Bayesian and Quasi-Bayesian Estimators for Mutual Information from Discrete Data

Maximally predictive states: From partial observations to long timescales

Isolating slower dynamics from fast fluctuations has proven remarkably powerful, but how do we proceed from partial observations of dynamical systems for which we lack underlying equations? Here, we construct maximally predictive states by concatenating measurements in time, partitioning the resulting sequences using maximum entropy, and choosing the sequence length to maximize short-time predictive information. Transitions between these states yield a simple approximation of the transfer operator, which we use to reveal timescale separation and long-lived collective modes through the operator spectrum. Applicable to both deterministic and stochastic processes, we illustrate our approach through partial observations of the Lorenz system and the stochastic dynamics of a particle in a double-well potential. We use our transfer operator approach to provide a new estimator of the Kolmogorov-Sinai entropy, which we demonstrate in discrete and continuous-time systems, as well as the movement behavior of the nematode worm C. elegans.

Exploring the Positive User Experience Possibilities Based on Product Emotion Theory: A Beverage Unmanned Retail Terminal Case

Since the last century, user experience has been regarded as a key concept in the process of product and service design. With the development of positive psychology, the transformation from negative to positive user experience has also taken place in the field of user experience; it emphasizes exploring the future possibility of positive user experience rather than just solving existing problems. Based on the research and analysis of existing literature, this study makes it clear that positive user experience research should be based on the “positive experience,” and arousing a positive emotion is conducive to improving positive user experience. On this basis, the product emotion theory is applied to the analysis process of “positive experience.” Through word frequency screening, thematic analysis, and correlation calculation, the relationship between product stimulus (object, activity, and identity) and user concern (goal, attitude, and standard) based on positive “user comments” is constructed, and positive user experience is understood from multiple levels. Based on the comment score, the positive user experience interval is divided in order to clarify the improvement direction. Finally, taking the “Angel Orange” unmanned retail terminal as an example, this study carried out an empirical analysis. As an exploratory study, this study can provide some insights into the quantitative research process of positive user experience design that evokes positive emotions from a user’s “positive experience” story.

Inferring a Property of a Large System from a Small Number of Samples

Inferring the value of a property of a large stochastic system is a difficult task when the number of samples is insufficient to reliably estimate the probability distribution. The Bayesian estimator of the property of interest requires the knowledge of the prior distribution, and in many situations, it is not clear which prior should be used. Several estimators have been developed so far in which the proposed prior us individually tailored for each property of interest; such is the case, for example, for the entropy, the amount of mutual information, or the correlation between pairs of variables. In this paper, we propose a general framework to select priors that is valid for arbitrary properties. We first demonstrate that only certain aspects of the prior distribution actually affect the inference process. We then expand the sought prior as a linear combination of a one-dimensional family of indexed priors, each of which is obtained through a maximum entropy approach with constrained mean values of the property under study. In many cases of interest, only one or very few components of the expansion turn out to contribute to the Bayesian estimator, so it is often valid to only keep a single component. The relevant component is selected by the data, so no handcrafted priors are required. We test the performance of this approximation with a few paradigmatic examples and show that it performs well in comparison to the ad-hoc methods previously proposed in the literature. Our method highlights the connection between Bayesian inference and equilibrium statistical mechanics, since the most relevant component of the expansion can be argued to be that with the right temperature.

Quantifying the compressibility of complex networks

Many complex networks depend upon biological entities for their preservation. Such entities, from human cognition to evolution, must first encode and then replicate those networks under marked resource constraints. Networks that survive are those that are amenable to constrained encoding-or, in other words, are compressible. But how compressible is a network? And what features make one network more compressible than another? Here, we answer these questions by modeling networks as information sources before compressing them using rate-distortion theory. Each network yields a unique rate-distortion curve, which specifies the minimal amount of information that remains at a given scale of description. A natural definition then emerges for the compressibility of a network: the amount of information that can be removed via compression, averaged across all scales. Analyzing an array of real and model networks, we demonstrate that compressibility increases with two common network properties: transitivity (or clustering) and degree heterogeneity. These results indicate that hierarchical organization-which is characterized by modular structure and heterogeneous degrees-facilitates compression in complex networks. Generally, our framework sheds light on the interplay between a network's structure and its capacity to be compressed, enabling investigations into the role of compression in shaping real-world networks.

The Reward-Complexity Trade-off in Schizophrenia

Action selection requires a policy that maps states of the world to a distribution over actions. The amount of memory needed to specify the policy (the policy complexity) increases with the state-dependence of the policy. If there is a capacity limit for policy complexity, then there will also be a trade-off between reward and complexity, since some reward will need to be sacrificed in order to satisfy the capacity constraint. This paper empirically characterizes the trade-off between reward and complexity for both schizophrenia patients and healthy controls. Schizophrenia patients adopt lower complexity policies on average, and these policies are more strongly biased away from the optimal reward-complexity trade-off curve compared to healthy controls. However, healthy controls are also biased away from the optimal trade-off curve, and both groups appear to lie on the same empirical trade-off curve. We explain these findings using a cost-sensitive actor-critic model. Our empirical and theoretical results shed new light on cognitive effort abnormalities in schizophrenia.

Asymptotic Information-Theoretic Detection of Dynamical Organization in Complex Systems

The identification of emergent structures in complex dynamical systems is a formidable challenge. We propose a computationally efficient methodology to address such a challenge, based on modeling the state of the system as a set of random variables. Specifically, we present a sieving algorithm to navigate the huge space of all subsets of variables and compare them in terms of a simple index that can be computed without resorting to simulations. We obtain such a simple index by studying the asymptotic distribution of an information-theoretic measure of coordination among variables, when there is no coordination at all, which allows us to fairly compare subsets of variables having different cardinalities. We show that increasing the number of observations allows the identification of larger and larger subsets. As an example of relevant application, we make use of a paradigmatic case regarding the identification of groups in autocatalytic sets of reactions, a chemical situation related to the origin of life problem.

Prioritizing Type 2 Diabetes Genes by Weighted PageRank on Bilayer Heterogeneous Networks

The prevalence of diabetes mellitus has been increasing rapidly in recent years. Type 2 diabetes makes up about 90 percent cases of diabetes. The interacting mixed effects of genetics and environments build possible interpretable pathogenesis. Thus, finding the causal disease genes is crucial in its clinical diagnosis and medical treatment. Currently, network-based computational method becomes a powerful tool of systematically analyzing complex diseases, such as the identification of candidate disease genes from networks. In this paper, we propose a bioinformatics framework of prioritizing type 2 diabetes genes by leveraging the modified PageRank algorithm on bilayer biomolecular networks consisting an ensemble gene-gene regulatory network and an integrative protein-protein interaction network. We specifically weigh the networks by differential mutual information for measuring the context specificities between genes and between proteins by transcriptomic and proteomic datasets, respectively. After formulating the network into two components of known disease genes and the other normal healthy genes, we rank the diabetes genes and others by bringing the orders in the bilayer network via an improved PageRank algorithm. We conclude that these known disease genes achieve significantly higher ranks compared to these randomly-selected normal genes, and the ranks are robust and consistent in multiple validation scenarios. In functional analysis, these high-ranked genes are identified to perform relevant risks and dysfunctions of type 2 diabetes.

Reconciling emergences: An information-theoretic approach to identify causal emergence in multivariate data

The broad concept of emergence is instrumental in various of the most challenging open scientific questions-yet, few quantitative theories of what constitutes emergent phenomena have been proposed. This article introduces a formal theory of causal emergence in multivariate systems, which studies the relationship between the dynamics of parts of a system and macroscopic features of interest. Our theory provides a quantitative definition of downward causation, and introduces a complementary modality of emergent behaviour-which we refer to as causal decoupling. Moreover, the theory allows practical criteria that can be efficiently calculated in large systems, making our framework applicable in a range of scenarios of practical interest. We illustrate our findings in a number of case studies, including Conway's Game of Life, Reynolds' flocking model, and neural activity as measured by electrocorticography.

Lossy-Context Surprisal: An Information-Theoretic Model of Memory Effects in Sentence Processing

A key component of research on human sentence processing is to characterize the processing difficulty associated with the comprehension of words in context. Models that explain and predict this difficulty can be broadly divided into two kinds, expectation-based and memory-based. In this work, we present a new model of incremental sentence processing difficulty that unifies and extends key features of both kinds of models. Our model, lossy-context surprisal, holds that the processing difficulty at a word in context is proportional to the surprisal of the word given a lossy memory representation of the context-that is, a memory representation that does not contain complete information about previous words. We show that this model provides an intuitive explanation for an outstanding puzzle involving interactions of memory and expectations: language-dependent structural forgetting, where the effects of memory on sentence processing appear to be moderated by language statistics. Furthermore, we demonstrate that dependency locality effects, a signature prediction of memory-based theories, can be derived from lossy-context surprisal as a special case of a novel, more general principle called information locality.

Estimating the Mutual Information between Two Discrete, Asymmetric Variables with Limited Samples

Determining the strength of nonlinear, statistical dependencies between two variables is a crucial matter in many research fields. The established measure for quantifying such relations is the mutual information. However, estimating mutual information from limited samples is a challenging task. Since the mutual information is the difference of two entropies, the existing Bayesian estimators of entropy may be used to estimate information. This procedure, however, is still biased in the severely under-sampled regime. Here, we propose an alternative estimator that is applicable to those cases in which the marginal distribution of one of the two variables—the one with minimal entropy—is well sampled. The other variable, as well as the joint and conditional distributions, can be severely undersampled. We obtain a consistent estimator that presents very low bias, outperforming previous methods even when the sampled data contain few coincidences. As with other Bayesian estimators, our proposal focuses on the strength of the interaction between the two variables, without seeking to model the specific way in which they are related. A distinctive property of our method is that the main data statistics determining the amount of mutual information is the inhomogeneity of the conditional distribution of the low-entropy variable in those states in which the large-entropy variable registers coincidences.

Information estimation using nonparametric copulas

Estimation of mutual information between random variables has become crucial in a range of fields, from physics to neuroscience to finance. Estimating information accurately over a wide range of conditions relies on the development of flexible methods to describe statistical dependencies among variables, without imposing potentially invalid assumptions on the data. Such methods are needed in cases that lack prior knowledge of their statistical properties and that have limited sample numbers. Here we propose a powerful and generally applicable information estimator based on non-parametric copulas. This estimator, called the non-parametric copula-based estimator (NPC), is tailored to take into account detailed stochastic relationships in the data independently of the data's marginal distributions. The NPC estimator can be used both for continuous and discrete numerical variables and thus provides a single framework for the mutual information estimation of both continuous and discrete data. By extensive validation on artificial samples drawn from various statistical distributions, we found that the NPC estimator compares well against commonly used alternatives. Unlike methods not based on copulas, it allows an estimation of information that is robust to changes of the details of the marginal distributions. Unlike parametric copula methods, it remains accurate regardless of the precise form of the interactions between the variables. In addition, the NPC estimator had accurate information estimates even at low sample numbers, in comparison to alternative estimators. The NPC estimator therefore provides a good balance between general applicability to arbitrarily shaped statistical dependencies in the data and shows accurate and robust performance when working with small sample sizes. We anticipate that the non-parametric copula information estimator will be a powerful tool in estimating mutual information between a broad range of data.

A statistical framework for neuroimaging data analysis based on mutual information estimated via a gaussian copula

We begin by reviewing the statistical framework of information theory as applicable to neuroimaging data analysis. A major factor hindering wider adoption of this framework in neuroimaging is the difficulty of estimating information theoretic quantities in practice. We present a novel estimation technique that combines the statistical theory of copulas with the closed form solution for the entropy of Gaussian variables. This results in a general, computationally efficient, flexible, and robust multivariate statistical framework that provides effect sizes on a common meaningful scale, allows for unified treatment of discrete, continuous, unidimensional and multidimensional variables, and enables direct comparisons of representations from behavioral and brain responses across any recording modality. We validate the use of this estimate as a statistical test within a neuroimaging context, considering both discrete stimulus classes and continuous stimulus features. We also present examples of analyses facilitated by these developments, including application of multivariate analyses to MEG planar magnetic field gradients, and pairwise temporal interactions in evoked EEG responses. We show the benefit of considering the instantaneous temporal derivative together with the raw values of M/EEG signals as a multivariate response, how we can separately quantify modulations of amplitude and direction for vector quantities, and how we can measure the emergence of novel information over time in evoked responses. Open-source Matlab and Python code implementing the new methods accompanies this article. Hum Brain Mapp 38:1541-1573, 2017. © 2016 Wiley Periodicals, Inc.

Information-theoretic measurements of coupling between structure and dynamics in glass formers

We analyze connections between structure and dynamics in two model glass formers, using the mutual information between an initial configuration and the ensuing dynamics to compare the predictive value of different structural observables. We consider the predictive power of normal modes, locally favored structures, and coarse-grained measurements of local energy and density. The mutual information allows the influence of the liquid structure on the dynamics to be analyzed quantitatively as a function of time, showing that normal modes give the most useful predictions on short time scales while local energy and density are most strongly predictive at long times.