Advanced mean field methods:theory and practice

A major problem in modern probabilistic modeling is the huge computational complexity involved in typical calculations with multivariate probability distributions when the number of random variables is large. Because exact computations are infeasible in such cases and Monte Carlo sampling techniques may reach their limits, there is a need for methods that allow for efficient approximate computations. One of the simplest approximations is based on the mean field method, which has a long history in statistical physics. The method is widely used, particularly in the growing field of graphical models. Researchers from disciplines such as statistical physics, computer science, and mathematical statistics are studying ways to improve this and related methods and are exploring novel application areas. Leading approaches include the variational approach, which goes beyond factorizable distributions to achieve systematic improvements; the TAP (Thouless-Anderson-Palmer) approach, which incorporates correlations by including effective reaction terms in the mean field theory; and the more general methods of graphical models. Bringing together ideas and techniques from these diverse disciplines, this book covers the theoretical foundations of advanced mean field methods, explores the relation between the different approaches, examines the quality of the approximation obtained, and demonstrates their application to various areas of probabilistic modeling.

Mean field methods for classification with Gaussian processes

We discuss the Application of TAP mean field methods known from Statistical Mechanics of disordered systems to Bayesian classification with Gaussian processes. In contrast to previous applications, no knowledge about the distribution of inputs is needed. Simulation results for the Sonar data set are given.

Mean field methods for Gaussian process classifiers

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TAP for parity check error correcting codes

We employ the methods presented in the previous chapter for decoding corrupted codewords, encoded using sparse parity check error correcting codes. We show the similarity between the equations derived from the TAP approach and those obtained from belief propagation, and examine their performance as practical decoding methods.

The TAP approach to intensive and extensive connectivity systems

The Thouless-Anderson-Palmer (TAP) approach was originally developed for analysing the Sherrington-Kirkpatrick model in the study of spin glass models and has been employed since then mainly in the context of extensively connected systems whereby each dynamical variable interacts weakly with the others. Recently, we extended this method for handling general intensively connected systems where each variable has only O(1) connections characterised by strong couplings. However, the new formulation looks quite different with respect to existing analyses and it is only natural to question whether it actually reproduces known results for systems of extensive connectivity. In this chapter, we apply our formulation of the TAP approach to an extensively connected system, the Hopfield associative memory model, showing that it produces identical results to those obtained by the conventional formulation.

Apparatus and test methods for measuring the impact of golf balls on turf and their application in the field

An apparatus was developed to project spinning golf balls directly onto golf greens. This employed a modified baseball/practice machine with two counter-rotating pneumatic wheels. The speed of the wheels could be varied independently allowing backspin to be given to the ball. The ball was projected into a darkened enclosure where the motion of the ball before and after impacting with the turf was recorded using a still camera and a stroboscope. The resulting photographs contained successive images of the ball on a single frame of film. The apparatus was tested on eighteen golf courses resulting in 721 photographs of impacts. Statistical analysis was carried out on the results of the photographs and from this, two types of green emerged. On the first, the ball tended to rebound with topspin, while on the second, the ball retained backspin after impact if the initial backspin was greater than about 350 rads-1. Eleven tests were devised to determine the characteristics of greens and statistical techniques were used to analyse the relationships between these tests. These showed the effects of the green characteristics on ball/turf impacts. It was found that the ball retained backspin on greens that were freely drained and had less than 60% of Poa annua (annual meadow grass) in their swards. Visco-elastic models were used to simulate the impact of the ball with the turf. Impacts were simulated by considering the ball to be rigid and the turf to be a two layered system consisting of springs and dampers. The model showed good agreement with experiment and was used to simulate impacts from two different shots onto two contrasting types of green.

Bayesian inference for wind field retrieval

In many problems in spatial statistics it is necessary to infer a global problem solution by combining local models. A principled approach to this problem is to develop a global probabilistic model for the relationships between local variables and to use this as the prior in a Bayesian inference procedure. We show how a Gaussian process with hyper-parameters estimated from Numerical Weather Prediction Models yields meteorologically convincing wind fields. We use neural networks to make local estimates of wind vector probabilities. The resulting inference problem cannot be solved analytically, but Markov Chain Monte Carlo methods allow us to retrieve accurate wind fields.

Evaluation of an open-field autorefractor's ability to measure refraction and hence potential to assess objective accommodation in pseudophakes

Background: To evaluate the accuracy of an open-field autorefractor compared with subjective refraction in pseudophakes and hence its ability to assess objective eye focus with intraocular lenses (IOLs). Methods: Objective refraction was measured at 6 m using the Shin-Nippon NVision-K 5001/Grand Seiko WR-5100K open-field autorefractor (five repeats) and by subjective refraction on 141 eyes implanted with a spherical (Softec1 n=53), aspherical (SoftecHD n=37) or accommodating (1CU n=22; Tetraflex n=29) IOL. Autorefraction was repeated 2 months later. Results: The autorefractor prescription was similar (average difference: 0.09±0.53 D; p=0.19) to that found by subjective refraction, with ~71% within ±0.50 D. The horizontal cylindrical components were similar (difference: 0.00±0.39 D; p=0.96), although the oblique (J45) autorefractor cylindrical vector was slightly more negative (by -0.06±0.25 D; p=0.06) than the subjective refraction. The results were similar for each of the IOL designs except for the spherical IOL, where the mean spherical equivalent difference between autorefraction and subjective was more hypermetropic than the Tetraflex accommodating IOL (F=2.77, p=0.04). The intrasession repeatability was

Sampling and measurement methods for a study of childhood refractive error in a UK population

Background There is a paucity of data describing the prevalence of childhood refractive error in the United Kingdom. The Northern Ireland Childhood Errors of Refraction study, along with its sister study the Aston Eye Study, are the first population-based surveys of children using both random cluster sampling and cycloplegic autorefraction to quantify levels of refractive error in the United Kingdom. Methods Children aged 6–7 years and 12–13 years were recruited from a stratified random sample of primary and post-primary schools, representative of the population of Northern Ireland as a whole. Measurements included assessment of visual acuity, oculomotor balance, ocular biometry and cycloplegic binocular open-field autorefraction. Questionnaires were used to identify putative risk factors for refractive error. Results 399 (57%) of 6–7 years and 669 (60%) of 12–13 years participated. School participation rates did not vary statistically significantly with the size of the school, whether the school is urban or rural, or whether it is in a deprived/non-deprived area. The gender balance, ethnicity and type of schooling of participants are reflective of the Northern Ireland population. Conclusions The study design, sample size and methodology will ensure accurate measures of the prevalence of refractive errors in the target population and will facilitate comparisons with other population-based refractive data.

Bayesian inference for wind field retrieval

In many problems in spatial statistics it is necessary to infer a global problem solution by combining local models. A principled approach to this problem is to develop a global probabilistic model for the relationships between local variables and to use this as the prior in a Bayesian inference procedure. We show how a Gaussian process with hyper-parameters estimated from Numerical Weather Prediction Models yields meteorologically convincing wind fields. We use neural networks to make local estimates of wind vector probabilities. The resulting inference problem cannot be solved analytically, but Markov Chain Monte Carlo methods allow us to retrieve accurate wind fields.

Description and critical appraisal of principal components analysis (PCA) methodology applied to pulsed-field gel electrophoresis profiles of methicillin-resistant Staphylococcus aureus isolates

Principal components analysis (PCA) has been described for over 50 years; however, it is rarely applied to the analysis of epidemiological data. In this study PCA was critically appraised in its ability to reveal relationships between pulsed-field gel electrophoresis (PFGE) profiles of methicillin- resistant Staphylococcus aureus (MRSA) in comparison to the more commonly employed cluster analysis and representation by dendrograms. The PFGE type following SmaI chromosomal digest was determined for 44 multidrug-resistant hospital-acquired methicillin-resistant S. aureus (MR-HA-MRSA) isolates, two multidrug-resistant community-acquired MRSA (MR-CA-MRSA), 50 hospital-acquired MRSA (HA-MRSA) isolates (from the University Hospital Birmingham, NHS Trust, UK) and 34 community-acquired MRSA (CA-MRSA) isolates (from general practitioners in Birmingham, UK). Strain relatedness was determined using Dice band-matching with UPGMA clustering and PCA. The results indicated that PCA revealed relationships between MRSA strains, which were more strongly correlated with known epidemiology, most likely because, unlike cluster analysis, PCA does not have the constraint of generating a hierarchic classification. In addition, PCA provides the opportunity for further analysis to identify key polymorphic bands within complex genotypic profiles, which is not always possible with dendrograms. Here we provide a detailed description of a PCA method for the analysis of PFGE profiles to complement further the epidemiological study of infectious disease. © 2005 Elsevier B.V. All rights reserved.

Visual field and structural alterations in age-related macular degeneration

The thesis investigated progression of the central 10° visual field with structural changes at the macula in a cross-section of patients with varying degrees of agerelated macular degeneration (AMD). The relationships between structure and function were investigated for both standard and short-wavelength automated perimetry (SWAP). Factors known to influence the measure of visual field progression were considered, including the accuracy of the refractive correction on SWAP thresholds and the learning effect. Techniques of assessing the structure to function relationships between fundus images and the visual field were developed with computer programming and evaluated for repeatability. Drusen quantification of fundus photographs and retro-mode scanning laser ophthalmoscopic images was performed. Visual field progression was related to structural changes derived from both manual and automated methods. Principal Findings: • Visual field sensitivity declined with advancing stage of AMD. SWAP showed greater sensitivity to progressive changes than standard perimetry. • Defects were confined to the central 5°. SWAP defects occurred at similar locations but were deeper and wider than corresponding standard perimetry defects. • The central field became less uniform as severity of AMD increased. SWAP visual field indices of focal loss were of more importance when detecting early change in AMD, than indices of diffuse loss. • The decline in visual field sensitivity over stage of severity of AMD was not uniform, whereas a linear relationship was found between the automated measure of drusen area and visual field parameters. • Perimetry exhibited a stronger relationship with drusen area than other measures of visual function. • Overcorrection of the refraction for the working distance in SWAP should be avoided in subjects with insufficient accommodative facility. • The perimetric learning effect in the 10° field did not differ significantly between normal subjects and AMD patients. • Subretinal deposits appeared more numerous in retro-mode imaging than in fundus photography.

The application of physiological observation methods to emotion research

Purpose - The purpose of this paper is to examine consumer emotions and the social science and observation measures that can be utilised to capture the emotional experiences of consumers. The paper is not setting out to solve the theoretical debate surrounding emotion research, rather to provide an assessment of methodological options available to researchers to aid their investigation into both the structure and content of the consumer emotional experience, acknowledging both the conscious and subconscious elements of that experience. Design/methodology/approach - A review of a wide range of prior research from the fields of marketing, consumer behaviour, psychology and neuroscience are examined to identify the different observation methods available to marketing researchers in the study of consumer emotion. This review also considers the self report measures available to researchers and identifies the main theoretical debates concerning emotion to provide a comprehensive overview of the issues surrounding the capture of emotional responses in a marketing context and to highlight the benefits that observation methods offer this area of research. Findings - This paper evaluates three observation methods and four widely used self report measures of emotion used in a marketing context. Whilst it is recognised that marketers have shown preference for the use of self report measures in prior research, mainly due to ease of implementation, it is posited that the benefits of observation methodology and the wealth of data that can be obtained using such methods can compliment prior research. In addition, the use of observation methods cannot only enhance our understanding of the consumer emotion experience but also enable us to collaborate with researchers from other fields in order to make progress in understanding emotion. Originality/value - This paper brings perspectives and methods together to provide an up to date consideration of emotion research for marketers. In order to generate valuable research in this area there is an identified need for discussion and implementation of the observation techniques available to marketing researchers working in this field. An evaluation of a variety of methods is undertaken as a point to start discussion or consideration of different observation techniques and how they can be utilised.

The feasibility of neuroimaging methods in marketing research

On July 17, 1990, President George Bush ssued “Proclamation #6158" which boldly declared the following ten years would be called the “Decade of the Brain” (Bush, 1990). Accordingly, the research mandates of all US federal biomedical institutions worldwide were redirected towards the study of the brain in general and cognitive neuroscience specifically. In 2008, one of the greatest legacies of this “Decade of the Brain” is the impressive array of techniques that can be used to study cortical activity. We now stand at a juncture where cognitive function can be mapped in the time, space and frequency domains, as and when such activity occurs. These advanced techniques have led to discoveries in many fields of research and clinical science, including psychology and psychiatry. Unfortunately, neuroscientific techniques have yet to be enthusiastically adopted by the social sciences. Market researchers, as specialized social scientists, have an unparalleled opportunity to adopt cognitive neuroscientific techniques and significantly redefine the field and possibly even cause substantial dislocations in business models. Following from this is a significant opportunity for more commercially-oriented researchers to employ such techniques in their own offerings. This report examines the feasibility of these techniques.