Statistical Causal Modelling and Decision Theory

Abstract

Mathematical formalisms of causal inference usually depend on theories of causation, and are often used to analyse problems of data-driven decision making. We show that it is possible to formalise data-driven decision problems and analyse key assumptions using a more minimal theory that aims only to satisfy the requirements of decision makers, and not to additionally offer an account of causation.

Motivated by the literature on decision theory, we consider maps from a decision maker's set of options to probability distributions on a common sample space to be the object of our study, which we call a decision model. We extend standard probability theory to a theory of probability sets to support reasoning with models of this type. We also make use of a string diagram notation for stochastic functions.

Drawing nontrivial conclusions from decision making models requires nontrivial assumptions. Such assumptions are usually formulated using a theory of causation. We propose that symmetries of decision models may cut out this ``causal detour''. In particular, we investigate the assumption that a sequence of pairs is related by conditionally independent and identical responses (henceforth: CIIR sequences). We show that this assumption is equivalent to the assumption that different infinite sequences of pairs are, in a certain sense, interchangeable -- an assumption that we argue is usually unreasonable if the pairs in question are observable.

We show how causal models formulated using both the causal Bayesian network and potential outcomes approach can be represented as decision models with CIIR sequences involving latent variables. The two approaches each require a different extra assumption in order to be made compatible with ours. Causal Bayesian networks require a specification of how to ``unroll'' a structural model into a sequential model. A potential outcomes model requires a specification of how the decision maker's options relate to the rest of the model. Both approaches avoid the criticism of CIIR sequences we raise as the pairs in question are not fully observable.

The assumption of precedent is the assumption that ``whatever we can do has been done before'', and is weaker than the assumption of CIIR sequences. We show that the assumption of precedent in conjunction with a technical condition of regular relationships between conditionals can yield a conclusion of CIIR sequences when the data displays the right kind of conditional independence. The aforementioned technical condition is similar to a number of assumptions found in the literature that license the conclusion of a directed causal relationship from certain features of the given data. We speculate the assumption of precedent may offer an alternative way to understand directed causal relationships.