Category Archives: Mathematics

Study of the explanation of probability and reasoning in the human mind through mental models, probability logic and classical logic

May 4, 2015 09:55 ,

P.N. Johnson-Laird, Sangeet S. Khemlani, Geoffrey P. Goodwin, Logic, probability, and human reasoning, Trends in Cognitive Sciences, Volume 19, Issue 4, April 2015, Pages 201-214, ISSN 1364-6613, DOI: 10.1016/j.tics.2015.02.006.

This review addresses the long-standing puzzle of how logic and probability fit together in human reasoning. Many cognitive scientists argue that conventional logic cannot underlie deductions, because it never requires valid conclusions to be withdrawn – not even if they are false; it treats conditional assertions implausibly; and it yields many vapid, although valid, conclusions. A new paradigm of probability logic allows conclusions to be withdrawn and treats conditionals more plausibly, although it does not address the problem of vapidity. The theory of mental models solves all of these problems. It explains how people reason about probabilities and postulates that the machinery for reasoning is itself probabilistic. Recent investigations accordingly suggest a way to integrate probability and deduction.

Posted in: Artificial Intelligence, Mathematical logic, Probability and statistics, Psycho-physiological bases of engineering , Tagged: ,

On the not-so-domain-generic nature of statistical learning in the human brain

March 11, 2015 15:09 ,

Ram Frost, Blair C. Armstrong, Noam Siegelman, Morten H. Christiansen, 2015, Domain generality versus modality specificity: the paradox of statistical learning, Trends in Cognitive Sciences, Volume 19, Issue 3, March 2015, Pages 117-125, DOI: 10.1016/j.tics.2014.12.010.

Statistical learning (SL) is typically considered to be a domain-general mechanism by which cognitive systems discover the underlying distributional properties of the input. However, recent studies examining whether there are commonalities in the learning of distributional information across different domains or modalities consistently reveal modality and stimulus specificity. Therefore, important questions are how and why a hypothesized domain-general learning mechanism systematically produces such effects. Here, we offer a theoretical framework according to which SL is not a unitary mechanism, but a set of domain-general computational principles that operate in different modalities and, therefore, are subject to the specific constraints characteristic of their respective brain regions. This framework offers testable predictions and we discuss its computational and neurobiological plausibility.

Posted in: Machine learning, Psycho-physiological bases of engineering , Tagged: ,

Novel algorithm for inexact graph matching of moderate size graphs based on Gaussian process regression

February 10, 2015 19:54 ,

Serradell, E.; Pinheiro, M.A.; Sznitman, R.; Kybic, J.; Moreno-Noguer, F.; Fua, P., (2015), Non-Rigid Graph Registration Using Active Testing Search, Pattern Analysis and Machine Intelligence, IEEE Transactions on , vol.37, no.3, pp.625,638. DOI: http://doi.org/10.1109/TPAMI.2014.2343235

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We present a new approach for matching sets of branching curvilinear structures that form graphs embedded in R^2 or R^3 and may be subject to deformations. Unlike earlier methods, ours does not rely on local appearance similarity nor does require a good initial alignment. Furthermore, it can cope with non-linear deformations, topological differences, and partial graphs. To handle arbitrary non-linear deformations, we use Gaussian process regressions to represent the geometrical mapping relating the two graphs. In the absence of appearance information, we iteratively establish correspondences between points, update the mapping accordingly, and use it to estimate where to find the most likely correspondences that will be used in the next step. To make the computation tractable for large graphs, the set of new potential matches considered at each iteration is not selected at random as with many RANSAC-based algorithms. Instead, we introduce a so-called Active Testing Search strategy that performs a priority search to favor the most likely matches and speed-up the process. We demonstrate the effectiveness of our approach first on synthetic cases and then on angiography data, retinal fundus images, and microscopy image stacks acquired at very different resolutions.

Posted in: Graph theory , Tagged: ,

Partially observable reinforcement learning and the problem of representing the history of the learning process efficiently

January 23, 2015 17:58 ,

Doshi-Velez, F.; Pfau, D.; Wood, F.; Roy, N., Bayesian Nonparametric Methods for Partially-Observable Reinforcement Learning, Pattern Analysis and Machine Intelligence, IEEE Transactions on , vol.37, no.2, pp.394,407, Feb. 2015, DOI: 10.1109/TPAMI.2013.191

Making intelligent decisions from incomplete information is critical in many applications: for example, robots must choose actions based on imperfect sensors, and speech-based interfaces must infer a user\u2019s needs from noisy microphone inputs. What makes these tasks hard is that often we do not have a natural representation with which to model the domain and use for choosing actions; we must learn about the domain\u2019s properties while simultaneously performing the task. Learning a representation also involves trade-offs between modeling the data that we have seen previously and being able to make predictions about new data. This article explores learning representations of stochastic systems using Bayesian nonparametric statistics. Bayesian nonparametric methods allow the sophistication of a representation to scale gracefully with the complexity in the data. Our main contribution is a careful empirical evaluation of how representations learned using Bayesian nonparametric methods compare to other standard learning approaches, especially in support of planning and control. We show that the Bayesian aspects of the methods result in achieving state-of-the-art performance in decision making with relatively few samples, while the nonparametric aspects often result in fewer computations. These results hold across a variety of different techniques for choosing actions given a representation.

Posted in: Reinforcement learning theory , Tagged: ,

Estimating an empirical distribution from a number of estimates distributed among several agents, minimizing the information exchange between the agents

January 23, 2015 17:25 ,

Sarwate, A.D.; Javidi, T., Distributed Learning of Distributions via Social Sampling, Automatic Control, IEEE Transactions on , vol.60, no.1, pp.34,45, Jan. 2015, DOI: 10.1109/TAC.2014.2329611

A protocol for distributed estimation of discrete distributions is proposed. Each agent begins with a single sample from the distribution, and the goal is to learn the empirical distribution of the samples. The protocol is based on a simple message-passing model motivated by communication in social networks. Agents sample a message randomly from their current estimates of the distribution, resulting in a protocol with quantized messages. Using tools from stochastic approximation, the algorithm is shown to converge almost surely. Examples illustrate three regimes with different consensus phenomena. Simulations demonstrate this convergence and give some insight into the effect of network topology.

Posted in: Probability and statistics , Tagged: