This is essentially by definition--intervention on Z doesn't change the potential outcomes, so it doesn't change the value of f (X). If f is a counterfactually invariant predictor: 1. Let L be either square error or cross entropy loss. Suppose that the target distribution Q is causally compatible with the training distribution P . Suppose that any of the following conditions hold: 1. the data obeys the anti-causal graph 2. the data obeys the causal-direction graph, there is no confounding (but possibly selection), and the association is purely spurious, Y X | X We begin with the anti-causal case.

The paper targets the problem of measuring the representation power of Graph neural networks (GNNs), an interesting and important topic, that has become popular recently (partially due to two prominent works (Xu et al. There are three main contributions: 1. Establishing the equivalence between two methods for measuring GNN representation power: (i) their ability to approximate permutation invariant functions (ii) their ability to distinguish non-isomorphic graphs. Although not very surprising, this is a nice observation. The authors show that these sigma algebras are an equivalent way to measure representation power of GNNs, for instance, the inclusion of sigma algebras originating from two models is equivalent to saying one model is more powerful than the other. This is a potentially useful observation.

We present a new game-theoretic framework in which Bayesian players with bounded rationality engage in a Markov game and each has private but incomplete information regarding other players' types. Instead of utilizing Harsanyi's abstract types and a common prior, we construct intentional player types whose structure is explicit and induces a {\em finite-level} belief hierarchy. We characterize an equilibrium in this game and establish the conditions for existence of the equilibrium. The computation of finding such equilibria is formalized as a constraint satisfaction problem and its effectiveness is demonstrated on two cooperative domains.

In impromptu or ad hoc settings, participating players are precluded from precoordination. Subsequently, each player's own model is private and includes some uncertainty about the others' types or behaviors. Harsanyi's formulation of a Bayesian game lays emphasis on this uncertainty while the players each play exactly one turn. We propose a new game-theoretic framework where Bayesian players engage in a Markov game and each has private but imperfect information regarding other players' types. Consequently, we construct player types whose structure is explicit and includes a finite level belief hierarchy instead of utilizing Harsanyi's abstract types and a common prior distribution. We formalize this new framework and demonstrate its effectiveness on two standard ad hoc teamwork domains involving two or more ad hoc players.

In 1998, it was proved by Ben-David and Litman that a concept space has a sample compression scheme of size d if and only if every finite subspace has a sample compression scheme of size d. In the compactness theorem, measurability of the hypotheses of the created sample compression scheme is not guaranteed; at the same time measurability of the hypotheses is a necessary condition for learnability. In this thesis we discuss when a sample compression scheme, created from compression schemes on finite subspaces via the compactness theorem, have measurable hypotheses. We show that if X is a standard Borel space with a d-maximum and universally separable concept class C, then (X, C) has a sample compression scheme of size d with universally Borel measurable hypotheses. Additionally we introduce a new variant of compression scheme called a copy sample compression scheme.