MIRI Research Associate Vadim Kosoy has a paper out on issues in naturalized induction: “Forecasting using incomplete models”. Abstract:
We consider the task of forecasting an infinite sequence of future observations based on some number of past observations, where the probability measure generating the observations is “suspected” to satisfy one or more of a set of incomplete models, i.e., convex sets in the space of probability measures.
This setting is in some sense intermediate between the realizable setting where the probability measure comes from some known set of probability measures (which can be addressed using e.g. Bayesian inference) and the unrealizable setting where the probability measure is completely arbitrary.
We demonstrate a method of forecasting which guarantees that, whenever the true probability measure satisfies an incomplete model in a given countable set, the forecast converges to the same incomplete model in the (appropriately normalized) Kantorovich-Rubinstein metric. This is analogous to merging of opinions for Bayesian inference, except that convergence in the Kantorovich-Rubinstein metric is weaker than convergence in total variation.
Kosoy’s work builds on logical inductors to create a cleaner (purely learning-theoretic) formalism for modeling complex environments, showing that the methods developed in “Logical induction” are useful for applications in classical sequence prediction unrelated to logic.
“Forecasting using incomplete models” also shows that the intuitive concept of an “incomplete” or “partial” model has an elegant and useful formalization related to Knightian uncertainty. Additionally, Kosoy shows that using incomplete models to generalize Bayesian inference allows an agent to make predictions about environments that can be as complex as the agent itself, or more complex — as contrasted with classical Bayesian inference.
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