MAP is the problem of finding a most probable instantiation of a set of variables given evidence. MAP has always been perceived to be significantly harder than the related problems of computing the probability of a variable instantiation Pr, or the problem of computing the most probable explanation (MPE). This paper investigates the complexity of MAP in Bayesian networks. Specifically, we show that MAP is complete for NP^PP and provide further negative complexity results for algorithms based on variable elimination. We also show that MAP remains hard even when MPE and Pr become easy. For example, we show that MAP is NP-complete when the networks are restricted to polytrees, and even then can not be effectively approximated. Given the difficulty of computing MAP exactly, and the difficulty of approximating MAP while providing useful guarantees on the resulting approximation, we investigate best effort approximations. We introduce a generic MAP approximation framework. We provide two instantiations of the framework; one for networks which are amenable to exact inference Pr, and one for networks for which even exact inference is too hard. This allows MAP approximation on networks that are too complex to even exactly solve the easier problems, Pr and MPE. Experimental results indicate that using these approximation algorithms provides much better solutions than standard techniques, and provide accurate MAP estimates in many cases.

This paper is concerned with algorithms for prediction of discrete sequences over a finite alphabet, using variable order Markov models. The class of such algorithms is large and in principle includes any lossless compression algorithm. We focus on six prominent prediction algorithms, including Context Tree Weighting (CTW), Prediction by Partial Match (PPM) and Probabilistic Suffix Trees (PSTs). We discuss the properties of these algorithms and compare their performance using real life sequences from three domains: proteins, English text and music pieces. The comparison is made with respect to prediction quality as measured by the average log-loss. We also compare classification algorithms based on these predictors with respect to a number of large protein classification tasks. Our results indicate that a "decomposed" CTW (a variant of the CTW algorithm) and PPM outperform all other algorithms in sequence prediction tasks. Somewhat surprisingly, a different algorithm, which is a modification of the Lempel-Ziv compression algorithm, significantly outperforms all algorithms on the protein classification problems.

Hierarchical latent class (HLC) models are tree-structured Bayesian networks where leaf nodes are observed while internal nodes are latent. There are no theoretically well justified model selection criteria for HLC models in particular and Bayesian networks with latent nodes in general. Nonetheless, empirical studies suggest that the BIC score is a reasonable criterion to use in practice for learning HLC models. Empirical studies also suggest that sometimes model selection can be improved if standard model dimension is replaced with effective model dimension in the penalty term of the BIC score. Effective dimensions are difficult to compute. In this paper, we prove a theorem that relates the effective dimension of an HLC model to the effective dimensions of a number of latent class models. The theorem makes it computationally feasible to compute the effective dimensions of large HLC models. The theorem can also be used to compute the effective dimensions of general tree models.

Inductive learning is based on inferring a general rule from a finite data set and using it to label new data. In transduction one attempts to solve the problem of using a labeled training set to label a set of unlabeled points, which are given to the learner prior to learning. Although transduction seems at the outset to be an easier task than induction, there have not been many provably useful algorithms for transduction. Moreover, the precise relation between induction and transduction has not yet been determined. The main theoretical developments related to transduction were presented by Vapnik more than twenty years ago. One of Vapnik's basic results is a rather tight error bound for transductive classification based on an exact computation of the hypergeometric tail. While tight, this bound is given implicitly via a computational routine. Our first contribution is a somewhat looser but explicit characterization of a slightly extended PAC-Bayesian version of Vapnik's transductive bound. This characterization is obtained using concentration inequalities for the tail of sums of random variables obtained by sampling without replacement. We then derive error bounds for compression schemes such as (transductive) support vector machines and for transduction algorithms based on clustering. The main observation used for deriving these new error bounds and algorithms is that the unlabeled test points, which in the transductive setting are known in advance, can be used in order to construct useful data dependent prior distributions over the hypothesis space.

Value iteration is a popular algorithm for finding near optimal policies for POMDPs. It is inefficient due to the need to account for the entire belief space, which necessitates the solution of large numbers of linear programs. In this paper, we study value iteration restricted to belief subsets. We show that, together with properly chosen belief subsets, restricted value iteration yields near-optimal policies and we give a condition for determining whether a given belief subset would bring about savings in space and time. We also apply restricted value iteration to two interesting classes of POMDPs, namely informative POMDPs and near-discernible POMDPs.

We propose a model for errors in sung queries, a variant of the hidden Markov model (HMM). This is a solution to the problem of identifying the degree of similarity between a (typically error-laden) sung query and a potential target in a database of musical works, an important problem in the field of music information retrieval. Similarity metrics are a critical component of query-by-humming (QBH) applications which search audio and multimedia databases for strong matches to oral queries. Our model comprehensively expresses the types of error or variation between target and query: cumulative and non-cumulative local errors, transposition, tempo and tempo changes, insertions, deletions and modulation. The model is not only expressive, but automatically trainable, or able to learn and generalize from query examples. We present results of simulations, designed to assess the discriminatory potential of the model, and tests with real sung queries, to demonstrate relevance to real-world applications.

We state the problem of inverse reinforcement learning in terms of preference elicitation, resulting in a principled (Bayesian) statistical formulation. This generalises previous work on Bayesian inverse reinforcement learning and allows us to obtain a posterior distribution on the agent's preferences, policy and optionally, the obtained reward sequence, from observations. We examine the relation of the resulting approach to other statistical methods for inverse reinforcement learning via analysis and experimental results. We show that preferences can be determined accurately, even if the observed agent's policy is sub-optimal with respect to its own preferences. In that case, significantly improved policies with respect to the agent's preferences are obtained, compared to both other methods and to the performance of the demonstrated policy.

Human intuition has been simulated by several research projects using artificial intelligence techniques. Most of these algorithms or models lack the ability to handle complications or diversions. Moreover, they also do not explain the factors influencing intuition and the accuracy of the results from this process. In this paper, we present a simple series based model for implementation of human-like intuition using the principles of connectivity and unknown entities. By using Poker hand datasets and Car evaluation datasets, we compare the performance of some well-known models with our intuition model. The aim of the experiment was to predict the maximum accurate answers using intuition based models. We found that the presence of unknown entities, diversion from the current problem scenario, and identifying weakness without the normal logic based execution, greatly affects the reliability of the answers. Generally, the intuition based models cannot be a substitute for the logic based mechanisms in handling such problems. The intuition can only act as a support for an ongoing logic based model that processes all the steps in a sequential manner. However, when time and computational cost are very strict constraints, this intuition based model becomes extremely important and useful, because it can give a reasonably good performance. Factors affecting intuition are analyzed and interpreted through our model.

As computational agents are developed for increasingly complicated e-commerce applications, the complexity of the decisions they face demands advances in artificial intelligence techniques. For example, an agent representing a seller in an auction should try to maximize the seller's profit by reasoning about a variety of possibly uncertain pieces of information, such as the maximum prices various buyers might be willing to pay, the possible prices being offered by competing sellers, the rules by which the auction operates, the dynamic arrival and matching of offers to buy and sell, and so on. A naive application of multiagent reasoning techniques would require the seller's agent to explicitly model all of the other agents through an extended time horizon, rendering the problem intractable for many realistically-sized problems. We have instead devised a new strategy that an agent can use to determine its bid price based on a more tractable Markov chain model of the auction process. We have experimentally identified the conditions under which our new strategy works well, as well as how well it works in comparison to the optimal performance the agent could have achieved had it known the future. Our results show that our new strategy in general performs well, outperforming other tractable heuristic strategies in a majority of experiments, and is particularly effective in a 'seller?s market', where many buy offers are available.

The aim of this chapter is twofold. In the first part we will provide a brief overview of the mathematical and statistical foundations of graphical models, along with their fundamental properties, estimation and basic inference procedures. In particular we will develop Markov networks (also known as Markov random fields) and Bayesian networks, which comprise most past and current literature on graphical models. In the second part we will review some applications of graphical models in systems biology.