the coin is fair.  That's just a true statement about the long-run frequency, with no probabilities involved.  It's how you reduce claims about probability to simpler claims that don't have probabilities in them, by looking at something which actually has a 22/1024 proportion of whatever.

This true statement about the hypothetical is derived from reasoning steps that go through probabilities.  After you've flipped the coin fifty times, for five runs of the program, there's thirty-two possible sequences of beeping and not beeping.  Extending these sequence lengths out towards an infinite limit, the sequences more than delta away from 22/1024 beep frequencies will tend to go under epsilon fraction of the total probability.  This is the only meaningful statement that can be made here about a 'long-run frequency', unless he wants to start taking direct ratios of countable infinities.  When the sequences are actually infinite, we can no longer directly speak of things having 22/1024 ratios inside them.

This limit is being obtained by considering the probability of the possible finite beep sequences, which is supposedly 'subjective' and forbidden.

Right, but the final conclusion is a certain one, which makes it objective, unlike the statement that a fair coin has a 2% probability of yielding less than 2 or more than 8 heads on the next occasion, when either it does or it doesn't.

Thellim might, under other circumstances, try to ask if his rules of metaphysics permit one to derive an 'objective' truth via a series of reasoning steps whose intermediates are 'not objective'.

She reminds herself that this person has in fact been driven crazy by the moon and she is not actually going to be able to argue him out of anything.  The main thing Thellim should do is note down that a manifestation of the compulsion appears to be a feeling that some complicated derivations of the probability axioms are 'objective', and the simpler steps of reasoning leading up to that conclusion are 'subjective', and that then the person refuses to accept the 'subjective' parts even though in principle they're how the 'objective' truth is derived.  They skip over the derivation, and feel like they can just go from the supposed premise to the 'objective' conclusion about long-runs, without any 'subjective' statements about short runs along the way.

Presumably this is what it feels like from the inside, after the moon has excluded from your mind all the simple and obvious ways to do science, and you've had to resort to fantastically complicated methods outside the boundary of what the moon is excluding.  The word 'subjective' is really just referring to whatever the moon prevents them from believing.  If you point out that their 'objective' statements are derived from 'subjective' intermediates in principle, they'll just jump all the way to the conclusion in what they claim to be one jump, maybe...?

Let's try a different tack.

Well, Thellim does in fact concede that the long-run frequency of occasions where this hypothetical fair-coin-flips-10-times program beeps, will tend towards 22/1024 as a limit.  She just thinks that it's not particularly plausible to have an epistemology of probability where you can say that, and exclude the idea that the next run of the program has a 2% probability of beeping.  In both cases you're just talking about a logical consequence of an uncertainty-containing model where the coin has an independent 50% probability of coming up heads each time, that your mind imagines being compared to reality using a logloss correspondence, and which you can actually score once you make the actual observations.

That this is not something you can derive from the long-run frequency of the coin yielding heads being 50%, as opposed to the statement that the coin has independently a 50% chance of showing heads on each occasion, is readily shown by considering the case of a coin that perfectly alternates heads and tails inside every run.  This coin has a long-run frequency of 50% heads, and also, the head-counting program run over it will never beep -

They're really getting a bit outside his area of expertise here, he's a scientist, not an epistemologist.  But presumably the definition of a long-run frequency also includes, for example, the idea that subsequences of two coinflips, HH, HT, TH, and TT, all have 25% frequency, within the long run.

This is either an infinite set of independent postulates none of which can be derived from each other or any simpler postulates, or they are the collection of simple deductive consequences of a model in which the coin has an independent 50% probability of coming up heads each time.

(Can he not see where his axioms are coming from, the excluded space inside his own mind... he can see the consequences of excluded beliefs, so they're probably still in there in some sense... it's just that when he thinks about them consciously, the moon forces him to label those thoughts as meaningless and 'subjective' and therefore bad, but that doesn't mean he can't intuit the consequences...)

Look, he's not actually a philosophy-of-science guy.

Being one of the smartest people on the planet, and the literally actually sanest person on it, is something that Thellim is still having trouble coming to grips with.  She isn't even a professional in any particular subspecialty of science, like report-writing or experiment-performance or experiment-analysis.  She just knows what eight-year-olds are supposed to know about science so they can read newspaper stories about it.

Which makes her all that this Earth has in the way of sane scientific methodology literally at all, and if she's going to do anything with that, she may need to get it done before the next lunar eclipse, which is now only three months away.

Can she possibly sell him on the idea that it would maybe be a better idea for experimental reports to report some different consequences of the probability axioms, even if those consequences are 'subjective'?  She realizes it probably feels very bad to talk about a consequence of the probability axioms that feels 'subjective'.  But, if it were super important, like maybe there were a lot of money at stake or something, people could talk about other deductive consequences of probability models that could be matched up against experimental realities?  Even though they seem to be 'subjective' and that's very bad?

Like what?

Likelihoods.  Likelihood functions.  Likelihood ratios.

If you flip a coin and see eight heads and two tails, you report, first of all, the actual exact sequence you observed including the ordering, and as summary, that the likelihood of this data given any underlying independent propensity p for the coin to yield heads, is equal to (p)^8 * (1-p)^2.

For 80% propensity it'd be... 2^26/10^10, and for 20% propensity it'd be 2^14/10^10, for example.

This is an axiomatic, deductive, fully valid and determined consequence of that hypothesis about the coin.

He knows what 'likelihoods' are, which makes him very smart and well-educated for an Earthling scientist!

He's heard that they have to be combined with 'priors' in order to make any use of them, though.  Priors are very subjective.