The very expressiveness of Bayesian networks can introduce fresh challenges due to the large number of relationships they often model. In many domains, it is thus often essential to supplement any available data with elicited expert judgements. This in turn leads to two key challenges: the cognitive burden of these judgements is often very high, and there are a very large number of judgements required to obtain a full probability model. We can mitigate both issues by introducing assumptions such as independence of causal influences (ICI) on the local structures throughout the network, restricting the parameter space of the model. However, the assumption of ICI is often unjustified and overly strong. In this paper, we introduce the surjective independence of causal influences (SICI) model which relaxes the ICI assumption and provides a more viable, practical alternative local structure model that facilitates efficient Bayesian network parameterisation.

Classical approaches for approximate inference depend on cleverly designed variational distributions and bounds. Modern approaches employ amortized variational inference, which uses a neural network to approximate any posterior without leveraging the structures of the generative models. In this paper, we propose Amortized Conjugate Posterior (ACP), a hybrid approach taking advantages of both types of approaches. Specifically, we use the classical methods to derive specific forms of posterior distributions and then learn the variational parameters using amortized inference. We study the effectiveness of the proposed approach on the Noisy-OR model and compare to both the classical and the modern approaches for approximate inference and parameter learning. Our results show that ACP outperforms other methods when there is a limited amount of training data.

A continuous time Bayesian network is a graphical model capable of describing discrete state systems that evolve in continuous time. Unfortunately, the number of parameters required for each node in the graph is exponential in the number of parents of the node, which can be prohibitively large for many real-world systems. To mitigate this problem, we propose a Noisy-OR model for continuous time Bayesian networks, which can reduce the number of required parameters from exponential to linear. We describe the model, as well as the process required to compute the remaining unspecified parameters. Finally, we experimentally validate the correctness of the proposed Noisy-OR formulation.

The Noisy-Or model is convenient for describing a class of uncertain relationships in Bayesian networks [Pearl 1988]. Pearl describes the Noisy-Or model for Boolean variables. Here we generalize the model to nary input and output variables and to arbitrary functions other than the Boolean OR function. This generalization is a useful modeling aid for construction of Bayesian networks. We illustrate with some examples including digital circuit diagnosis and network reliability analysis.

Given a set of experiments in which varying subsets of observed variables are subject to intervention, we consider the problem of identifiability of causal models exhibiting latent confounding. While identifiability is trivial when each experiment intervenes on a large number of variables, the situation is more complicated when only one or a few variables are subject to intervention per experiment. For linear causal models with latent variables Hyttinen et al. (2010) gave precise conditions for when such data are sufficient to identify the full model. While their result cannot be extended to discrete-valued variables with arbitrary cause-effect relationships, we show that a similar result can be obtained for the class of causal models whose conditional probability distributions are restricted to a `noisy-OR' parameterization. We further show that identification is preserved under an extension of the model that allows for negative influences, and present learning algorithms that we test for accuracy, scalability and robustness.

The Indian Buffet Process is a Bayesian nonparametric approach that models objects as arising from an infinite number of latent factors. Here we extend the latent factor model framework to two or more unbounded layers of latent factors. From a generative perspective, each layer defines a conditional \emph{factorial} prior distribution over the binary latent variables of the layer below via a noisy-or mechanism. We explore the properties of the model with two empirical studies, one digit recognition task and one music tag data experiment.

We show that linear generalizations of Rescorla-Wagner can perform Maximum Likelihood estimation of the parameters of all generative models for causal reasoning. Our approach involves augmenting variables to deal with conjunctions of causes, similar to the agumented model of Rescorla. Our results involve genericity assumptions on the distributions of causes. If these assumptions are violated, for example for the Cheng causal power theory, then we show that a linear Rescorla-Wagner can estimate the parameters of the model up to a nonlinear transformtion. Moreover, a nonlinear Rescorla-Wagner is able to estimate the parameters directly to within arbitrary accuracy. Previous results can be used to determine convergence and to estimate convergence rates.

We show that linear generalizations of Rescorla-Wagner can perform Maximum Likelihood estimation of the parameters of all generative models for causal reasoning. Our approach involves augmenting variables to deal with conjunctions of causes, similar to the agumented model of Rescorla. Our results involve genericity assumptions on the distributions of causes. If these assumptions are violated, for example for the Cheng causal power theory, then we show that a linear Rescorla-Wagner can estimate the parameters of the model up to a nonlinear transformtion. Moreover, a nonlinear Rescorla-Wagner is able to estimate the parameters directly to within arbitrary accuracy. Previous results can be used to determine convergence and to estimate convergence rates.

Moreover, a nonlinear Rescorla-Wagner is able to estimate the parameters directly to within arbitrary accuracy.

In this paper we explore two quantitative approaches to the modelling of counterfactual reasoning - a linear and a noisy-OR model - based on information contained in conceptual dependency networks. Empirical data is acquired in a study and the fit of the models compared to it. We conclude by considering the appropriateness of nonparametric approaches to counterfactual reasoning, and examining the prospects for other parametric approaches in the future.