Doya,
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Doya, et
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The Bayesian approach
is a mathematically
rigorous computational mechanism for combining prior knowledge with incoming evidence. |
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Doya, et
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A classical example of Bayesian inference is the Kalman filter, which has been extensively used in engineering,
communication, and control over the past few decades. |
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Doya, et
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The Kalman filter utilizes knowledge
about the noise in sensory observations and the dynamics of the observed
system to keep track of the best estimate of the system's current state and its variance. |
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Doya, et
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Although Kalman
filters assume linear dynamics and Gaussian noise, recent Bayesian
filters such as particle filters have extended the
basic idea to nonlinear,
non-Gaussian systems. |
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Doya, et
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A Bayesian approach can contribute to an understanding of the brain at
multiple levels. |
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Doya, et
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Spike trains
are the primary means of communication in the nervous system. |
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Doya, et
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Neurons
exhibit stochastic variability. Even for repeated presentations of a fixed stimulus, a neuron's spike response cannot be predicted with certainty. |
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Doya, et
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Many variables in the brain are encoded with population
codes. |
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Doya, et
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A major focus of theoretical neuroscience has been
understanding how populations of neurons encode information about single
variables and how this
information can be decoded from the population activity. |
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Doya, et
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One of the few things in systems neuroscience we are fairly certain about is that information is encoded in population
activity. |
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Doya, et
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The brain
receives information about the outside world; that information is represented in population activity at the sensory level; and to perform an action, such as reaching for an object, population
codes in motor cortex must be generated to drive the appropriate joint
movements. |
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Doya, et
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Networks of neurons implement input-output functions: they take as input population activity from one
set of neurons and produce as output population activity on another. |
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Doya, et
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We will assume that the activity of each neuron is described by one number, its firing rate. |
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Doya, et
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Noise is
partially inherited from the input and is partially generated from internal noise in the network. |
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Doya, et
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The noise has zero mean and covariance matrix R. |
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Doya, et
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We will assume that the noise is
small, linearize the equations of motion around an equilibrium on the attractor, and solve explicitly for the variance. |
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Doya, et
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Networks compute with population codes by taking as input one set of noisy population codes and producing as output another set. |
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Doya, et
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Areas V1 and V2,
in addition to encoding fine details of images in terms of filter responses, compute a segmentation of images which
allows a more compact and parsimonious encoding of images in terms of the
properties of regions and surfaces in the visual scene. |
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Doya, et
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Neurons and
their retinotopic arrangement in visual areas can represent information
precisely, thus furnishing an appropriate computational and representational infrastructure for the task. |
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Doya, et
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Segmentation
detects and extracts coherent regions in an image and then encodes the image in terms of probabilistic models of surfaces and regions in it. |
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Doya, et
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Bayesian probability theory provides a framework for modeling
how an observer should combine information from multiple cues and from prior knowledge about objects in the world to make perceptual inferences. |
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Doya, et
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Experiments
that have quantitatively tested the prediction of Bayesian models of cue integration have largely supported the hypothesis that human observers are "Bayes optimal" in their
interpretation of image data. |
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Doya, et
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A decision is a commitment to a proposition among multiple options. |
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Doya, et
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Sequential sampling has been used to explain a variety of reaction-time decision tasks. This framework has a rich history in statistical decision theory and mathematical psychology. |
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Doya, et
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We include recent data that
suggests a possible neural implementation of the computational principles of reaction time decision tasks in the parietal cortex. |
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Doya, et
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Optimal control theory is a mature mathematical
discipline with numerous applications in both science and engineering. It is emerging as a computational framework of choice
for studying the neural control of movement, in much the same way that probabilistic
inference is emerging as the computational framework of choice
for studying sensory information processing. |
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Doya, et
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Duality of optimal control and probabilistic inference. |
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Doya, et
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Neural information processing in sensory and motor areas. |
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Doya, et
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Recursive estimation, Kalman filter. |
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Doya, et
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Duality of optimal control and optimal estimation. |
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Doya, et
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Markoff property of stochastic
processes -- a process is Markoff if its future is conditionally
independent of the past, given the present state. |
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Doya, et
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Optimal control theory is based on two fundamental ideas -- One is dynamic programming and the associated optimality
principle introduced by Bellman in the United States.
The other is the maximum principle, introduced by Pontryagin in the Soviet Union. |
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Doya, et
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Kalman filter |
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Doya, et
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When the estimation
problem involves nonlinear
dynamics or non-Gaussian noise the solution
must rely on numerical approximations, the most widely used being the extended
Kalman filter (EKF). |
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Doya, et
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Optimal control and optimal estimation are closely
related mathematical problems, the best-known
example being the linear quadratic regulator and the Kalman filter. |
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Doya, et
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The brain
generates the best
behavior it can, subject
to the constraints imposed by the body and environment. This makes optimal control theory an
appealing computational framework for studying the neural control of movement. |
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Doya, et
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Decision Theory -- the decision
theory framework combines a probabilistic Bayesian model of the world with desires or goals
of an agent that are formalized by a utility
function. |
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Uncertainty
that stems from noise
of the lack of complete knowledge places estimation of attributes of the world and control of our
actuators firmly within a statistical framework. The knowledge about movement outcomes must be described by a probability
distribution. |
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Doya, et
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Bayesian statistics is such a powerful and versatile concept that we should expect the neural networks to use it in a large range of circumstances. |
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Doya, et
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Perception
does not supply us with perfect knowledge about events, making it necessary to estimate the characteristics of events. |
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Doya, et
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Research studies have shown that
people's behavior is well predicted by the assumption that they use optimal Bayesian statistics as a strategy. |
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Doya, et
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Bayesian statistics not only specifies how to combine
new information, likelihood, and prior knowledge, but just as well
specifies how two sources of information should be combined into a joint estimate. |
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Doya, et
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Bayesian statistics is a general framework that states how people's neural networks could optimally combine different sources of information into a joint estimate. Human performance
indicates that in many cases they operate very close to the theoretical optimum. |
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Doya, et
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Doya, et
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