Causal models defined in terms of a collection of equations, as defined by Pearl, are axiomatized here. Axiomatizations are provided for three successively more general classes of causal models: (1) the class of recursive theories (those without feedback), (2) the class of theories where the solutions to the equations are unique, (3) arbitrary theories (where the equations may not have solutions and, if they do, they are not necessarily unique). It is shown that to reason about causality in the most general third class, we must extend the language used by Galles and Pearl (1997, 1998). In addition, the complexity of the decision procedures is characterized for all the languages and classes of models considered.

We show how to find a minimum weight loop cutset in a Bayesian network with high probability. Finding such a loop cutset is the first step in the method of conditioning for inference. Our randomized algorithm for finding a loop cutset outputs a minimum loop cutset after O(c 6^k kn) steps with probability at least 1 - (1 - 1/(6^k))^c6^k, where c > 1 is a constant specified by the user, k is the minimal size of a minimum weight loop cutset, and n is the number of vertices. We also show empirically that a variant of this algorithm often finds a loop cutset that is closer to the minimum weight loop cutset than the ones found by the best deterministic algorithms known.

Decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental prior probability distribution is known. Solomonoff's theory of universal induction formally solves the problem of sequence prediction for unknown prior distribution. We combine both ideas and get a parameterless theory of universal Artificial Intelligence. We give strong arguments that the resulting AIXI model is the most intelligent unbiased agent possible. We outline for a number of problem classes, including sequence prediction, strategic games, function minimization, reinforcement and supervised learning, how the AIXI model can formally solve them. The major drawback of the AIXI model is that it is uncomputable. To overcome this problem, we construct a modified algorithm AIXI-tl, which is still effectively more intelligent than any other time t and space l bounded agent. The computation time of AIXI-tl is of the order tx2^l. Other discussed topics are formal definitions of intelligence order relations, the horizon problem and relations of the AIXI theory to other AI approaches.

A major problem in machine learning is that of inductive bias: how to choose a learner's hypothesis space so that it is large enough to contain a solution to the problem being learnt, yet small enough to ensure reliable generalization from reasonably-sized training sets. Typically such bias is supplied by hand through the skill and insights of experts. In this paper a model for automatically learning bias is investigated. The central assumption of the model is that the learner is embedded within an environment of related learning tasks. Within such an environment the learner can sample from multiple tasks, and hence it can search for a hypothesis space that contains good solutions to many of the problems in the environment. Under certain restrictions on the set of all hypothesis spaces available to the learner, we show that a hypothesis space that performs well on a sufficiently large number of training tasks will also perform well when learning novel tasks in the same environment. Explicit bounds are also derived demonstrating that learning multiple tasks within an environment of related tasks can potentially give much better generalization than learning a single task.

Pearl and Dechter (1996) claimed that the d-separation criterion for conditional independence in acyclic causal networks also applies to networks of discrete variables that have feedback cycles, provided that the variables of the system are uniquely determined by the random disturbances. I show by example that this is not true in general. Some condition stronger than uniqueness is needed, such as the existence of a causal dynamics guaranteed to lead to the unique solution.

We show the similarity between belief propagation and TAP, for decoding corrupted messages encoded by Sourlas's method.

We show the similarity between belief propagation and TAP, for decoding corrupted messages encoded by Sourlas's method.

We show the similarity between belief propagation and TAP, for decoding corrupted messages encoded by Sourlas's method.

One of the simplest methods is to use linear transformations of the observed data.

There has been much recent work on measuring image statistics and on learning probability distributions on images. We observe that the mapping from images to statistics is many-to-one and show it can be quantified by a phase space factor. This phase space approach throws light on the Minimax Entropy technique for learning Gibbs distributions on images with potentials derived from image statistics and elucidates the ambiguities that are inherent to determining the potentials. In addition, it shows that if the phase factor can be approximated by an analytic distribution then this approximation yields a swift "Minutemax" algorithm that vastly reduces the computation time for Minimax entropy learning. An illustration of this concept, using a Gaussian to approximate the phase factor, gives a good approximation to the results of Zhu and Mumford (1997) in just seconds of CPU time. The phase space approach also gives insight into the multi-scale potentials found by Zhu and Mumford (1997) and suggests that the forms of the potentials are influenced greatly by phase space considerations. Finally, we prove that probability distributions learned in feature space alone are equivalent to Minimax Entropy learning with a multinomial approximation of the phase factor. 1 Introduction Bayesian probability theory gives a powerful framework for visual perception (Knill and Richards 1996). This approach, however, requires specifying prior probabilities and likelihood functions. Learning these probabilities is difficult because it requires estimating distributions on random variables of very high dimensions (for example, images with 200 x 200 pixels, or shape curves of length 400 pixels).