When estimating the causal function between a vector of covariates X and a response Y in the presence of unobserved confounding, standard regression procedures such as ordinary least squares (OLS) are even asymptotically biased. Instrumental variable approaches (Wright, 1928; Imbens and Angrist, 1994; Newey, 2013) exploit the existence of exogenous heterogeneity in the form of an instrumental variable (IV) Z and estimate, under suitable conditions, the causal function consistently. Importantly, the errors in Y and the hidden confounders U should be uncorrelated with the instruments Z. Usually, this has to be argued for with background knowledge. When the data generating process is modeled by a structural causal model (SCM) (Pearl, 2009; Bongers et al., 2021) (so that the distribution is Markov with respect to the induced graph), then the above condition is satisfied if Y and U are d-separated from Z in the graph obtained by removing the edge from X to Y. Furthermore, in this case the errors in Y and U are even independent from Z. Using that the errors and instruments are not only uncorrelated but also independent comes with several benefits. For example, even in settings, where the causal function can be identified by classical approaches based on uncorrelatedness, the independence can be exploited to construct estimators that achieve the semiparametric efficiency bound, at least when the error distribution comes from a known, parametric family (Hansen et al., 2010). Furthermore, the independence constraint is stronger than uncorrelatedness and therefore yields stronger identifiability results, which has been reported in the field of econometrics (e.g., Imbens and Newey, 2009; Chesher, 2003).

Non-convex optimization is a critical component in modern machine learning. Unfortunately, theoretical guarantees for nonconvex optimization have been mostly negative, and the problems are computationally hard in the worst case. Nevertheless, simple local-search algorithms such as stochastic gradient descent have enjoyed great empirical success in areas such as deep learning. As such, recent research efforts have attempted to bridge this gap between theory and practice. For example, one property that can guarantee the success of local search methods over nonconvex functions is when all local minima are also the global minima. Interestingly, it has been recently proven that many well known nonconvex problems do have this property, under mild conditions. Consequently, local-search methods, which are designed to find a local optimum, automatically achieve global optimality. Examples of such problems include matrix completion [1], orthogonal tensor decomposition [2, 3], phase retrieval [4], complete dictionary learning [5], and so on. However, such a class of nonconvex problems is limited, and there are many practical problems with poor local optima, where local search methods can fail.

We show that the only parameter prior for complete Gaussian DAG models that satisfies global parameter independence, complete model equivalence, and some weak regularity assumptions, is the normal-Wishart distribution. Our analysis is based on the following new characterization of the Wishart distribution: let W be an n x n, n >= 3, positive-definite symmetric matrix of random variables and f(W) be a pdf of W. Then, f(W) is a Wishart distribution if and only if W_{11}-W_{12}W_{22}^{-1}W_{12}' is independent of {W_{12}, W_{22}} for every block partitioning W_{11}, W_{12}, W_{12}', W_{22} of W. Similar characterizations of the normal and normal-Wishart distributions are provided as well. We also show how to construct a prior for every DAG model over X from the prior of a single regression model.

We introduce a new interpretation of two related notions - conditional utility and utility independence. Unlike the traditional interpretation, the new interpretation renders the notions the direct analogues of their probabilistic counterparts. To capture these notions formally, we appeal to the notion of utility distribution, introduced in previous paper. We show that utility distributions, which have a structure that is identical to that of probability distributions, can be viewed as a special case of an additive multiattribute utility functions, and show how this special case permits us to capture the novel senses of conditional utility and utility independence. Finally, we present the notion of utility networks, which do for utilities what Bayesian networks do for probabilities. Specifically, utility networks exploit the new interpretation of conditional utility and utility independence to compactly represent a utility distribution.

Programs that make decisions need mechanisms for representing and reasoning about the desirability of the possible consequences of their choices. This work is an exploration of preference models based on utility theory. The framework presented is distinguished by a qualitative view of preferences and a knowledge-based approach to the application of utility theory. The design for a comprehensive preference modeler is implemented in part by the U tility R easoning P ackage (URP), a collection of facilities for constructing and analyzing preference models. Qualitative mathematical reasoning techniques are employed to develop partial specifications of single-attribute utility functions from qualitative preference assertions.