Infra-Bayesianism doesn’t consider the worst case, since, even though each hypothesis is treated using the maximin decision rule, there is still a prior over many hypotheses[1]. One such hypothesis can upper bound the probability you will get a stroke in the next few seconds. An infra-Bayesian agent would learn this hypothesis and plan accordingly.
We might say that infra-Bayesianism assumes the worst only of that which is not only unknown but unknowable. To make a somewhat informal analogy with logic, we assume the worst model of the theory and thereby make any gain that can be gained provably.
One justification often given for Solomonoff induction is: we live in a simple universe. However, Solomonoff induction is uncomputable, so a simple universe cannot contain it. Instead, it might contain something like bounded Solomonoff induction. However, in order to justify bounded Solomonoff induction, we would need to assume that the universe is simple and cheap, which is false. In other words, postulating an “average-case” entails postulating a false dogmatic belief. Bounded “infra-Solomonoff” induction solves the problem by relying instead on the following assumption: the universe has some simple and cheap properties that can be exploited.
Like in the Bayesian case, you can alternatively think of the prior as just a single infradistribution, which is the mixture of all the hypotheses it is comprised of. This is an equivalent view.
Infra-Bayesianism doesn’t consider the worst case, since, even though each hypothesis is treated using the maximin decision rule, there is still a prior over many hypotheses[1]. One such hypothesis can upper bound the probability you will get a stroke in the next few seconds. An infra-Bayesian agent would learn this hypothesis and plan accordingly.
We might say that infra-Bayesianism assumes the worst only of that which is not only unknown but unknowable. To make a somewhat informal analogy with logic, we assume the worst model of the theory and thereby make any gain that can be gained provably.
One justification often given for Solomonoff induction is: we live in a simple universe. However, Solomonoff induction is uncomputable, so a simple universe cannot contain it. Instead, it might contain something like bounded Solomonoff induction. However, in order to justify bounded Solomonoff induction, we would need to assume that the universe is simple and cheap, which is false. In other words, postulating an “average-case” entails postulating a false dogmatic belief. Bounded “infra-Solomonoff” induction solves the problem by relying instead on the following assumption: the universe has some simple and cheap properties that can be exploited.
Like in the Bayesian case, you can alternatively think of the prior as just a single infradistribution, which is the mixture of all the hypotheses it is comprised of. This is an equivalent view.