Multisensory Oddity Detection as Bayesian Inference

A key goal for the perceptual system is to optimally combine information from all the senses that may be available in order to develop the most accurate and unified picture possible of the outside world. The contemporary theoretical framework of ideal observer maximum likelihood integration (MLI) has been highly successful in modelling how the human brain combines information from a variety of different sensory modalities. However, in various recent experiments involving multisensory stimuli of uncertain correspondence, MLI breaks down as a successful model of sensory combination. Within the paradigm of direct stimulus estimation, perceptual models which use Bayesian inference to resolve correspondence have recently been shown to generalize successfully to these cases where MLI fails. This approach has been known variously as model inference, causal inference or structure inference. In this paper, we examine causal uncertainty in another important class of multi-sensory perception paradigm – that of oddity detection and demonstrate how a Bayesian ideal observer also treats oddity detection as a structure inference problem. We validate this approach by showing that it provides an intuitive and quantitative explanation of an important pair of multi-sensory oddity detection experiments – involving cues across and within modalities – for which MLI previously failed dramatically, allowing a novel unifying treatment of within and cross modal multisensory perception. Our successful application of structure inference models to the new ‘oddity detection’ paradigm, and the resultant unified explanation of across and within modality cases provide further evidence to suggest that structure inference may be a commonly evolved principle for combining perceptual information in the brain.     

Comparison of Bayesian and empirical ranking approaches to visual perception

Much current vision research is predicated on the idea--and a rapidly growing body of evidence--that visual percepts are generated according to the empirical significance of light stimuli rather than their physical characteristics. As a result, an increasing number of investigators have asked how visual perception can be rationalized in these terms. Here, we compare two different theoretical frameworks for predicting what observers actually see in response to visual stimuli: Bayesian decision theory and empirical ranking theory. Deciding which of these approaches has greater merit is likely to determine how the statistical operations that apparently underlie visual perception are eventually understood.     

How Recent History Affects Perception: The Normative Approach and Its Heuristic Approximation

There is accumulating evidence that prior knowledge about expectations plays an important role in perception. The Bayesian framework is the standard computational approach to explain how prior knowledge about the distribution of expected stimuli is incorporated with noisy observations in order to improve performance. However, it is unclear what information about the prior distribution is acquired by the perceptual system over short periods of time and how this information is utilized in the process of perceptual decision making. Here we address this question using a simple two-tone discrimination task. We find that the “contraction bias”, in which small magnitudes are overestimated and large magnitudes are underestimated, dominates the pattern of responses of human participants. This contraction bias is consistent with the Bayesian hypothesis in which the true prior information is available to the decision-maker. However, a trial-by-trial analysis of the pattern of responses reveals that the contribution of most recent trials to performance is overweighted compared with the predictions of a standard Bayesian model. Moreover, we study participants'' performance in a-typical distributions of stimuli and demonstrate substantial deviations from the ideal Bayesian detector, suggesting that the brain utilizes a heuristic approximation of the Bayesian inference. We propose a biologically plausible model, in which decision in the two-tone discrimination task is based on a comparison between the second tone and an exponentially-decaying average of the first tone and past tones. We show that this model accounts for both the contraction bias and the deviations from the ideal Bayesian detector hypothesis. These findings demonstrate the power of Bayesian-like heuristics in the brain, as well as their limitations in their failure to fully adapt to novel environments.     

Human wavelength discrimination of monochromatic light explained by optimal wavelength decoding of light of unknown intensity

We show that human ability to discriminate the wavelength of monochromatic light can be understood as maximum likelihood decoding of the cone absorptions, with a signal processing efficiency that is independent of the wavelength. This work is built on the framework of ideal observer analysis of visual discrimination used in many previous works. A distinctive aspect of our work is that we highlight a perceptual confound that observers should confuse a change in input light wavelength with a change in input intensity. Hence a simple ideal observer model which assumes that an observer has a full knowledge of input intensity should over-estimate human ability in discriminating wavelengths of two inputs of unequal intensity. This confound also makes it difficult to consistently measure human ability in wavelength discrimination by asking observers to distinguish two input colors while matching their brightness. We argue that the best experimental method for reliable measurement of discrimination thresholds is the one of Pokorny and Smith, in which observers only need to distinguish two inputs, regardless of whether they differ in hue or brightness. We mathematically formulate wavelength discrimination under this wavelength-intensity confound and show a good agreement between our theoretical prediction and the behavioral data. Our analysis explains why the discrimination threshold varies with the input wavelength, and shows how sensitively the threshold depends on the relative densities of the three types of cones in the retina (and in particular predict discriminations in dichromats). Our mathematical formulation and solution can be applied to general problems of sensory discrimination when there is a perceptual confound from other sensory feature dimensions.     

Posterior Probability Matching and Human Perceptual Decision Making

Probability matching is a classic theory of decision making that was first developed in models of cognition. Posterior probability matching, a variant in which observers match their response probabilities to the posterior probability of each response being correct, is being used increasingly often in models of perception. However, little is known about whether posterior probability matching is consistent with the vast literature on vision and hearing that has developed within signal detection theory. Here we test posterior probability matching models using two tools from detection theory. First, we examine the models’ performance in a two-pass experiment, where each block of trials is presented twice, and we measure the proportion of times that the model gives the same response twice to repeated stimuli. We show that at low performance levels, posterior probability matching models give highly inconsistent responses across repeated presentations of identical trials. We find that practised human observers are more consistent across repeated trials than these models predict, and we find some evidence that less practised observers more consistent as well. Second, we compare the performance of posterior probability matching models on a discrimination task to the performance of a theoretical ideal observer that achieves the best possible performance. We find that posterior probability matching is very inefficient at low-to-moderate performance levels, and that human observers can be more efficient than is ever possible according to posterior probability matching models. These findings support classic signal detection models, and rule out a broad class of posterior probability matching models for expert performance on perceptual tasks that range in complexity from contrast discrimination to symmetry detection. However, our findings leave open the possibility that inexperienced observers may show posterior probability matching behaviour, and our methods provide new tools for testing for such a strategy.     

Where is the light? Bayesian perceptual priors for lighting direction

Perception of shaded three-dimensional figures is inherently ambiguous, but this ambiguity can be resolved if the brain assumes that figures are lit from a specific direction. Under the Bayesian framework, the visual system assigns a weighting to each possible direction, and these weightings define a prior probability distribution for light-source direction. Here, we describe a non-parametric maximum-likelihood estimation method for finding the prior distribution for lighting direction. Our results suggest that each observer has a distinct prior distribution, with non-zero values in all directions, but with a peak which indicates observers are biased to expect light to come from above left. The implications of these results for estimating general perceptual priors are discussed.     

Maximum likelihood and Bayesian methods for estimating the distribution of selective effects among classes of mutations using DNA polymorphism data

Maximum likelihood and Bayesian approaches are presented for analyzing hierarchical statistical models of natural selection operating on DNA polymorphism within a panmictic population. For analyzing Bayesian models, we present Markov chain Monte-Carlo (MCMC) methods for sampling from the joint posterior distribution of parameters. For frequentist analysis, an Expectation-Maximization (EM) algorithm is presented for finding the maximum likelihood estimate of the genome wide mean and variance in selection intensity among classes of mutations. The framework presented here provides an ideal setting for modeling mutations dispersed through the genome and, in particular, for the analysis of how natural selection operates on different classes of single nucleotide polymorphisms (SNPs).     

Formalizing Darwinism and inclusive fitness theory

Inclusive fitness maximization is a basic building block for biological contributions to any theory of the evolution of society. There is a view in mathematical population genetics that nothing is caused to be maximized in the process of natural selection, but this is explained as arising from a misunderstanding about the meaning of fitness maximization. Current theoretical work on inclusive fitness is discussed, with emphasis on the author''s ‘formal Darwinism project’. Generally, favourable conclusions are drawn about the validity of assuming fitness maximization, but the need for continuing work is emphasized, along with the possibility that substantive exceptions may be uncovered. The formal Darwinism project aims more ambitiously to represent in a formal mathematical framework the central point of Darwin''s Origin of Species, that the mechanical processes of inheritance and reproduction can give rise to the appearance of design, and it is a fitting ambition in Darwin''s bicentenary year to capture his most profound discovery in the lingua franca of science.     

Formal Darwinism

Two questions are raised for Grafen’s formal darwinism project of aligning evolutionary dynamics under natural selection with the optimization of phenotypes for individuals of a population. The first question concerns mean fitness maximization during frequency-dependent selection; in such selection regimes, not only is mean fitness typically not maximized but it is implausible that any parameter closely related to fitness is being maximized. The second question concerns whether natural selection on inclusive fitness differences can be regarded as individual selection or whether it leads to a departure from the central motivation that led to the formal darwinism project, viz., to show that “Darwinian” evolution through individual selection leads to “good design” or phenotypic adaptation through trait optimization.     

Tracking the contribution of inductive bias to individualised internal models

Internal models capture the regularities of the environment and are central to understanding how humans adapt to environmental statistics. In general, the correct internal model is unknown to observers, instead they rely on an approximate model that is continually adapted throughout learning. However, experimenters assume an ideal observer model, which captures stimulus structure but ignores the diverging hypotheses that humans form during learning. We combine non-parametric Bayesian methods and probabilistic programming to infer rich and dynamic individualised internal models from response times. We demonstrate that the approach is capable of characterizing the discrepancy between the internal model maintained by individuals and the ideal observer model and to track the evolution of the contribution of the ideal observer model to the internal model throughout training. In particular, in an implicit visuomotor sequence learning task the identified discrepancy revealed an inductive bias that was consistent across individuals but varied in strength and persistence.     

Cultural transmission and the evolution of human behaviour: a general approach based on the Price equation

Transmitted culture can be viewed as an inheritance system somewhat independent of genes that is subject to processes of descent with modification in its own right. Although many authors have conceptualized cultural change as a Darwinian process, there is no generally agreed formal framework for defining key concepts such as natural selection, fitness, relatedness and altruism for the cultural case. Here, we present and explore such a framework using the Price equation. Assuming an isolated, independently measurable culturally transmitted trait, we show that cultural natural selection maximizes cultural fitness, a distinct quantity from genetic fitness, and also that cultural relatedness and cultural altruism are not reducible to or necessarily related to their genetic counterparts. We show that antagonistic coevolution will occur between genes and culture whenever cultural fitness is not perfectly aligned with genetic fitness, as genetic selection will shape psychological mechanisms to avoid susceptibility to cultural traits that bear a genetic fitness cost. We discuss the difficulties with conceptualizing cultural change using the framework of evolutionary theory, the degree to which cultural evolution is autonomous from genetic evolution, and the extent to which cultural change should be seen as a Darwinian process. We argue that the nonselection components of evolutionary change are much more important for culture than for genes, and that this and other important differences from the genetic case mean that different approaches and emphases are needed for cultural than genetic processes.     

Levels of selection and the formal Darwinism project

Understanding good design requires addressing the question of what units undergo natural selection, thereby becoming adapted. There is, therefore, a natural connection between the formal Darwinism project (which aims to connect population genetics with the evolution of design and fitness maximization) and levels of selection issues. We argue that the formal Darwinism project offers contradictory and confusing lines of thinking concerning level(s) of selection. The project favors multicellular organisms over both the lower (cell) and higher (social group) levels as the level of adaptation. Grafen offers four reasons for giving such special status to multicellular organisms: (1) they lack appreciable within-organism cell selection, (2) they have multiple features that appear contrived for the same purpose, (3) they possess a set of phenotypes, and (4) they leave offspring according to their phenotypes. We discuss why these rationales are not compelling and suggest that a more even-handed approach, in which multicellular organisms are not assumed to have special status, would be desirable for a project that aims to make progress on the foundations of evolutionary theory.     

ESTIMATING AND VISUALIZING FITNESS SURFACES USING MARK–RECAPTURE DATA

Understanding how selection operates on a set of phenotypic traits is central to evolutionary biology. Often, it requires estimating survival (or other fitness‐related life‐history traits) which can be difficult to obtain for natural populations because individuals cannot be exhaustively followed. To cope with this issue of imperfect detection, we advocate the use of mark‐recapture data and we provide a general framework for both the estimation of linear and nonlinear selection gradients and the visualization of fitness surfaces. To quantify the strength of selection, the standard second‐order polynomial regression method is integrated in mark‐recapture models. To visualize the form of selection, we use splines to display selection acting on multivariate phenotypes in the most flexible way. We employ Markov chain Monte Carlo sampling in a Bayesian framework to estimate model parameters, assessing traits relevance and calculating the optimal amount of smoothing. We illustrate our approach using data from a wild population of Common blackbirds (Turdus merula) to investigate survival in relation to morphological traits, and provide evidence for correlational selection using the new methodology. Overall, the framework we propose will help in exploring the full potential of mark‐recapture data to study natural selection.     

Developmental selection and self-organization

Developmental selection is the differential survival and proliferation of developmental units, such as cellular lineages. This type of internal selection has been proposed as an explanation for diverse examples of self-organization, from the wiring of brains to the formation of pores on leaf surfaces. A general understanding of developmental selection has been slowed by failure to understand its relationship to familiar forms of genetical selection and evolution. I show the formal analogies between models of developmental selection and genetical selection. The general method I outline for the analysis of selective systems partitions self-organizing selective systems into generative rules that create variation and selective filters that move the population toward a target design. The method also emphasizes aggregate statistical measures of evolving systems, such as the covariance between particular traits and fitness. The identification of useful aggregate measures is a crucial step in the analysis of selective systems. I apply these concepts to a model of self-organization in ant colonies.     

Quantifying the causal pathways contributing to natural selection

The consequences of natural selection can be understood from a purely statistical perspective. In contrast, an explicitly causal approach is required to understand why trait values covary with fitness. In particular, key evolutionary constructs, such as sexual selection, fecundity selection, and so on, are best understood as selection via particular fitness components. To formalize and operationalize these concepts, we must disentangle the various causal pathways contributing to selection. Such decompositions are currently only known for linear models, where they are sometimes referred to as “Wright's rules.” Here, we provide a general framework, based on path analysis, for partitioning selection among its contributing causal pathways. We show how the extended selection gradient—which represents selection arising from a trait's causal effects on fitness—can be decomposed into path-specific selection gradients, which correspond to distinct causal mechanisms of selection. This framework allows for nonlinear effects and nonadditive interactions among variables, which may be estimated using standard statistical methods (e.g., generalized linear [mixed] models or generalized additive models). We thus provide a generalization of Wright's path rules that accommodates the nonlinear and nonadditive mechanisms by which natural selection commonly arises.     

Generalized movement strategies for constrained consumers: ignoring fitness can be adaptive

Abstract Movements made by real organisms--such as movements involved in dispersal, migration, and habitat selection--are expected to occasionally be suboptimal because of realistic constraints imposed by incomplete information, perceptual limitations, and stochasticity. Previous theory considering such constraints has shown that movements appropriately conditioned on habitat or resource characteristics can balance out suboptimal components of movement and thereby lead organisms to ideal free distributions and fitness maxima, whereas movements conditioned on fitness differentials cannot. These findings suggest a somewhat paradoxical hypothesis: even if organisms have information about their fitness, movement strategies that maximize fitness may be conditioned on something other than fitness per se. We test this hypothesis by investigating the evolutionary stability of generalized, conditional movement strategies that vary in their use of information on fitness versus information on habitat characteristics. We show that when costs of sensory machinery are included, natural selection should favor movement strategies that completely ignore fitness information. Finally, we synthesize previous work by showing how several previous important theoretical results for adaptive movement strategies are united under our one general model.     

Observers exploit stochastic models of sensory change to help judge the passage of time

Sensory stimulation can systematically bias the perceived passage of time, but why and how this happens is mysterious. In this report, we provide evidence that such biases may ultimately derive from an innate and adaptive use of stochastically evolving dynamic stimuli to help refine estimates derived from internal timekeeping mechanisms. A simplified statistical model based on probabilistic expectations of stimulus change derived from the second-order temporal statistics of the natural environment makes three predictions. First, random noise-like stimuli whose statistics violate natural expectations should induce timing bias. Second, a previously unexplored obverse of this effect is that similar noise stimuli with natural statistics should reduce the variability of timing estimates. Finally, this reduction in variability should scale with the interval being timed, so as to preserve the overall Weber law of interval timing. All three predictions are borne out experimentally. Thus, in the context of our novel theoretical framework, these results suggest that observers routinely rely on sensory input to augment their sense of the passage of time, through a process of Bayesian inference based on expectations of change in the natural environment.     

ESTIMATING UNCERTAINTY IN MULTIVARIATE RESPONSES TO SELECTION

Predicting the responses to natural selection is one of the key goals of evolutionary biology. Two of the challenges in fulfilling this goal have been the realization that many estimates of natural selection might be highly biased by environmentally induced covariances between traits and fitness, and that many estimated responses to selection do not incorporate or report uncertainty in the estimates. Here we describe the application of a framework that blends the merits of the Robertson–Price Identity approach and the multivariate breeder's equation to address these challenges. The approach allows genetic covariance matrices, selection differentials, selection gradients, and responses to selection to be estimated without environmentally induced bias, direct and indirect selection and responses to selection to be distinguished, and if implemented in a Bayesian‐MCMC framework, statistically robust estimates of uncertainty on all of these parameters to be made. We illustrate our approach with a worked example of previously published data. More generally, we suggest that applying both the Robertson–Price Identity and the multivariate breeder's equation will facilitate hypothesis testing about natural selection, genetic constraints, and evolutionary responses.