Mutual information is widely used in artificial intelligence, in a descriptive way, to measure the stochastic dependence of discrete random variables. In order to address questions such as the reliability of the empirical value, one must consider sample-to-population inferential approaches. This paper deals with the distribution of mutual information, as obtained in a Bayesian framework by a second-order Dirichlet prior distribution. The exact analytical expression for the mean and an analytical approximation of the variance are reported. Asymptotic approximations of the distribution are proposed. The results are applied to the problem of selecting features for incremental learning and classification of the naive Bayes classifier. A fast, newly defined method is shown to outperform the traditional approach based on empirical mutual information on a number of real data sets. Finally, a theoretical development is reported that allows one to efficiently extend the above methods to incomplete samples in an easy and effective way.

* Appears in Proceedings of the Eighteenth Conference on Uncertainty in Artificial Intelligence (UAI2002)
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This paper is concerned with the reliable inference of optimal tree-approximations to the dependency structure of an unknown distribution generating data. The traditional approach to the problem measures the dependency strength between random variables by the index called mutual information. In this paper reliability is achieved by Walley's imprecise Dirichlet model, which generalizes Bayesian learning with Dirichlet priors. Adopting the imprecise Dirichlet model results in posterior interval expectation for mutual information, and in a set of plausible trees consistent with the data. Reliable inference about the actual tree is achieved by focusing on the substructure common to all the plausible trees. We develop an exact algorithm that infers the substructure in time O(m^4), m being the number of random variables. The new algorithm is applied to a set of data sampled from a known distribution. The method is shown to reliably infer edges of the actual tree even when the data are very scarce, unlike the traditional approach. Finally, we provide lower and upper credibility limits for mutual information under the imprecise Dirichlet model. These enable the previous developments to be extended to a full inferential method for trees.

* Annals of Mathematics and Artificial Intelligence, 45 (2005) 215-239
* 26 pages, 7 figures
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Mutual information is widely used, in a descriptive way, to measure the stochastic dependence of categorical random variables. In order to address questions such as the reliability of the descriptive value, one must consider sample-to-population inferential approaches. This paper deals with the posterior distribution of mutual information, as obtained in a Bayesian framework by a second-order Dirichlet prior distribution. The exact analytical expression for the mean, and analytical approximations for the variance, skewness and kurtosis are derived. These approximations have a guaranteed accuracy level of the order O(1/n^3), where n is the sample size. Leading order approximations for the mean and the variance are derived in the case of incomplete samples. The derived analytical expressions allow the distribution of mutual information to be approximated reliably and quickly. In fact, the derived expressions can be computed with the same order of complexity needed for descriptive mutual information. This makes the distribution of mutual information become a concrete alternative to descriptive mutual information in many applications which would benefit from moving to the inductive side. Some of these prospective applications are discussed, and one of them, namely feature selection, is shown to perform significantly better when inductive mutual information is used.

* Computational Statistics & Data Analysis, Vol.48, No.3, March 2005, pages 633--657
* 26 pages, LaTeX, 5 figures, 4 tables
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Given the joint chances of a pair of random variables one can compute quantities of interest, like the mutual information. The Bayesian treatment of unknown chances involves computing, from a second order prior distribution and the data likelihood, a posterior distribution of the chances. A common treatment of incomplete data is to assume ignorability and determine the chances by the expectation maximization (EM) algorithm. The two different methods above are well established but typically separated. This paper joins the two approaches in the case of Dirichlet priors, and derives efficient approximations for the mean, mode and the (co)variance of the chances and the mutual information. Furthermore, we prove the unimodality of the posterior distribution, whence the important property of convergence of EM to the global maximum in the chosen framework. These results are applied to the problem of selecting features for incremental learning and naive Bayes classification. A fast filter based on the distribution of mutual information is shown to outperform the traditional filter based on empirical mutual information on a number of incomplete real data sets.

* Proceedings of the 26th German Conference on Artificial Intelligence (KI-2003) 396-406
* 11 pages, 1 figure
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The ability to recover from a fall is an essential feature for a legged robot to navigate in challenging environments robustly. Until today, there has been very little progress on this topic. Current solutions mostly build upon (heuristically) predefined trajectories, resulting in unnatural behaviors and requiring considerable effort in engineering system-specific components. In this paper, we present an approach based on model-free Deep Reinforcement Learning (RL) to control recovery maneuvers of quadrupedal robots using a hierarchical behavior-based controller. The controller consists of four neural network policies including three behaviors and one behavior selector to coordinate them. Each of them is trained individually in simulation and deployed directly on a real system. We experimentally validate our approach on the quadrupedal robot ANYmal, which is a dog-sized quadrupedal system with 12 degrees of freedom. With our method, ANYmal manifests dynamic and reactive recovery behaviors to recover from an arbitrary fall configuration within less than 5 seconds. We tested the recovery maneuver more than 100 times, and the success rate was higher than 97 %.

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In this paper, we consider the coherent theory of (epistemic) uncertainty of Walley, in which beliefs are represented through sets of probability distributions, and we focus on the problem of modeling prior ignorance about a categorical random variable. In this setting, it is a known result that a state of prior ignorance is not compatible with learning. To overcome this problem, another state of beliefs, called \emph{near-ignorance}, has been proposed. Near-ignorance resembles ignorance very closely, by satisfying some principles that can arguably be regarded as necessary in a state of ignorance, and allows learning to take place. What this paper does, is to provide new and substantial evidence that also near-ignorance cannot be really regarded as a way out of the problem of starting statistical inference in conditions of very weak beliefs. The key to this result is focusing on a setting characterized by a variable of interest that is \emph{latent}. We argue that such a setting is by far the most common case in practice, and we provide, for the case of categorical latent variables (and general \emph{manifest} variables) a condition that, if satisfied, prevents learning to take place under prior near-ignorance. This condition is shown to be easily satisfied even in the most common statistical problems. We regard these results as a strong form of evidence against the possibility to adopt a condition of prior near-ignorance in real statistical problems.

* International Journal of Approximate Reasoning, 50:4 (2009) pages 597-611
* 27 LaTeX pages
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Legged robots pose one of the greatest challenges in robotics. Dynamic and agile maneuvers of animals cannot be imitated by existing methods that are crafted by humans. A compelling alternative is reinforcement learning, which requires minimal craftsmanship and promotes the natural evolution of a control policy. However, so far, reinforcement learning research for legged robots is mainly limited to simulation, and only few and comparably simple examples have been deployed on real systems. The primary reason is that training with real robots, particularly with dynamically balancing systems, is complicated and expensive. In the present work, we introduce a method for training a neural network policy in simulation and transferring it to a state-of-the-art legged system, thereby leveraging fast, automated, and cost-effective data generation schemes. The approach is applied to the ANYmal robot, a sophisticated medium-dog-sized quadrupedal system. Using policies trained in simulation, the quadrupedal machine achieves locomotion skills that go beyond what had been achieved with prior methods: ANYmal is capable of precisely and energy-efficiently following high-level body velocity commands, running faster than before, and recovering from falling even in complex configurations.

* Science Robotics 4.26 (2019): eaau5872
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