Process-Conditioned Investing with Incomplete Information using Maximum Causal Entropy

Investing to optimally maximize the growth rate of wealth based on sequences of event outcomes has many information-theoretic interpretations. Namely, the mutual information characterizes the benefit of additional side information being available when making investment decisions [1] in settings where the probabilistic relationships between side information and event outcomes are known. Additionally, the relative variant of the principle of maximum entropy [2] provides the optimal investment allocation in the more general setting where the relationships between side information and event outcomes are only partially known [3]. In this paper, we build upon recent work characterizing the growth rates of investment in settings with inter-dependent side information and event outcome sequences [4]. We consider the extension to settings with inter-dependent event outcomes and side information where the probabilistic relationships between side information and event outcomes are only partially known. We introduce the principle of minimum relative causal entropy to obtain the optimal worst-case investment allocations for this setting. We present efficient algorithms for obtaining these investment allocations using convex optimization techniques and dynamic programming that illustrates a close connection to optimal control theory.