The inferential method that solves the problems with frequentism — and, more importantly, follows deductively from the axioms of probability theory — is Bayesian inference.
You seem to be conflating Bayesian inference with Bayes Theorem. Bayesian inference is a method, not a proposition, so cannot be the conclusion of a deductive argument. Perhaps the conclusion you have in mind is something like “We should use Bayesian inference for...” or “Bayesian inference is the best method for...”. But such propositions cannot follow from mathematical axioms alone.
Moreover, the fact that Bayes Theorem follows from certain axioms of probability doesn’t automatically show that it’s true. Axiomatic systems have no relevance to the real world unless we have established (whether explicitly or implicitly) some mapping of the language of that system onto the real world. Unless we’ve done that, the word “probability” as used in Bayes Theorem is just a symbol without relevance to the world, and to say that Bayes Theorem is “true” is merely to say that it is a valid statement in the language of that axiomatic system.
In practice, we are liable to take the word “probability” (as used in the mathematical axioms of probability) as having the same meaning as “probability” (as we previously used that word). That meaning has some relevance to the real world. But if we do that, we cannot simply take the axioms (and consequently Bayes Theorem) as automatically true. We must consider whether they are true given our meaning of the word “probability”. But “probability” is a notoriously tricky word, with multiple “interpretations” (i.e. meanings). We may have good reason to think that the axioms of probability (and hence Bayes Theorem) are true for one meaning of “probability” (e.g. frequentist). But it doesn’t automatically follow that they are also true for other meanings of “probability” (e.g. Bayesian).
I’m not denying that Bayesian inference is a valuable method, or that it has some sort of justification. But justifying it is not nearly so straightforward as your comment suggests, Luke.
I’m not denying that Bayesian inference is a valuable method, or that it has some sort of justification. But justifying it is not nearly so straightforward as your comment suggests, Luke.
It stands on the foundations of probability theory, and while foundational stuff like Cox’s theorem takes some slogging through, once that’s in place, it is quite straightforward to justify Bayesian inference.
It’s actually somewhat tricky to establish that the rules of probability apply to the Frequentist meaning of probability. You have to mess around with long run frequencies and infinite limits. Even once that’s done, it hard to make the case that the Frequentist meaning has anything to do with the real world—there are no such thing as infinitely repeatable experiments.
In contrast, a few simple desiderata for “logical reasoning under uncertainty” establish probability theory as the only consistent way to do so that satisfy those criteria. Sure, other criteria may suggest some other way of doing so, but no one has put forward any such reasonable way.
In contrast, a few simple desiderata for “logical reasoning under uncertainty” establish probability theory as the only consistent way to do so that satisfy those criteria. Sure, other criteria may suggest some other way of doing so, but no one has put forward any such reasonable way.
Could Dempster-Shafer theory count? I haven’t seen anyone do a Cox-style derivation of it, but I would guess there’s something analogous in Shafer’s original book.
P.S. Bayes Theorem is derived from a basic statement about conditional probability, such as the following:
P(S/T) = P(S&T)/P(T)
According to the SEP (http://plato.stanford.edu/entries/epistemology-bayesian/) this is usually taken as a “definition”, not an axiom, and Bayesians usually give conditional probability some real-world significance by adding a Principle of Conditionalization. In that case it’s the Principle of Conditionalization that requires justification in order to establish that Bayes Theorem is true in the sense that Bayesians require.
Just to follow up on the previous replies to this line of thought, see Wikipedia’s article on Cox’s theorem and especially reference 6 of that article.
On the Principle of Conditionalization, it might be argued that Cox’s theorem assumes it as a premise; the easiest way to derive it from more basic considerations is through a diachronic Dutch book argument.
You seem to be conflating Bayesian inference with Bayes Theorem. Bayesian inference is a method, not a proposition, so cannot be the conclusion of a deductive argument. Perhaps the conclusion you have in mind is something like “We should use Bayesian inference for...” or “Bayesian inference is the best method for...”. But such propositions cannot follow from mathematical axioms alone.
Moreover, the fact that Bayes Theorem follows from certain axioms of probability doesn’t automatically show that it’s true. Axiomatic systems have no relevance to the real world unless we have established (whether explicitly or implicitly) some mapping of the language of that system onto the real world. Unless we’ve done that, the word “probability” as used in Bayes Theorem is just a symbol without relevance to the world, and to say that Bayes Theorem is “true” is merely to say that it is a valid statement in the language of that axiomatic system.
In practice, we are liable to take the word “probability” (as used in the mathematical axioms of probability) as having the same meaning as “probability” (as we previously used that word). That meaning has some relevance to the real world. But if we do that, we cannot simply take the axioms (and consequently Bayes Theorem) as automatically true. We must consider whether they are true given our meaning of the word “probability”. But “probability” is a notoriously tricky word, with multiple “interpretations” (i.e. meanings). We may have good reason to think that the axioms of probability (and hence Bayes Theorem) are true for one meaning of “probability” (e.g. frequentist). But it doesn’t automatically follow that they are also true for other meanings of “probability” (e.g. Bayesian).
I’m not denying that Bayesian inference is a valuable method, or that it has some sort of justification. But justifying it is not nearly so straightforward as your comment suggests, Luke.
It stands on the foundations of probability theory, and while foundational stuff like Cox’s theorem takes some slogging through, once that’s in place, it is quite straightforward to justify Bayesian inference.
It’s actually somewhat tricky to establish that the rules of probability apply to the Frequentist meaning of probability. You have to mess around with long run frequencies and infinite limits. Even once that’s done, it hard to make the case that the Frequentist meaning has anything to do with the real world—there are no such thing as infinitely repeatable experiments.
In contrast, a few simple desiderata for “logical reasoning under uncertainty” establish probability theory as the only consistent way to do so that satisfy those criteria. Sure, other criteria may suggest some other way of doing so, but no one has put forward any such reasonable way.
Could Dempster-Shafer theory count? I haven’t seen anyone do a Cox-style derivation of it, but I would guess there’s something analogous in Shafer’s original book.
I would be quite interested in seeing such. Unfortunately I don’t have any time to look for such in the foreseeable future.
P.S. Bayes Theorem is derived from a basic statement about conditional probability, such as the following:
P(S/T) = P(S&T)/P(T)
According to the SEP (http://plato.stanford.edu/entries/epistemology-bayesian/) this is usually taken as a “definition”, not an axiom, and Bayesians usually give conditional probability some real-world significance by adding a Principle of Conditionalization. In that case it’s the Principle of Conditionalization that requires justification in order to establish that Bayes Theorem is true in the sense that Bayesians require.
Just to follow up on the previous replies to this line of thought, see Wikipedia’s article on Cox’s theorem and especially reference 6 of that article.
On the Principle of Conditionalization, it might be argued that Cox’s theorem assumes it as a premise; the easiest way to derive it from more basic considerations is through a diachronic Dutch book argument.