PHYSICAL LIMITS OF INFERENCE

David Wolpert, NASA Ames Research Center

I show that physical devices that perform observation, prediction, or recollection share an underlying mathematical structure. I call devices with that structure ``inference devices''. I present a set of existence and impossibility results concerning inference devices. These results hold independent of the precise physical laws governing our universe. In a limited sense, the impossibility results establish that Laplace was wrong to claim that even in a classical, non-chaotic universe the future can be unerringly predicted, given sufficient knowledge of the present. Alternatively, these impossibility results can be viewed as a non-quantum mechanical ``uncertainty principle''.

The mathematics of inference devices has close connections to the mathematics of Turing Machines (TM's). In particular, the impossibility results for inference devices are similar to the Halting theorem for TM's. Furthermore, one can define an analog of Universal TM's (UTM's) for inference devices. I call those analogs ``strong inference devices''. I use strong inference devices to define the ``inference complexity'' of an inference task, which is the analog of the Kolmogorov complexity of computing a string. A task-independent bound is derived on how much the inference complexity of an inference task can differ for two different inference devices. This is analogous to the ``encoding'' bound governing how much the Kolmogorov complexity of a string can differ between two UTM's used to compute that string. However no universe can contain more than one strong inference device. So whereas the Kolmogorov complexity of a string is arbitrary up to specification of the UTM, there is no such arbitrariness in the inference complexity of an inference task.

I informally discuss the philosophical implications of these results, e.g., for whether the universe ``is'' a computer. I also derive some graph-theoretic properties governing any set of multiple inference devices. I also present an extension of the framework to address physical devices used for control. I end with an extension of the framework to address probabilistic inference.

I explore a Bayesian version of such a Predictive Game Theory (PGT) using the entropic prior and a likelihood that quantifies the rationalities of the players. A popular game theory equilibrium concept parameterized by player rationalities is the Quantal Response Equilibrium concept (QRE). I show that for some games the local peaks of the posterior density over joint strategies approximate the associated QRE's, and derive the associated correction terms. I also discuss how to estimate parameters of the likelihood from observational data. I end by showing how PGT specifies a unique equilibrium of any game, thereby solving a long-standing problem of conventional game theory.