> "degrees of belief have to be represented by probability measures", "the philosophical thesis that graded rational belief is based on a probability measure"
Of course it all depends on how we want to define things, we agree on that. There is some "justification" for Bayesian inference if we accept some constraints. And even if there are alternatives - or extensions - to Bayesian epistemology I don't think they have produced a better inference method (or any, really). [I know your comment was about the philosophical foundations, not about the statistical methods. But the alternative statistical methods do not have better philosophical foundations.]
Sorry, I can't agree with you on that one at all. It doesn't "...all depend on how we want to define things." Whether the representation of an epistemic state -- any state, really -- is suitable and adequate for a task is not just a matter of definition, it depends on the reality of what you want to describe. You cannot represent the throw of a six-sided die with a set {1, 2, 3, 4, 5}, for example. If you model in ideal rational agent's belief with a probability measure, then you cannot adequately represent lack of belief. Whether that's okay or not depends on the task.
> I know your comment was about the philosophical foundations, not about the statistical methods.
Absolutely, at the risk of sounding picky I have to say that you've answered to a comment I've never made.
> But the alternative statistical methods do not have better philosophical foundations.
Frequentism and the propensity view have better philosophical justifications, though. You may disagree, but that was the whole point of my first comment. We know that there are genuine stochastic processes with corresponding objective probabilities, for example. Frequentism also prevents incorrect applications of probability such as using statistics to predict the outcome of singular events based on mere conjecture about the priors. You can only do that with an analytic model.
> Frequentism and the propensity view have better philosophical justifications, though.
Not really if the knowledge we care about is related to a concrete situation (rather than the frequency of something under hypothethical replications defined in some ad-hoc way). As you said, whether that's okay or not depends on the task.
If we care about whether there was life on Mars or whether Aduhlem is an effective treatment for Alzheimer's I don't think that frequentist inference has good philosophical support. Frequentist epistemology is not directly applicable.
Of course if you consider the frequentist methods themselves as genuine stochastic processes with corresponding objective probabilities (which also requires a valid model, by the way) you have good philosophical support to say things about those methods and their long-term frequency properties.
But this knowledge about the statistical methods used doesn't translate into knowledge about the existence of life on Mars or the efficacy of Aduhlem unless you are ready to make additional assumptions - 'philosophically unjustified' as they may be.
You're involuntarily confirming my negative criticism of Bayesianism by suggesting Bayesian methods could tell us whether there is life on Mars. Sometimes you really need to gather more information and/or develop an analytic model. It seems that a lot of Bayesianism consists of wishful thinking and trying to take shortcuts (e.g. trying to avoid randomized controlled trials for new drugs).
> suggesting Bayesian methods could tell us whether there is life on Mars.
What I suggest is that Bayesian methods provide a framework to reason about the plausability of some statement about the world in a systematic way (unlike Frequentist methods, whatever the limitations in Bayesian methods).
> Sometimes you really need to gather more information and/or develop an analytic model.
Bayesian methods are definitely not a way to escape the need for an analytic model (including all the prior knowledge) and data gathering. What they provide is a mechanism to integrate the data using the model and calculate the impact of incremental information on our knowledge / uncertainty.
I’m not saying that it’s easy to have a good model and useful data for complex questions. But with Frequentist methods in addition to the model and the data you’d be missing the mechanism to use them in a meaningful way.
I wonder why do you say that Bayesians try to avoid randomized controlled trials for new drugs, by the way. Bayesian methods are increasingly used in randomized clinical trials.
Of course it all depends on how we want to define things, we agree on that. There is some "justification" for Bayesian inference if we accept some constraints. And even if there are alternatives - or extensions - to Bayesian epistemology I don't think they have produced a better inference method (or any, really). [I know your comment was about the philosophical foundations, not about the statistical methods. But the alternative statistical methods do not have better philosophical foundations.]