Missingness-MDPs: Bridging the Theory of Missing Data and POMDPs

TL;DR

This paper introduces missingness-MDPs, a new subclass of POMDPs that models missing data patterns, and develops PAC algorithms to learn and plan with these models, showing improved performance over existing methods.

Contribution

It formalizes missing data within POMDPs as missingness-MDPs and provides PAC algorithms for learning and planning in this setting, bridging missing data theory and decision processes.

Findings

PAC algorithms effectively learn missingness functions from data.

Proposed methods achieve near-optimal policies with high probability.

Empirical results outperform two model-free POMDP approaches.

Abstract

We introduce missingness-MDPs (miss-MDPs), a novel subclass of partially observable Markov decision processes (POMDPs) that incorporates the theory of missing data. A miss-MDP is a POMDP whose observation function is a missingness function, specifying the probability that individual state features are missing (i.e., unobserved) at a time step. The literature distinguishes three canonical missingness types: missing (1) completely at random (MCAR), (2) at random (MAR), and (3) not at random (MNAR). Our planning problem is to compute near-optimal policies for a miss-MDP with an unknown missingness function, given a dataset of action-observation trajectories. Achieving such optimality guarantees for policies requires learning the missingness function from data, which is infeasible for general POMDPs. To overcome this challenge, we exploit the structural properties of different missingness types to derive probably approximately correct (PAC) algorithms for learning the missingness function. These algorithms yield an approximate but fully specified miss-MDP that we solve using off-the-shelf planning methods. We prove that, with high probability, the resulting policies are epsilon-optimal in the true miss-MDP. Empirical results confirm the theory and demonstrate superior performance of our approach over two model-free POMDP methods.