Cascaded Bayesian processes: an account of bias in orientation perception

Following adaptation to an oriented (1-d) signal in central vision, the orientation of subsequently viewed test signals may appear repelled away from or attracted towards the adapting orientation. Small angular differences between the adaptor and test yield 'repulsive' shifts, while large angular differences yield 'attractive' shifts. In peripheral vision, however, both small and large angular differences yield repulsive shifts. To account for these tilt after-effects (TAEs), a cascaded model of orientation estimation that is optimized using hierarchical Bayesian methods is proposed. The model accounts for orientation bias through adaptation-induced losses in information that arise because of signal uncertainties and neural constraints placed upon the propagation of visual information. Repulsive (direct) TAEs arise at early stages of visual processing from adaptation of orientation-selective units with peak sensitivity at the orientation of the adaptor (theta). Attractive (indirect) TAEs result from adaptation of second-stage units with peak sensitivity at theta and theta+90 degrees , which arise from an efficient stage of linear compression that pools across the responses of the first-stage orientation-selective units. A spatial orientation vector is estimated from the transformed oriented unit responses. The change from attractive to repulsive TAEs in peripheral vision can be explained by the differing harmonic biases resulting from constraints on signal power (in central vision) versus signal uncertainties in orientation (in peripheral vision). The proposed model is consistent with recent work by computational neuroscientists in supposing that visual bias reflects the adjustment of a rational system in the light of uncertain signals and system constraints.

The relationship between self-awareness of attentional status, behavioral performance and oscillatory brain rhythms

High-level cognitive factors, including self-awareness, are believed to play an important role in human visual perception. The principal aim of this study was to determine whether oscillatory brain rhythms play a role in the neural processes involved in self-monitoring attentional status. To do so we measured cortical activity using magnetoencephalography (MEG) and functional magnetic resonance imaging (fMRI) while participants were asked to self-monitor their internal status, only initiating the presentation of a stimulus when they perceived their attentional focus to be maximal. We employed a hierarchical Bayesian method that uses fMRI results as soft-constrained spatial information to solve the MEG inverse problem, allowing us to estimate cortical currents in the order of millimeters and milliseconds. Our results show that, during self-monitoring of internal status, there was a sustained decrease in power within the 7-13 Hz (alpha) range in the rostral cingulate motor area (rCMA) on the human medial wall, beginning approximately 430 msec after the trial start (p < 0.05, FDR corrected). We also show that gamma-band power (41-47 Hz) within this area was positively correlated with task performance from 40-640 msec after the trial start (r = 0.71, p < 0.05). We conclude: (1) the rCMA is involved in processes governing self-monitoring of internal status; and (2) the qualitative differences between alpha and gamma activity are reflective of their different roles in self-monitoring internal states. We suggest that alpha suppression may reflect a strengthening of top-down interareal connections, while a positive correlation between gamma activity and task performance indicates that gamma may play an important role in guiding visuomotor behavior. © 2013 Yamagishi et al.

Training Bayesian networks for image segmentation

We are concerned with the problem of image segmentation in which each pixel is assigned to one of a predefined finite number of classes. In Bayesian image analysis, this requires fusing together local predictions for the class labels with a prior model of segmentations. Markov Random Fields (MRFs) have been used to incorporate some of this prior knowledge, but this not entirely satisfactory as inference in MRFs is NP-hard. The multiscale quadtree model of Bouman and Shapiro (1994) is an attractive alternative, as this is a tree-structured belief network in which inference can be carried out in linear time (Pearl 1988). It is an hierarchical model where the bottom-level nodes are pixels, and higher levels correspond to downsampled versions of the image. The conditional-probability tables (CPTs) in the belief network encode the knowledge of how the levels interact. In this paper we discuss two methods of learning the CPTs given training data, using (a) maximum likelihood and the EM algorithm and (b) emphconditional maximum likelihood (CML). Segmentations obtained using networks trained by CML show a statistically-significant improvement in performance on synthetic images. We also demonstrate the methods on a real-world outdoor-scene segmentation task.

Data assimilation for precipitation nowcasting using Bayesian inference

This work introduces a new variational Bayes data assimilation method for the stochastic estimation of precipitation dynamics using radar observations for short term probabilistic forecasting (nowcasting). A previously developed spatial rainfall model based on the decomposition of the observed precipitation field using a basis function expansion captures the precipitation intensity from radar images as a set of ‘rain cells’. The prior distributions for the basis function parameters are carefully chosen to have a conjugate structure for the precipitation field model to allow a novel variational Bayes method to be applied to estimate the posterior distributions in closed form, based on solving an optimisation problem, in a spirit similar to 3D VAR analysis, but seeking approximations to the posterior distribution rather than simply the most probable state. A hierarchical Kalman filter is used to estimate the advection field based on the assimilated precipitation fields at two times. The model is applied to tracking precipitation dynamics in a realistic setting, using UK Met Office radar data from both a summer convective event and a winter frontal event. The performance of the model is assessed both traditionally and using probabilistic measures of fit based on ROC curves. The model is shown to provide very good assimilation characteristics, and promising forecast skill. Improvements to the forecasting scheme are discussed

An unsupervised Bayesian modelling approach to storyline detection from news articles

Storyline detection from news articles aims at summarizing events described under a certain news topic and revealing how those events evolve over time. It is a difficult task because it requires first the detection of events from news articles published in different time periods and then the construction of storylines by linking events into coherent news stories. Moreover, each storyline has different hierarchical structures which are dependent across epochs. Existing approaches often ignore the dependency of hierarchical structures in storyline generation. In this paper, we propose an unsupervised Bayesian model, called dynamic storyline detection model, to extract structured representations and evolution patterns of storylines. The proposed model is evaluated on a large scale news corpus. Experimental results show that our proposed model outperforms several baseline approaches.

Bayesian retrieval of scatterometer wind fields

The retrieval of wind fields from scatterometer observations has traditionally been separated into two phases; local wind vector retrieval and ambiguity removal. Operationally, a forward model relating wind vector to backscatter is inverted, typically using look up tables, to retrieve up to four local wind vector solutions. A heuristic procedure, using numerical weather prediction forecast wind vectors and, often, some neighbourhood comparison is then used to select the correct solution. In this paper we develop a Bayesian method for wind field retrieval, and show how a direct local inverse model, relating backscatter to wind vector, improves the wind vector retrieval accuracy. We compare these results with the operational U.K. Meteorological Office retrievals, our own CMOD4 retrievals and a neural network based local forward model retrieval. We suggest that the neural network based inverse model, which is extremely fast to use, improves upon current forward models when used in a variational data assimilation scheme.

Bayesian invariant measurements of generalisation for discrete distributions

Neural network learning rules can be viewed as statistical estimators. They should be studied in Bayesian framework even if they are not Bayesian estimators. Generalisation should be measured by the divergence between the true distribution and the estimated distribution. Information divergences are invariant measurements of the divergence between two distributions. The posterior average information divergence is used to measure the generalisation ability of a network. The optimal estimators for multinomial distributions with Dirichlet priors are studied in detail. This confirms that the definition is compatible with intuition. The results also show that many commonly used methods can be put under this unified framework, by assume special priors and special divergences.

Bayesian invariant measurements of generalisation for continuous distributions

A family of measurements of generalisation is proposed for estimators of continuous distributions. In particular, they apply to neural network learning rules associated with continuous neural networks. The optimal estimators (learning rules) in this sense are Bayesian decision methods with information divergence as loss function. The Bayesian framework guarantees internal coherence of such measurements, while the information geometric loss function guarantees invariance. The theoretical solution for the optimal estimator is derived by a variational method. It is applied to the family of Gaussian distributions and the implications are discussed. This is one in a series of technical reports on this topic; it generalises the results of ¸iteZhu95:prob.discrete to continuous distributions and serve as a concrete example of a larger picture ¸iteZhu95:generalisation.

Two Bayesian treatments of the n-tuple recognition method

Two probabilistic interpretations of the n-tuple recognition method are put forward in order to allow this technique to be analysed with the same Bayesian methods used in connection with other neural network models. Elementary demonstrations are then given of the use of maximum likelihood and maximum entropy methods for tuning the model parameters and assisting their interpretation. One of the models can be used to illustrate the significance of overlapping n-tuple samples with respect to correlations in the patterns.

A Bayesian formulation of search, control and the exploration/exploitation trade-off

A new approach to optimisation is introduced based on a precise probabilistic statement of what is ideally required of an optimisation method. It is convenient to express the formalism in terms of the control of a stationary environment. This leads to an objective function for the controller which unifies the objectives of exploration and exploitation, thereby providing a quantitative principle for managing this trade-off. This is demonstrated using a variant of the multi-armed bandit problem. This approach opens new possibilities for optimisation algorithms, particularly by using neural network or other adaptive methods for the adaptive controller. It also opens possibilities for deepening understanding of existing methods. The realisation of these possibilities requires research into practical approximations of the exact formalism.

Bayesian invariant measurements of generalisation

The problem of evaluating different learning rules and other statistical estimators is analysed. A new general theory of statistical inference is developed by combining Bayesian decision theory with information geometry. It is coherent and invariant. For each sample a unique ideal estimate exists and is given by an average over the posterior. An optimal estimate within a model is given by a projection of the ideal estimate. The ideal estimate is a sufficient statistic of the posterior, so practical learning rules are functions of the ideal estimator. If the sole purpose of learning is to extract information from the data, the learning rule must also approximate the ideal estimator. This framework is applicable to both Bayesian and non-Bayesian methods, with arbitrary statistical models, and to supervised, unsupervised and reinforcement learning schemes.

An upper bound on the Bayesian error bars for generalized linear regression

In the Bayesian framework, predictions for a regression problem are expressed in terms of a distribution of output values. The mode of this distribution corresponds to the most probable output, while the uncertainty associated with the predictions can conveniently be expressed in terms of error bars. In this paper we consider the evaluation of error bars in the context of the class of generalized linear regression models. We provide insights into the dependence of the error bars on the location of the data points and we derive an upper bound on the true error bars in terms of the contributions from individual data points which are themselves easily evaluated.

Bayesian regression filter and the issue of priors

We propose a Bayesian framework for regression problems, which covers areas which are usually dealt with by function approximation. An online learning algorithm is derived which solves regression problems with a Kalman filter. Its solution always improves with increasing model complexity, without the risk of over-fitting. In the infinite dimension limit it approaches the true Bayesian posterior. The issues of prior selection and over-fitting are also discussed, showing that some of the commonly held beliefs are misleading. The practical implementation is summarised. Simulations using 13 popular publicly available data sets are used to demonstrate the method and highlight important issues concerning the choice of priors.

On the relationship between Bayesian error bars and the input data density

We investigate the dependence of Bayesian error bars on the distribution of data in input space. For generalized linear regression models we derive an upper bound on the error bars which shows that, in the neighbourhood of the data points, the error bars are substantially reduced from their prior values. For regions of high data density we also show that the contribution to the output variance due to the uncertainty in the weights can exhibit an approximate inverse proportionality to the probability density. Empirical results support these conclusions.

A hierarchical latent variable model for data visualization

Visualization has proven to be a powerful and widely-applicable tool the analysis and interpretation of data. Most visualization algorithms aim to find a projection from the data space down to a two-dimensional visualization space. However, for complex data sets living in a high-dimensional space it is unlikely that a single two-dimensional projection can reveal all of the interesting structure. We therefore introduce a hierarchical visualization algorithm which allows the complete data set to be visualized at the top level, with clusters and sub-clusters of data points visualized at deeper levels. The algorithm is based on a hierarchical mixture of latent variable models, whose parameters are estimated using the expectation-maximization algorithm. We demonstrate the principle of the approach first on a toy data set, and then apply the algorithm to the visualization of a synthetic data set in 12 dimensions obtained from a simulation of multi-phase flows in oil pipelines and to data in 36 dimensions derived from satellite images.