Interpretation of confidence intervals and Bayesian credible intervals

If we live in a materialist deterministic world - which many would cite as an axiom for simplicity - then there really is no probability. Everything happens with 100% certainty.

Then, what is probability? If everything will happen with 100% certainty, but probability certainly appears to exist, then probability must reflect something about our information about something occurring.

The author refers to two foundational approaches to our state of knowledge. The first is the objectivist approach, which states that everyone who has the same state of knowledge about a system will evaluate the same probability of something occurring. The second is the subjectivist approach, which states that a given individual with a certain state of knowledge will evaluate some probability of something occurring. To me, these appear to be the same thing except insofar as the former requires a consensus of many while the latter a consensus of one.

The author asks how we might actually define Bayesian probability without resorting to the frequentist approach (i.e. hypothetically simulating many trials of the same event, however infrequent in reality it may be).

First, he says this would mean "interpreting [the credible interval] like a confidence interval". I am no statistician, but is that necessarily true? I don't see why confidence intervals would suddenly emerge in order to interpret a credible interval.

Second, I am not sure the frequentist interpretation is so problematic. When we interpret the plain-English definition of a probability, are we not mentally simulating repeated trials in order to evaluate something's occurrence? What else could a probability imply? If something has a 20% chance of occurring, then it does not occur 80% of the time, and so we must envision 80% of universes (part of the hypothetical trials) where it does not occur. I don't see any other way around this, frequentist or not.

(Note: I am not a statistician, while the author is, and the above is simply my layman's understanding of the article.)

1. The outcome of the election here is not a probability. It is the population value - the ratio of people voting for candidate X on the election date. It doesn't have to be repeated in the same way measurements of height for all people in United States would not have to be repeated, if instead of vote we were measuring heights.

2. Frequentist probability doesn't require to physically repeat things. It can reason about what would happen in the repeated sampling under certain conditions, and then draw inferences about those assumed conditions. With the election example: if you get a survey of 100 people with 70% voting for candidate "A" we don't need to repeat this survey in order to know the likelihood (frequency) of this result happening if the real proportion of people voting for candidate "A" across the US is 50%.

This article kind of helps in establishing that it is a hard question to answer. Clearly harder with intervals.

I can't help but think much of this gets overcomplicated because we don't take everything in intervals. In large because it is hard, yes; but we should be more comfortable with things not getting known to an exact value.

Of course, frequentist and Bayesian stats are completely mathematically equivalent. The choice just affects our mental patterns.

[0] https://en.wikipedia.org/wiki/Probability_interpretations#Ph...

I'm not clear on what it is that you [the post's author] don't understand about interpretation of Bayesian credible intervals.

Both "objective" and "subjective" Bayesians interpret them as degrees of belief - that, for instance, one would use to make bets (supposing, of course, that you have no moral objection to gambling, etc.).

The difference is that that "objective" Bayesians think that one can formalize "what one knows" and then create an "objective" prior on that basis, that everyone "with the same knowledge" would agree is correct. I don't buy this. Formalizing "what one knows" by any means other than specifying a prior (which would defeat the point) seems impossible. And supposing one did, there is disagreement about what an "objective" prior based on it would be. To joke, "The best thing about objective priors is there are so many of them to choose from!".

Many simple examples can illustrate that the objective Bayesian framework just isn't going to work. One example is the one-way random effects model, where the prior on the variance of the random effects will sometimes have a large influence on the inference (eg, on the posterior probability that the overall mean is positive), but where there is no sensible "objective" prior - you just have to subjectively specify how likely it is that the variance is very close to zero. Another even simpler example is inference for theta given an observation x~N(theta,1), when it is known (with certainty) that theta is non-negative, and the observed x is -1. There's just no alternative to subjectively deciding how likely a priori it is that theta is close to zero.

Frequentist methods also don't give sensible answers in these examples. Subjective Bayesianism is the only way.