Hidden Markov Models With Set-Valued Parameters

Hidden Markov models (HMMs) are widely used probabilistic models of sequential data. As with other probabilistic models, they require the specification of local conditional probability distributions, whose assessment can be too difficult and error-prone, especially when data are scarce or costly to acquire. The imprecise HMM (iHMM) generalizes HMMs by allowing the quantification to be done by sets of, instead of single, probability distributions. iHMMs have the ability to suspend judgment when there is not enough statistical evidence, and can serve as a sensitivity analysis tool for standard non-stationary HMMs. In this paper, we consider iHMMs under the strong independence interpretation, for which we develop efficient inference algorithms to address standard HMM usage such as the computation of likelihoods and most probable explanations, as well as performing filtering and predictive inference. Experiments with real data show that iHMMs produce more reliable inferences without compromising the computational efficiency.

The Inferential Complexity of Bayesian and Credal Networks

This paper presents new results on the complexity of graph-theoretical models that represent probabilities (Bayesian networks) and that represent interval and set valued probabilities (credal networks). We define a new class of networks with bounded width, and introduce a new decision problem for Bayesian networks, the maximin a posteriori. We present new links between the Bayesian and credal networks, and present new results both for Bayesian networks (most probable explanation with observations, maximin a posteriori) and for credal networks (bounds on probabilities a posteriori, most probable explanation with and without observations, maximum a posteriori).

Belief Updating and Learning in Semi-Qualitative Probabilistic Networks

This paper explores semi-qualitative probabilistic networks (SQPNs) that combine numeric and qualitative information. We first show that exact inferences with SQPNs are NPPP-Complete. We then show that existing qualitative relations in SQPNs (plus probabilistic logic and imprecise assessments) can be dealt effectively through multilinear programming. We then discuss learning: we consider a maximum likelihood method that generates point estimates given a SQPN and empirical data, and we describe a Bayesian-minded method that employs the Imprecise Dirichlet Model to generate set-valued estimates.

Multilinear and Integer Programming for Markov Decision Processes with Imprecise Probabilities

Markov Decision Processes (MDPs) are extensively used to encode sequences of decisions with probabilistic effects. Markov Decision Processes with Imprecise Probabilities (MDPIPs) encode sequences of decisions whose effects are modeled using sets of probability distributions. In this paper we examine the computation of Γ-maximin policies for MDPIPs using multilinear and integer programming. We discuss the application of our algorithms to “factored” models and to a recent proposal, Markov Decision Processes with Set-valued Transitions (MDPSTs), that unifies the fields of probabilistic and “nondeterministic” planning in artificial intelligence research. 

Cartographic veracity in medieval mapping: analyzing geographical variation in the Gough Map of Great Britain

This article explores statistical approaches for assessing the relative accuracy of medieval mapping. It focuses on one particular map, the Gough Map of Great Britain. This is an early and remarkable example of a medieval “national” map covering Plantagenet Britain. Conventionally dated to c. 1360, the map shows the position of places in and coastal outline of Great Britain to a considerable degree of spatial accuracy. In this article, aspects of the map's content are subjected to a systematic analysis to identify geographical variations in the map's veracity, or truthfulness. It thus contributes to debates among historical geographers and cartographic historians on the nature of medieval maps and mapping and, in particular, questions of their distortion of geographic space. Based on a newly developed digital version of the Gough Map, several regression-based approaches are used here to explore the degree and nature of spatial distortion in the Gough Map. This demonstrates that not only are there marked variations in the positional accuracy of places shown on the map between regions (i.e., England, Scotland, and Wales), but there are also fine-scale geographical variations in the spatial accuracy of the map within these regions. The article concludes by suggesting that the map was constructed using a range of sources, and that the Gough Map is a composite of multiscale representations of places in Great Britain. The article details a set of approaches that could be transferred to other contexts and add value to historic maps by enhancing understanding of their contents.

Analyticity of smooth eigenfunctions and spectral analysis of the Gauss map

We provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form Uf(x) = integral a(x, y) f(b( x, y)) mu(dy) acting on functions f: [u, v] --> C ( evolution operators of one-dimensional dynamical systems and Markov processes have this form). We estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the Frobenius-Perron operator of the continuous fraction Gauss map. We prove that any infinitely differentiable eigenfunction f of this Frobenius-Perron operator, corresponding to a non-zero eigenvalue admits a (unique) analytic extension to the set C\(-infinity, 1]. Analyzing the spectrum of the Frobenius Perron operator in spaces of smooth functions, we extend significantly the domain of validity of the Mayer and Ropstorff asymptotic formula for the decay of correlations of the Gauss map.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

Demonstrating the Transferability and the Descriptive Power of Sketch-Map

Increasingly, it is recognized that new automated forms of analysis are required to understand the high-dimensional output obtained from atomistic simulations. Recently, we introduced a new dimensionality reduction algorithm, sketch-map, that was designed specifically to work with data from molecular dynamics trajectories. In what follows, we provide more details on how this algorithm works and on how to set its parameters. We also test it on two well-studied Lennard-Jones clusters and show that the coordinates we extract using this algorithm are extremely robust. In particular, we demonstrate that the coordinates constructed for one particular Lennard-Jones cluster can be used to describe the configurations adopted by a second, different cluster and even to tell apart different phases of bulk Lennard-Jonesium.

Simplifying the representation of complex free-energy landscapes using sketch-map

A new scheme, sketch-map, for obtaining a low-dimensional representation of the region of phase space explored during an enhanced dynamics simulation is proposed. We show evidence, from an examination of the distribution of pairwise distances between frames, that some features of the free-energy surface are inherently high-dimensional. This makes dimensionality reduction problematic because the data does not satisfy the assumptions made in conventional manifold learning algorithms We therefore propose that when dimensionality reduction is performed on trajectory data one should think of the resultant embedding as a quickly sketched set of directions rather than a road map. In other words, the embedding tells one about the connectivity between states but does not provide the vectors that correspond to the slow degrees of freedom. This realization informs the development of sketch-map, which endeavors to reproduce the proximity information from the high-dimensionality description in a space of lower dimensionality even when a faithful embedding is not possible.

The Map of Connections 3.1: Unofficial and previously unmapped Irish border crossings

I charted unofficial border-crossings along Ireland's border, those not found on any other map.

During many surveys of Ireland's border I discovered that it is often perforated. Gates are set in hedgerows for the convenience of farmers, stepping stones and community-built bridges span rivers, walkers’ routes and muddy by-ways go wherever they please. These kinds of connections have always been there, although I think it is fair to say that their numbers have increased during the Peace Process. Roads blocked or cratered during the Troubles are being re-connected at a rate too fast for the Ordnance Survey to keep up with. On the local level cross-border movement is quietly happening, unchecked and often unmapped, until now.

This map attempts to throw the borderline in perpendicular, showing it as a place of connection rather than division.

What defines a geochemical map? Implications for environmental assessment.

The environmental quality of land is often assessed by the calculation of threshold values which aim to differentiate between concentrations of elements based on whether the soils are in residential or industrial sites. In Europe, for example, soil guideline values exist for agricultural and grazing land. A threshold is often set to differentiate between concentrations of the element that naturally occur in the soil and concentrations that result from diffuse anthropogenic sources. Regional geochemistry and, in particular, single component geochemical maps are increasingly being used to determine these baseline environmental assessments. The key question raised in this paper is whether the geochemical map can provide an accurate interpretation on its own. Implicit is the thought that single component geochemical maps represent absolute abundances. However,because of the compositional (closed) nature of the data univariate geochemical maps cannot be compared directly with one another.. As a result, any interpretation based on them is vulnerable to spurious correlation problems. What does this mean for soil geochemistry mapping, baseline quality documentation, soil resource assessment or risk evaluation? Despite the limitation of relative abundances, individual raw geochemical maps are deemed fundamental to several applications of geochemical maps including environmental assessments. However, element toxicity is related to its bioavailable concentration, which is lowered if its source is mixed with another source. Elements interact, for example under reducing conditions with iron oxides, its solid state is lost and arsenic becomes soluble and mobile. Both of these matters may be more adequately dealt with if a single component map is not interpreted in isolation to determine baseline and threshold assessments. A range of alternative compositionally compliant representations based on log-ratio and log-contrast approaches are explored to supplement the classical single component maps for environmental assessment. Case study examples are shown based on the Tellus soil geochemical dataset, covering Northern Ireland and the results of in vitro oral bioaccessibility testing carried out on a sub-set of archived Tellus Survey shallow soils following the Unified BARGE (Bioaccessibility Research Group of Europe).

Rapid tryptophan depletion improves decision-making cognition in healthy humans without affecting reversal learning or set shifting

Rapid tryptophan (Trp) depletion (RTD) has been reported to cause deterioration in the quality of decision making and impaired reversal learning, while leaving attentional set shifting relatively unimpaired. These findings have been attributed to a more powerful neuromodulatory effect of reduced 5-HT on ventral prefrontal cortex (PFC) than on dorsolateral PFC. In view of the limited number of reports, the aim of this study was to independently replicate these findings using the same test paradigms. Healthy human subjects without a personal or family history of affective disorder were assessed using a computerized decision making/gambling task and the CANTAB ID/ED attentional set-shifting task under Trp-depleted (n=17; nine males and eight females) or control (n=15; seven males and eight females) conditions, in a double-blind, randomized, parallel-group design. There was no significant effect of RTD on set shifting, reversal learning, risk taking, impulsivity, or subjective mood. However, RTD significantly altered decision making such that depleted subjects chose the more likely of two possible outcomes significantly more often than controls. This is in direct contrast to the previous report that subjects chose the more likely outcome significantly less often following RTD. In the terminology of that report, our result may be interpreted as improvement in the quality of decision making following RTD. This contrast between studies highlights the variability in the cognitive effects of RTD between apparently similar groups of healthy subjects, and suggests the need for future RTD studies to control for a range of personality, family history, and genetic factors that may be associated with 5-HT function.