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> Ultimately it's a question of how much each bit of evidence would cause you to update (that is, difference of degree, rather than kind).

Isn't this only because you have strong priors for the coin and weak ones for Mars life? Suppose you didn't have your example's information about the coin. Then the situations are the same and a new piece of evidence in either case (e.g., discovering whether the coin is round, has two heads, or where its center of gravity is).

There is no such thing as a "strong" prior or a "weak" prior. A prior is jut a number between 0 and 1.

But the coin example is really not very apt because there is potential information that will change your belief about the coin. For example, you might discover that the coin is magnetic, and there is an external field being generated somewhere in just the right way to make the coin land heads every time. Or something like that. Whatever the case, if you get (say) 100 heads in a row you should start to have some serious doubts about your initial analysis of the situation.

But ultimately, it should take more surprising evidence to shake up your beliefs about the coin that it would to shake up your beliefs about Mars, right?

So, the prior which is strong/weak is the probability distribution of probability distributions. There is a <<1% chance that I will ever believe there is a >90% chance of flipping heads. But there is a much greater probability that I will one day believe there is a >90% chance of life on Mars.

What terminology would you use to describe that, if not strong/weak priors?

> it should take more surprising evidence to shake up your beliefs about the coin that it would to shake up your beliefs about Mars, right?

I would say it's the exact opposite. Encountering an unfair coin would not surprise me a bit because I know, for example, that double-headed coins exist. But encountering life on Mars would surprise me a lot because I've accumulated a lot of evidence already that there isn't any. (There might have been in the past, but there isn't any now.)

The only thing that I consider "surprising" is encountering evidence that moves one of my priors from being close to 0 to being close to 1 or vice versa. That happens less and less as time goes by because (I hypothesize) my beliefs converge to reality, and reality is stable in the sense that there is something out there towards which it is possible to converge. But at no stage in this process are my beliefs about propositions modeled by anything other than numbers between 0 and 1, with 0 meaning, "I am absolutely certain this proposition is false", 1 meaning, "I am absolutely certain this proposition is true", and 0.5 meaning, "I have absolutely no idea whether or not this proposition is true."

How does this work for the reasoning you just gave? Was there some evidence that moved your priors about Bayesian reasoning from near 0 to close to 1?

Apply that to any philosophical or mathematical reasoning. Let's take an interpretation of QM. How would you assign a probability to Many Worlds Interpretation? Do you have some way of assigning your degree of belief between 0 and 1? Or are there just arguments you find more or less convincing for it?

There was a thread on here the other day about black hole information paradox and how we may never be able to test any mathematical reasoning regarding what happens to the information inside a black hole. Can you assign a Bayesian probability to that?

> Was there some evidence that moved your priors about Bayesian reasoning from near 0 to close to 1?

I think you misinterpreted what I said. My belief in Bayesian reasoning was very nearly 1 from the first time I encountered Bayes's theorem. Nothing I've seen since has changed this, i.e. nothing about Bayes's theorem has surprised me.

The last time I can remember being surprised (in a Bayesian sense) about anything was when Donald Trump won the 2016 election. My prior on his winning was very close to zero when he announced his candidacy (and frankly I think his was too), and obviously my posterior after the election was very close to 1. (Contrast this with the reasoning currently being deployed by proponents of the Big Lie.) That event really rocked my world, but I can't think of anything that has surprised me in the same way since then.

> How would you assign a probability to Many Worlds Interpretation?

The same way you assign a probability to any proposition: you look at the evidence and decide whether the MWI is a good explanation that accounts for all the evidence. I did a very deep dive into this exact question about three years ago:

https://blog.rongarret.info/2019/07/the-trouble-with-many-wo...

> Can you assign a Bayesian probability to that?

It depends on what you mean by "that". You can assign a Bayesian probability to any proposition, but you have to be careful because not all statements are propositions. For example: "Penguins are beautiful" is not a proposition. It makes no sense to say that it is "true" or "false" because it is ambiguous (do you mean all penguins or just some penguins?) and it involves a value judgement.

The MWI is particularly thorny because it is really easy to hide ambiguity and value judgements in seemingly innocuous language. For example, you will often hear proponents of the MWI say something like, "The universe splits every time a quantum measurement is made." But that is ambiguous because it doesn't tell you what a quantum measurement is. It also has a tacit "value judgement" hidden in the phrase "the universe". The whole point of the MWI is that there is not one universe, there are unfathomably many of them. So which universe is "the" universe? Well, to me, "the" universe is the universe that "I" live in. But what does the word "I" mean in the context of the MWI? Because according to the MWI, every time the universe splits, I split along with it and become multiple me's in multiple universes. So which "me" is the real me?

The MWI is very strongly analogous to the beauty of penguins.

> I think you misinterpreted what I said. My belief in Bayesian reasoning was very nearly 1 from the first time I encountered Bayes's theorem. Nothing I've seen since has changed this, i.e. nothing about Bayes's theorem has surprised me.

My question was whether you used Bayesian reasoning to accept Bayesian reasoning as your epistemology. And I don't think you did. Instead, I think you were given a persuasive argument for why it's a good way to think about evaluating claims or making predictions, and then you started to think that way.

MWI is an example where there isn't a good way to assign a probability, because so far it's not testable. Instead, people like Sean Carrol are convinced by the mathematics that it's true, or they're skeptical for some reason like it violating Occam's razor or the measurement problem you pointed out. At any rate, it's isn't because there is some probability attached.

IOW, Bayesian reasoning isn't the only reason we believe things.

> My question was whether you used Bayesian reasoning to accept Bayesian reasoning as your epistemology. And I don't think you did.

Ah. Yes, I agree. In fact, I very rarely actually do the Bayesian math.

My core epistemology is Popperian.

> MWI is an example where there isn't a good way to assign a probability, because so far it's not testable.

It's even worse than that. One of the predictions of the MWI is that it is not testable! The MWI is "not even wrong".

Priors are not just numbers between 0 and 1. They are probability distributions. If they were not, it would be impossible to update beliefs based on evidence.

My prior beliefs about the coin include the possibility that the coin has a bias other than 50%. This distribution may be centered at 50%, but I may assign more or less probability to extreme biases, and this determines how I update my belief when I observe several heads in a row.

Of course it is possible to update beliefs based on evidence if priors are numbers. If I flip a coin but don't show you the outcome, you will rationally assign a prior of 0.5 to the proposition "The coin landed heads-up". If I then show you that the coin did indeed land heads-up (or tails-up) you will update your assessment of the probability of "The coin landed heads up" to something close to 1 (or 0).

Now, it is possible to do Bayesian analysis on probability distributions rather than "bare" priors, but that is just a way of assigning priors to a family of propositions rather than single propositions. For example, "This is a fair coin" is not a proposition to which one can assign a prior because it isn't a well-defined proposition. The intuition behind "fair coin" is something like "the ratio of heads to tails will approach 1 as the number of flips goes to infinity". But no coin can be flipped an infinite number of times, so this is a meaningless definition. Furthermore, consider a coin that always landed on the face opposite that of the previous flip. The ratio of heads to tails for such a bizarre coin would indeed approach 1 as the number of flips approached infinity, but that is not what is generally meant by a "fair coin". So the usual way of dealing with coins is assigning priors to the family of propositions, "If I flip this coin N times the probability that I will get M heads is X" for all possible values of N and M. But each of the propositions in that family gets a prior which is a number.

That framing actually stops holding for continuous distributions. Like in a distribution on heights of people there isn't a probability or prior associated with a particular height. Probabilities are assigned to subsets of this distribution (sometimes called sigma algebras). Priors stands for prior distribution and it really is best to see Bayesian analysis as a machine that consumes probability distributions as input and spits out probability distributions as output.

> That framing actually stops holding for continuous distributions.

No, it doesn't. A continuous distribution is just a way of assigning probabilities to a parameterized set of propositions where the parameters are continuous and so the set is infinite. But for any given proposition in that set the prior is a number.

> Priors stands for prior distribution

No. Bayes's theorem is:

P(H|E) = P(E|H) * P(H) / P(E)

The "prior" is P(H) which is a probability, i.e. a number between 0 and 1, as are all the other "P" values. The "*" and "/" operations in Bayes's theorems are scalar multiplication and division.

> No, it doesn't. A continuous distribution is just a way of assigning probabilities to a parameterized set of propositions where the parameters are continuous and so the set is infinite. But for any given proposition in that set the prior is a number.

A continuous distribution does not assign probabilities to each proposition but subsets of them. To see this concretely consider the continuous Uniform distribution from 1 to 1.5. It will for all values of its support have a probability density of 2. Most people would not consider 2 a probability.

For continuous distributions, Bayes's theorem becomes about probability densities and not probabilities.

> A continuous distribution does not assign probabilities to each proposition but subsets of them.

Nope. It assigns probabilities to individual propositions.

> Most people would not consider 2 a probability.

That's true, but in your example 2 is not a probability but a probability density, and a probability density is not the same as a probability. To get a probability out of a probability density you have to integrate. For each possible interval over which you could integrate there is a corresponding proposition whose probability of being true is exactly the value of the integral.

Ok I see where my confusion was. For you propositions are things that can be associated with measurable sets.

No, propositions are simply statements for which is it meaningful to assign a truth value. It might be possible to come up with a proposition that is associated with a non-measurable set, though I can't offhand think of an example. But remember: Bayesian probabilities are models of belief. Priors are peronsal. And so in order to assign a prior to a proposition, the statement of the proposition must have some referent in your personal ontology. You can't hold a belief about the truth value of a statement unless you know (or at least think you know) what that statement means. So unless a person has something in their ontology that corresponds to a non-measurable set (and I suspect most people don't) then that person cannot assign a Bayesian prior to a statement associated with a non-measurable set. But that's a limitation of that particular individual, not a limitation of Bayesian reasoning. For example: you cannot assign a Bayesian prior to the statement, "The frobnostication of any integer is even" because you don't know what a frobnostication is.

(Note that there are all kinds of ways that statements can fail to be propositions. For example, you can't assign a Bayesian prior to the statement: "All even integers are green" despite the fact that you know what all the words mean.)

But priors are not *just* numbers between 0 and 1, as GP claimed.

It is true that in your example, the outcome is binary: there is only a single proposition (the coin landed heads). And so the prior distribution is a Bernoulli distribution, but I would agree you could think of this prior as a single number between 0 and 1.

But in such a scenario, the only possibilities for updating beliefs based on evidence is to observe heads, then update the posterior to P(Heads)=100%, or observe tails, then update posterior to P(Heads)=0%. I might say that this is updating your beliefs based on evidence, but only trivially so.

If you observe heads and form a posterior of P(Heads)=99.999%, then it means that your prior probability distribution was more complex: it included the possibility that "the coin landed heads but I mistakenly observed tails". In this case, your priors include a family of propositions, not a single number.

In GPs example, the prior included the possibility that the coin was magnetic, etc. So it could not be represented by a single number.

> the prior distribution is a Bernoulli distribution

No, it isn't. The prior on the outcome of multiple tosses is a Bernoulli distribution, but the prior on the outcome of a single toss is a number. And for a fair coin, that number is 0.5 before you see the result, and it is either 0 or 1 after.

Note that "This coin will land/landed heads" is a different proposition than "this coin is a fair coin." The latter is much more problematic because it is hard to define what a "fair coin" is.

> your prior probability distribution was more complex

No. You are confusing my prior on a single proposition (a number) with the set of priors on all propositions which I have considered in my lifetime. The latter is obviously a set of numbers, one for each proposition.

> "the coin landed heads but I mistakenly observed tails"

Yes, that is two separate propositions:

1. The coin landed heads.

2. My senses are reliable.

All evidence must be assessed relative to some "background model", a set of propositions whose priors are very nearly 1. And it is also possible to model an infinite family of propositions about continuous properties, e.g. The length of that rod is between X and Y for any two real numbers X and Y. But none of this changes the fact that for any single proposition the prior is a number.

> it could not be represented by a single number

That depends on what you mean by "it". The probability that the coin landed heads certainly can be represented by a single number, as can the probability that the coin is magnetic. Two different propositions, hence two different priors.

But both of those cases are concerning the “Bayesian epistemology” side of things, not using probability to reason about how to bet in games of chance with well-established rules and mechanics. The usage of probability Deutsch is approving of is the latter, and is unrelated to a particular observer’s level of confidence that the cards are designed fairly or that the dealer and other players are playing fairly.

> suppose you are asked to assign a probability to the proposition that there was once life on Mars

As soon as you accept this question as meaningful, you have already come out as a Bayesian. Deutsch would simply say (if I understand him right) that this question is not a matter of probabilities, whereas the coin question is.

Suppose you were a bookie and somebody wanted to make a million dollar bet on it. You'd love to take the money, but obviously you need a line. How do you pick it?

A bookie would not care at all what the real likelihood of life on mars would be. A bookie would care about the likelihood that would attract approximately equal bets on both sides of the line, since that way they guarantee that they make money. So they would do whatever sociological research would be required to estimate that and set the line accordingly and probably not do an ounce of research on Mars itself.

This assumes that there is a chance to create a matched book. This is usually possible for something "liquid" like a big league sporting event, but not always for poorly-followed sports or weird prop bets on low-interest games. Even for liquid games, the book is not always perfectly matched and they may end up with some actual exposure to the outcome of the event.

In any case, in this scenario its just one guy looking to make a bet. There is definitely SOME line, possibly million-to-one or higher, that a smart book (or any rational actor) should be willing to take the bet.