Prior and posterior linear pooling for combining expert opinions: Uses and impact on Bayesian Networks

The use of expert knowledge to quantify a Bayesian Network (BN) is necessary when data is not available. This however raises questions regarding how opinions from multiple experts can be used in a BN. Linear pooling is a popular method for combining probability assessments from multiple experts. In particular, Prior Linear Pooling (PrLP), which pools opinions then places them into the BN is a common method. This paper firstly proposes an alternative pooling method, Posterior Linear Pooling (PoLP). This method constructs a BN for each expert, then pools the resulting probabilities at the nodes of interest. Secondly, it investigates the advantages and disadvantages of using these pooling methods to combine the opinions of multiple experts. Finally, the methods are applied to an existing BN, the Wayfinding Bayesian Network Model, to investigate the behaviour of different groups of people and how these different methods may be able to capture such differences. The paper focusses on 6 nodes Human Factors, Environmental Factors, Wayfinding, Communication, Visual Elements of Communication and Navigation Pathway, and three subgroups Gender (female, male),Travel Experience (experienced, inexperienced), and Travel Purpose (business, personal) and finds that different behaviors can indeed be captured by the different methods.

Biologically Plausible Neural Circuits for Realization of Maximum Operations

Object recognition in the visual cortex is based on a hierarchical architecture, in which specialized brain regions along the ventral pathway extract object features of increasing levels of complexity, accompanied by greater invariance in stimulus size, position, and orientation. Recent theoretical studies postulate a non-linear pooling function, such as the maximum (MAX) operation could be fundamental in achieving such invariance. In this paper, we are concerned with neurally plausible mechanisms that may be involved in realizing the MAX operation. Four canonical circuits are proposed, each based on neural mechanisms that have been previously discussed in the context of cortical processing. Through simulations and mathematical analysis, we examine the relative performance and robustness of these mechanisms. We derive experimentally verifiable predictions for each circuit and discuss their respective physiological considerations.

Theory and data for area summation of contrast with and without uncertainty:evidence for a noisy energy model

Contrast sensitivity improves with the area of a sine-wave grating, but why? Here we assess this phenomenon against contemporary models involving spatial summation, probability summation, uncertainty, and stochastic noise. Using a two-interval forced-choice procedure we measured contrast sensitivity for circular patches of sine-wave gratings with various diameters that were blocked or interleaved across trials to produce low and high extrinsic uncertainty, respectively. Summation curves were steep initially, becoming shallower thereafter. For the smaller stimuli, sensitivity was slightly worse for the interleaved design than for the blocked design. Neither area nor blocking affected the slope of the psychometric function. We derived model predictions for noisy mechanisms and extrinsic uncertainty that was either low or high. The contrast transducer was either linear (c1.0) or nonlinear (c2.0), and pooling was either linear or a MAX operation. There was either no intrinsic uncertainty, or it was fixed or proportional to stimulus size. Of these 10 canonical models, only the nonlinear transducer with linear pooling (the noisy energy model) described the main forms of the data for both experimental designs. We also show how a cross-correlator can be modified to fit our results and provide a contemporary presentation of the relation between summation and the slope of the psychometric function.

RcppArmadillo : accelerating R with high-performance C++ linear algebra

The R statistical environment and language has demonstrated particular strengths for interactive development of statistical algorithms, as well as data modelling and visualisation. Its current implementation has an interpreter at its core which may result in a performance penalty in comparison to directly executing user algorithms in the native machine code of the host CPU. In contrast, the C++ language has no built-in visualisation capabilities, handling of linear algebra or even basic statistical algorithms; however, user programs are converted to high-performance machine code, ahead of execution. A new method avoids possible speed penalties in R by using the Rcpp extension package in conjunction with the Armadillo C++ matrix library. In addition to the inherent performance advantages of compiled code, Armadillo provides an easy-to-use template-based meta-programming framework, allowing the automatic pooling of several linear algebra operations into one, which in turn can lead to further speedups. With the aid of Rcpp and Armadillo, conversion of linear algebra centered algorithms from R to C++ becomes straightforward. The algorithms retains the overall structure as well as readability, all while maintaining a bidirectional link with the host R environment. Empirical timing comparisons of R and C++ implementations of a Kalman filtering algorithm indicate a speedup of several orders of magnitude.

A common contrast pooling rule for suppression within and between the eyes

Recent work has revealed multiple pathways for cross-orientation suppression in cat and human vision. In particular, ipsiocular and interocular pathways appear to assert their influence before binocular summation in human but have different (1) spatial tuning, (2) temporal dependencies, and (3) adaptation after-effects. Here we use mask components that fall outside the excitatory passband of the detecting mechanism to investigate the rules for pooling multiple mask components within these pathways. We measured psychophysical contrast masking functions for vertical 1 cycle/deg sine-wave gratings in the presence of left or right oblique (645 deg) 3 cycles/deg mask gratings with contrast C%, or a plaid made from their sum, where each component (i) had contrast 0.5Ci%. Masks and targets were presented to two eyes (binocular), one eye (monoptic), or different eyes (dichoptic). Binocular-masking functions superimposed when plotted against C, but in the monoptic and dichoptic conditions, the grating produced slightly more suppression than the plaid when Ci $ 16%. We tested contrast gain control models involving two types of contrast combination on the denominator: (1) spatial pooling of the mask after a local nonlinearity (to calculate either root mean square contrast or energy) and (2) "linear suppression" (Holmes & Meese, 2004, Journal of Vision 4, 1080–1089), involving the linear sum of the mask component contrasts. Monoptic and dichoptic masking were typically better fit by the spatial pooling models, but binocular masking was not: it demanded strict linear summation of the Michelson contrast across mask orientation. Another scheme, in which suppressive pooling followed compressive contrast responses to the mask components (e.g., oriented cortical cells), was ruled out by all of our data. We conclude that the different processes that underlie monoptic and dichoptic masking use the same type of contrast pooling within their respective suppressive fields, but the effects do not sum to predict the binocular case.

Pooling and suppression in human spatial vision

A distinct feature of several recent models of contrast masking is that detecting mechanisms are divisively inhibited by a broadly tuned ‘gain pool’ of narrow-band spatial pattern mechanisms. The contrast gain control provided by this ‘cross-channel’ architecture achieves contrast normalisation of early pattern mechanisms, which is important for keeping them within the non-saturating part of their biological operating characteristic. These models superseded earlier ‘within-channel’ models, which had supposed that masking arose from direct stimulation of the detecting mechanism by the mask. To reveal the extent of masking, I measured the levels produced with large ranges of pattern spatial relationships that have not been explored before. Substantial interactions between channels tuned to different orientations and spatial frequencies were found. Differences in the masking levels produced with single and multiple component mask patterns provided insights into the summation rules within the gain pool. A widely used cross-channel masking model was tested on these data and was found to perform poorly. The model was developed and a version in which linear summation was allowed between all components within the gain pool but with the exception of the self-suppressing route typically provided the best account of the data. Subsequently, an adaptation paradigm was used to probe the processes underlying pooled responses in masking. This delivered less insight into the pooling than the other studies and areas were identified that require investigation for a new unifying model of masking and adaptation. In further experiments, levels of cross-channel masking were found to be greatly influenced by the spatio-temporal tuning of the channels involved. Old masking experiments and ideas relying on within-channel models were re-elevated in terms of contemporary cross-channel models (e.g. estimations of channel bandwidths from orientation masking functions) and this led to different conclusions than those originally arrived at. The investigation of effects with spatio-temporally superimposed patterns is focussed upon throughout this work, though it is shown how these enquiries might be extended to investigate effects across spatial and temporal position.

Components of learning and assessment in linear algebra

Linear algebra provides theory and technology that are the cornerstones of a range of cutting edge mathematical applications, from designing computer games to complex industrial problems, as well as more traditional applications in statistics and mathematical modelling. Once past introductions to matrices and vectors, the challenges of balancing theory, applications and computational work across mathematical and statistical topics and problems are considerable, particularly given the diversity of abilities and interests in typical cohorts. This paper considers two such cohorts in a second level linear algebra course in different years. The course objectives and materials were almost the same, but some changes were made in the assessment package. In addition to considering effects of these changes, the links with achievement in first year courses are analysed, together with achievement in a following computational mathematics course. Some results that may initially appear surprising provide insight into the components of student learning in linear algebra.