There is no decision theory that isn’t Bayesian... or is there?

David Manheim says in a comment under a blog post:

If you’re not making decisions, there’s no need for Bayes. If you are, you’re Bayesian whether you like it or not – there is no decision theory that isn’t Bayesian.

Let us ignore the first couple of statements here; they have been challenged by other commenters, and they are beyond the focus of this post. I am curious about the last statement: there is no decision theory that isn’t Bayesian. I am no expert in decision theory, but this sounds too general to be even approximately true. How crude a generalization is this? Are there no noteworthy decision theories that are nonbayesian?

• The statement is obviously false in any literal sense. I suspect the writer means he can characterize any decision rule as equivalent to a Bayesian rule with suitable choices (possibly very artificial) of prior, likelihood and loss function. With the implication that a rule that appears very artificial in this way may not be very good. In that sense the statement is probably not capable of being proved or disproved. Jul 30 at 8:51
• There are sub-optimal decision strategies ;o) Jul 30 at 9:01
• @GordonSmyth, this makes sense. My question is about the practical side of this. If every noteworthy decision theory is equivalent to a Bayesian one that itself is not too wacky, then I would say David Manheim's statement is "practically correct". If there exists a noteworthy nonbayesian theory and one has to bent over backwards to come up with an equivalent Bayesian one (and that one is kind of wacky), then the statement is "practically incorrect". What I just wrote is quite loose, but I hope the gist is more or less clear. Jul 30 at 9:06
• A fuzzy decision theory could not be expressed in Bayesian terms as it would violate one of the axioms of Bayesian probability. Jul 30 at 10:22
• @DikranMarsupial, thank you for a good example. Jul 30 at 12:25

The An Introduction to Decision Theory book by Martin Peterson (Cambridge, 2009) has chapter 10 titled Bayesian vs Non-Bayesian Decision Theory if this answers your question. Yes, there are non-Bayesian approaches, though I never found them interesting and don't feel competent enough for summarizing them.

• Thank you! The chapter is fairly short, I think I will be able to read it quickly and get back here. Jul 30 at 8:38
• OK, I read it. It helped a bit by explaining what a Bayesian decision theory is and what are its pros and cons. But it did not say much at all about alternative theories. Jul 30 at 9:57
• @RichardHardy yeah, it doesn't go into much detail. I don't remember, but if you have the book at hand you could check if maybe they discuss it in other chapters as well.
– Tim
Jul 30 at 10:44

Taking only the basic decision theory (theories?) based on maximization of expected utility, the answer depends on
(i) the definition of what is considered Bayesian and
(ii) whether a theory that has both a Bayesian and a nonbayesian interpretation or a nonbayesian theory that can be approximated by a Bayesian one count as counterexamples.

Chapter 10 of Martin Peterson's "An Introduction to Decision Theory" (Cambridge, 2009) from Tim's answer characterizes a Bayesian decision theory as follows:

1. Subjective degrees of belief can be represented by a probability function defined in terms of the decision maker's preferences over uncertain prospects.
2. Degrees of desire can be represented by a utility function defined in the same way, that is, in terms of preferences over uncertain prospects.
3. Rational decision makers act as if they maximise subjective expected utility.

Regarding (i): The step that seems particularly/genuinely Bayesian* to me is 1. but not necessarily 2. or 3. However, one could say all of them are as Bayesian as it gets because of the persons who developed and advanced such theories (Ramsey, de Finetti, Savage).

Regarding (ii): The framework above might be broad enough to cover theories of maximization of expected utility based on probability distributions arrived at without the use of Bayesian thinking. (I think this is commonly seen in microeconomics textbooks, at least on the introductory level; they often take probability distributions as given, leaving it up to the user to figure out how to come up with them in reality.) E.g. in step 1. one could develop a model of uncertain prospects and use frequentist or fiducial techniques to estimate its parameters without ever invoking Bayesian arguments. One could then proceed with steps 2. and 3. as above. Now, this could most likely be approximated by the framework above, and in that sense this would (almost?) count as a Bayesian decision theory. On the other hand, does that necessarily invalidate this as a counterexample?

(Work in progress)

*The definition and delimitation of the term genuinely Bayesian is certainly debatable and I may be making a mistake here.