Probabilities, Bayes and the Free-energy principle

Laurent Udo Perrinet, INT

Acknowledgements:

Laurent Pezard & Demian Battaglia, INS
Anna Montagnini, Nicole Malfait & Frédéric Chavane, INT
Stéphanie Ouine, PhD program

PhD program in Neuroscience, Marseille
March 27th, 2018
ⓦ https://invibe.net/LaurentPerrinet/Presentations/2018-03-26_cours-NeuroComp_FEP
Presentation made with slides.py

http://invibe.net/LaurentPerrinet/Presentations/2018-03-26_cours-NeuroComp_FEP

Probabilities, Bayes and the Free-energy principle

Laurent Udo Perrinet, INT

Acknowledgements:

Laurent Pezard & Demian Battaglia, INS
Anna Montagnini, Nicole Malfait & Frédéric Chavane, INT
Stéphanie Ouine, PhD program

PhD program in Neuroscience, Marseille
March 27th, 2018
ⓦ https://invibe.net/LaurentPerrinet/Presentations/2018-03-26_cours-NeuroComp_FEP
Presentation made with slides.py

Problem statement

(see this viperlib page)

Summary: the encoding / decoding problem

Examples of Bayesian mechanisms in perception

Ernst & Bülthoff (2004) Trends in Cognitive Sciences

Let's see some effects of priors in perception

* convex vs concave
* hollow mask
* cardinals
* cross-modal integration

Examples of Bayesian mechanisms in perception

Let's see some effects of priors in perception

* convex vs concave
* hollow mask
* cardinals
* cross-modal integration

Examples of Bayesian mechanisms in perception

Let's see some effects of priors in perception

* convex vs concave
* hollow mask
* cardinals
* cross-modal integration

Examples of Bayesian mechanisms in perception

Let's see some effects of priors in perception

* convex vs concave
* hollow mask
* cardinals
* cross-modal integration

Examples of Bayesian mechanisms in perception

LT Maloney (2002) Journal of Vision

Let's see some effects of priors in perception

* convex vs concave
* hollow mask
* cardinals
* cross-modal integration

Outline

Problem statement

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Practical example: How to decode neural activity?

Active inference, EMs & oculomotor delays

Take-home message

Probabilities and Bayesian inference

Lets try this using this notebook (solution)

Instead, let’s focus on one variable like the weather. We know how probable it is that it’s sunny or raining. For both cases, we can look at the conditional probabilities. How likely am I to wear a t-shirt if it’s sunny? How likely am I to wear a coat if it’s raining?

There’s a 25% chance that it’s raining. If it is raining, there’s a 75% chance that I’d wear a coat. So, the probability that it is raining and I’m wearing a coat is 25% times 75% which is approximately 19%. The probability that it’s raining and I’m wearing a coat is the probability that it is raining, times the probability that I’d wear a coat if it is raining. We write this:

p(rain,coat)=p(rain)⋅p(coat | rain)

This is a single case of one of the most fundamental identities of probability theory:

p(x,y)=p(x)⋅p(y|x)

We’re factoring the distribution, breaking it down into the product of two pieces. First we look at the probability that one variable, like the weather, will take on a certain value. Then we look at the probability that another variable, like my clothing, will take on a certain value conditioned on the first variable.

Probabilities and Bayesian inference

(see http://colah.github.io/posts/2015-09-Visual-Information/)

The key point is that the choice of which variable to start with is arbitrary. We could just as easily start by focusing on my clothing and then look at the weather conditioned on it. This might feel a bit less intuitive, because we understand that there’s a causal relationship of the weather influencing what I wear and not the other way around… but it is the basis of probabilistic inference: inverting the encoding / decoding problem.

Let’s go through an example. If we pick a random day, there’s a 38% chance that I’d be wearing a coat. If we know that I’m wearing a coat, how likely is it that it’s raining? Well, I’m more likely to wear a coat in the rain than in the sun, but rain is kind of rare in California, and so it works out that there’s a 50% chance that it’s raining. And so, the probability that it’s raining and I’m wearing a coat is the probability that I’m wearing a coat (38%), times the probability that it would be raining if I was wearing a coat (50%) which is approximately 19%.

p(rain,coat)=p(coat)⋅p(rain | coat)

This gives us a second way to visualize the exact same probability distribution.

Note that the labels have slightly different meanings than in the previous diagram: t-shirt and coat are now marginal probabilities, the probability of me wearing that clothing without consideration of the weather. On the other hand, there are now two rain and sunny labels, for the probabilities of them conditional on me wearing a t-shirt and me wearing a coat respectively.

(You may have heard of Bayes’ Theorem. If you want, you can think of it as the way to translate between these two different ways of displaying the probability distribution!)

Outline

Problem statement

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Practical example: How to decode neural activity?

Active inference, EMs & oculomotor delays

Take-home message

Variational Inference and the Free-energy principle

Friston (2010) Nat Neuro Reviews

Variational Inference and the Free-energy principle

Bogacz (2017) Journal of Mathematical Psychology

I enjoyed reading "A tutorial on the free-energy framework for modelling perception and learning" by Rafal Bogacz, which is freely available here. In particular, the author encourages to replicate the results in the paper. He is himself giving solutions in matlab, so I had to do the same in python all within a notebook...

      A tutorial on the free-energy framework for modelling perception and learning
    http://dx.doi.org/10.1016/j.jmp.2015.11.003

online
complex priors

The architecture of the model performing simple perceptual inference. Circles denote neural “nodes”, arrows denote excitatory connections, while lines ended with circles denote inhibitory connections. Labels above the connections encode their strength, and lack of label indicates the strength of 1. Rectangles indicate the values that need to be transmitted via the connections they label.

Variational Inference and the Free-energy principle

Lets try this using this notebook (solution)

First, let's see the application of Bayes theorem We start by considering in this section a simple perceptual problem in which a value of a single variable has to be inferred from a single observation. To make it more concrete, consider a simple organism that tries to infer the size or diameter of a food item, which we denote by $v$, on the basis of light intensity it observes. Let us assume that our simple animal has only one light sensitive receptor which provides it with a noisy estimate of light intensity, which we denote by $u$. Let g denote a non-linear function relating the average light intensity with the size. Since the amount of light reflected is related to the area of an object, in this example we will consider a simple function of $g(v)=v^2$. Let us further assume that the sensory input is noisy—in particular, when the size of food item is v, the perceived light intensity is normally distributed with mean g(v) , and variance $Σ_u$ (although a normal distribution is not the best choice for a distribution of light intensity, as it includes negative numbers, we will still use it for a simplicity): $$ p(u|v)=f(u;g(v),Σu). $$ In $f(x;μ,Σ) $ denotes the density of a normal distribution with mean μ and variance Σ

Due to the noise present in the observed light intensity, the animal can refine its guess for the size v by combining the sensory stimulus with the prior knowledge on how large the food items usually are, that it had learnt from experience. For simplicity, let us assume that our animal expects this size to be normally distributed with mean $v_p$ and variance $Σ_p$ (subscript p stands for “prior”), which we can write as: $$ p(v)=f(v;vp,Σp). $$

Let us now assume that our animal observed a particular value of light intensity, and attempts to estimate the size of the food item on the basis of this observation. We will first consider an exact solution to this problem, and illustrate why it would be difficult to compute it in a simple neural circuit. Then we will present an approximate solution that can be easily implemented in a simple network of neurons.

Exact solution

To compute how likely different sizes $v$ are given the observed sensory input $u$, we could use Bayes’ theorem: $$ p(v|u)=p(v)p(u|v)p(u). $$

Term $p(u)$ in the denominator of equation is a normalization term, which ensures that the posterior probabilities of all sizes $p(v|u)$ integrate to 1:

$$ p(u)=∫p(v)p(u|v)dv. $$

The integral in the above equation sums over the whole range of possible values of $v$, so it is a definite integral, but for brevity of notation we do not state the limits of integration in this and all other integrals in the paper.

Now combining Eqs. we can compute numerically how likely different sizes are given the sensory observation. For readers who are not familiar with such Bayesian inference we recommend doing the following exercise now.

Exercise 1.

Assume that our animal observed the light intensity $u=2$, the level of noise in its receptor is $Σ_u=1$, and the mean and variance of its prior expectation of size are $v_p=3$ and $Σ_p=1$. Write a computer program that computes the posterior probabilities of sizes from 0.01 to 5, and plots them.

Variational Inference and the Free-energy principle

Lets try this using this notebook (solution)

Finding the most likely feature value

Instead of finding the whole posterior distribution p(v|u) , let us try to find the most likely size of the food item v which maximizes p(v|u) . We will denote this most likely size by ϕ , and its posterior probability density by p(ϕ∣∣u) . It is reasonable to assume that in many cases the brain represents at a given moment of time only most likely values of features. For example in case of binocular rivalry, only one of the two possible interpretations of sensory inputs is represented.

We will look for the value ϕ which maximizes p(ϕ∣∣u) . According to Eq. (4), the posterior probability p(ϕ∣∣u) depends on a ratio of two quantities, but the denominator p(u) does not depend on ϕ . Thus the value of ϕ which maximizes p(ϕ∣∣u) is the same one which maximizes the numerator of Eq. (4). We will denote the logarithm of the numerator by F , as it is related to the negative of free energy (as we will describe in Section 3):

$$ F=lnp(ϕ)+lnp(u∣∣ϕ). $$ the definition of this model, one may derive a learning rule to estimate the best $u$

Let's define $F = \log( p(v |u) )$ after some derivation, we get

$$ rac{\partial F}{\partial v} = rac{v - v_p}{\Sigma_p} + \dot{g}(v) \cdot rac{u - g(v)}{\Sigma_u} $$

In the above equation we used the property of logarithm $ln(ab)=lna+lnb$. We will maximize the logarithm of the numerator of Eq. (4), because it has the same maximum as the numerator itself as ln is a monotonic function, and is easier to compute as the expressions for p(u∣∣ϕ) and p(ϕ) involve exponentiation.

To find the parameter ϕ that describes the most likely size of the food item, we will use a simple gradient ascent: i.e. we will modify ϕ proportionally to the gradient of F , which will turn out to be a very simple operation. It is relatively straightforward to compute F by substituting Eqs. (1), (2) ; (3) into Eq. (6) and then to compute the derivative of F (TRY IT YOURSELF).

equation(7) F=lnf(ϕ;vp,Σp)+lnf(u;g(ϕ),Σu)=ln[12πΣp√exp(−(ϕ−vp)22Σp)]+ln[12πΣu√exp(−(u−g(ϕ))22Σu)]=ln12π√−12lnΣp−(ϕ−vp)22Σp+ln12π√−12lnΣu−(u−g(ϕ))22Σu=12(−lnΣp−(ϕ−vp)2Σp−lnΣu−(u−g(ϕ))2Σu)+C. Turn MathJax off

We incorporated the constant terms in the 2nd line above into a constant C . Now we can compute the derivative of F over ϕ :

equation(8) ∂F∂ϕ=vp−ϕΣp+u−g(ϕ)Σug′(ϕ). Turn MathJax off

In the above equation we used the chain rule to compute the second term, and g′(ϕ) is a derivative of function g evaluated at ϕ , so in our example g′(ϕ)=2ϕ . We can find our best guess ϕ for v simply by changing ϕ in proportion to the gradient:

equation(9) ϕ̇ =∂F∂ϕ. Turn MathJax off

In the above equation ϕ̇ is the rate of change of ϕ with time. Let us note that the update of ϕ is very intuitive. It is driven by two terms in Eq. (8): the first moves it towards the mean of the prior, the second moves it according to the sensory stimulus, and both terms are weighted by the reliabilities of prior and sensory input respectively.

Now please note that the above procedure for finding the approximate distribution of distance to food item is computationally much simpler than the exact method presented at the start of the paper. To gain more appreciation for the simplicity of this computation we recommend doing the following exercise.

Exercise 2.

Write a computer program finding the most likely size of the food item ϕ for the situation described in Exercise 1. Initialize ϕ=vp , and then find its values in the next 5 time units (you can use Euler’s method, i.e. update $ϕ(t+Δt)=ϕ(t)+Δt∂F/∂ϕ$ with t=0.01$ ).

Fig. 2(a) shows a solution to Exercise 2. Please notice that it rapidly converges to the value of $ϕ≈1.6 $, which is also the value that maximizes the exact posterior probability $p(v|u)$ shown in Fig. 1.

Variational Inference and the Free-energy principle

Lets try this using this notebook (solution)

one may also learn the variance

The model parameters can be hence optimized by modifying them proportionally to the gradient of $F$. It is straightforward to find the derivatives of F over $v_p$, $Σ_p$ and $Σ_u$:

$$ rac{∂F}{∂v_p}=rac{ϕ−v_p}{Σ_p} $$ $$ rac{∂F}{∂Σp}=rac 1 2 ( rac {(ϕ−v_p)^2} {Σ^2_p}−rac{1}{Σ_p} ) $$ $$ rac{∂F}{∂Σu}=rac 1 2 ( rac {(u−g(ϕ))^2}{Σ^2_u}−rac{1}{Σ_u}. $$

Simulate learning of variance $Σ_i$ over trials. For simplicity, only simulate the network described by Eqs. (59)– (60), and assume that variables ϕ are constant. On each trial generate input $ϕ_i$ from a normal distribution with mean 5 and variance 2, while set $g_i(ϕ_i+1)=5$ (so that the upper level correctly predicts the mean of $ϕ_i$). Simulate the network for 20 time units, and then update weight $Σ_i$ with learning rate $α=0.01$. Simulate 1000 trials and plot how $Σ_i$ changes across trials.

Outline

Problem statement

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Practical example: How to decode neural activity?

Active inference, EMs & oculomotor delays

Take-home message

Practical example: How to decode neural activity?

Optimal representation of sensory information

Mehrdad Jazayeri & J Anthony Movshon (2007) Nature Neuroscience

Optimal representation of sensory information

Mehrdad Jazayeri & J Anthony Movshon (2007) Nature Neuroscience

Figure 2 Computing likelihood for the direction of motion. (a) A random-dot stimulus (bottom) activates a set of directionally tuned neurons in area MT. The smooth curves represent neuronal tuning curves, and small circles show the noise-perturbed population response on a particular trial. To represent likelihood, we recoded the sensory signals by weighting the inputs from the population of tuned ‘encoding’ neurons. For the example shown, the correct weighting function has a cosinusoidal form, and the weighted signals converge to an output neuron representing log likelihood for a leftward direction. (b) Same as a, except here the output layer consists of an ensemble of neurons. The weighted signals converge to this output layer where the neurons represent the log likelihood for all possible directions, the likelihood function. Here, at the output, the average likelihood profile is shown; the colored points represent the average likelihoods of four example directions. The peak of the average likelihood function—the expected maximum-likelihood estimate of the stimulus direction—is shown as orange.

Maximum likelihood decoding.

The decoding algorithm consists of maximizing the posterior probability $P({ heta}|Y)$ as a function of the estimated direction ${ heta}$, given a distribution hypothesis: The evidence term $P(Y)$ is a normalization term independent of ${ heta}$ $ o P(Y)$=cst There is no prior knowledge on ${ heta}$ (such that $ orall heta_1, heta_2$, $P( heta_1)$ = $P( heta_2)$)

Thus, maximizing the posterior $P( heta|Y)$ under the Poisson hypothesis is equivalent to maximizing the following likelihood function: egin{equation} L( heta) = P(Y|{ heta}) = \Pi ^N _{i=1} rac{f_i({ heta}) ^{k_i}e^{-f_i({ heta})}}{k_i!} \end{equation}

In practice, It is often the log-likelihood function that is considered:

egin{equation} LL( heta) = log(P(Y|{ heta})) = \sum_{i=1}^N{k_i\log[f_{i}( heta)]}-\sum_{i=1}^N{f_{i}( heta)}- \sum_{i=1}^N \log[{k_i!}] \end{equation}

In the end:

egin{equation} LL( heta) = \sum_{i=1}^N{k_i\log[f_{i}( heta)]} - \log(Z) \end{equation}

Outline

Problem statement

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Practical example: How to decode neural activity?

Active inference, EMs & oculomotor delays

Take-home message

* ... As a consequence, for a tennis player ---here (highly trained) Jo-Wilfried Tsonga at Wimbledon--- trying to intercept a passing-shot ball at a (conservative) speed of $20~m.s^{-1}$, the position sensed on the retinal space corresponds to the instant when its image formed on the photoreceptors of the retina and reaches our hypothetical motion perception area behind:

and at this instant, the sensed physical position is lagging behind (as represented here by $ au_s \cdot v 1~m$ ), that is, approximately at $45$ degrees of eccentricity (red dotted line),
while the position at the moment of emitting the motor command will be $.8~m$ ahead of its present physical position ($ au_m \cdot v$).
As a consequence, note that the player's gaze is directed to the ball at its present position (red line), in anticipatory fashion. Optimal control directs action (future motion of the eye) to the expected position (red dashed line) of the ball in the future --- and the racket (black dashed line) to the expected position of the ball when motor commands reach the periphery (muscles). This is obviously an interesting challenge for modelling an optimal control theory.

in a first study, we have proposed that PERCEPTION (following Helmhotz 1866) is an active process of hypothesis testing by which we seek to confirm our predictive models of the (hidden) world: Active inference: (cite TITLE). In theory, one could find any prior to fit any experimental data, but the beauty of the theory comes from the simplicity of the models chosen to model the data at hand, such as saccades...
... and even better if these models may find a possible correspondance into the neural anatomy and explain some deviation to a control behaviour, such as that we modelled for understanding some aspects of the EMs of schizophrenic patients
Today, I will again focus on the problem of sensorimotor delays in the optimal control of (smooth) eye movements under uncertainty. Specifically, we consider delays in the visuo-oculomotor loop and their implications for active inference and I will present the results presented in the following paper (show TITLE).

This schematic shows the dependencies among various quantities modelling exchanges of an agent with the environment. It shows the states of the environment and the system in terms of a probabilistic dependency graph, where connections denote directed (causal) dependencies. The quantities are described within the nodes of this graph -- with exemplar forms for their dependencies on other variables.
Hidden (external) and internal states of the agent are separated by action and sensory states. Both action and internal states -- encoding a conditional probability density function over hidden states -- minimise free energy. Note that hidden states in the real world and the form of their dynamics can be different from that assumed by the generative model; (this is why hidden states are in bold. )

Pursuit initiation

This figure reports the conditional estimates of hidden states and causes during the simulation of pursuit initiation, using a single rightward (positive) sweep of a visual target, while compensating for sensory motor delays. We will use the format of this figure in subsequent figures: the upper left panel shows the predicted sensory input (coloured lines) and sensory prediction errors (dotted red lines) along with the true values (broken black lines). Here, we see horizontal excursions of oculomotor angle (upper lines) and the angular position of the target in an intrinsic frame of reference (lower lines). This is effectively the distance of the target from the centre of gaze and reports the spatial lag of the target that is being followed (solid red line). One can see clearly the initial displacement of the target that is suppressed after a few hundred milliseconds. The sensory predictions are based upon the conditional expectations of hidden oculomotor (blue line) and target (red line) angular displacements shown on the upper right. The grey regions correspond to 90% Bayesian confidence intervals and the broken lines show the true values of these hidden states. One can see the motion that elicits following responses and the oculomotor excursion that follows with a short delay of about 64ms. The hidden cause of these displacements is shown with its conditional expectation on the lower left. The true cause and action are shown on the lower right. The action (blue line) is responsible for oculomotor displacements and is driven by the proprioceptive prediction errors.

This figure illustrates the effects of sensorimotor delays on pursuit initiation (red lines) in relation to compensated (optimal) active inference -- as shown in the previous figure (blue lines). The left panels show the true (solid lines) and estimated sensory input (dotted lines), while action is shown in the right panels. Under pure sensory delays (top row), one can see clearly the delay in sensory predictions, in relation to the true inputs. The thicker (solid and dotted) red lines correspond respectively to (true and predicted) proprioceptive input, reflecting oculomotor displacement. The middle row shows the equivalent results with pure motor delays and the lower row presents the results with combined sensorimotor delays. Of note here is the failure of optimal control with oscillatory fluctuations in oculomotor trajectories, which become unstable under combined sensorimotor delays.

This figure illustrates the effects of sensorimotor delays on pursuit initiation (red lines) in relation to compensated (optimal) active inference -- as shown in the previous figure (blue lines). The left panels show the true (solid lines) and estimated sensory input (dotted lines), while action is shown in the right panels. Under pure sensory delays (top row), one can see clearly the delay in sensory predictions, in relation to the true inputs. The thicker (solid and dotted) red lines correspond respectively to (true and predicted) proprioceptive input, reflecting oculomotor displacement. The middle row shows the equivalent results with pure motor delays and the lower row presents the results with combined sensorimotor delays. Of note here is the failure of optimal control with oscillatory fluctuations in oculomotor trajectories, which become unstable under combined sensorimotor delays.

This figure illustrates the effects of sensorimotor delays on pursuit initiation (red lines) in relation to compensated (optimal) active inference -- as shown in the previous figure (blue lines). The left panels show the true (solid lines) and estimated sensory input (dotted lines), while action is shown in the right panels. Under pure sensory delays (top row), one can see clearly the delay in sensory predictions, in relation to the true inputs. The thicker (solid and dotted) red lines correspond respectively to (true and predicted) proprioceptive input, reflecting oculomotor displacement. The middle row shows the equivalent results with pure motor delays and the lower row presents the results with combined sensorimotor delays. Of note here is the failure of optimal control with oscillatory fluctuations in oculomotor trajectories, which become unstable under combined sensorimotor delays.

This figure illustrates the effects of sensorimotor delays on pursuit initiation (red lines) in relation to compensated (optimal) active inference -- as shown in the previous figure (blue lines). The left panels show the true (solid lines) and estimated sensory input (dotted lines), while action is shown in the right panels. Under pure sensory delays (top row), one can see clearly the delay in sensory predictions, in relation to the true inputs. The thicker (solid and dotted) red lines correspond respectively to (true and predicted) proprioceptive input, reflecting oculomotor displacement. The middle row shows the equivalent results with pure motor delays and the lower row presents the results with combined sensorimotor delays. Of note here is the failure of optimal control with oscillatory fluctuations in oculomotor trajectories, which become unstable under combined sensorimotor delays.

Smooth Pursuit

We then consider an extension of the generative model to simulate smooth pursuit eye movements --- in which the visuo-oculomotor system believes both the target and its centre of gaze are attracted to a (hidden) point moving in the visual field.

This figure uses the same format as the previous figure -- the only difference is that the target motion has been rectified so that it is (approximately) hemi-sinusoidal. The thing to note here is that the improved accuracy of the pursuit previously apparent at the onset of the second cycle of motion has now disappeared -- because active inference does not have access to the immediately preceding trajectory. This failure of an anticipatory improvement in tracking is contrary to empirical predictions. \item \odot
the generative model has been equipped with a second hierarchical level that contains hidden states, modelling latent periodic behaviour of the (hidden) causes of target motion. With this addition, the improvement in pursuit accuracy apparent at the onset of the second cycle of motion is reinstated. This is because the model has an internal representation of latent causes of target motion that can be called upon even when these causes are not expressed explicitly in the target trajectory.

This figure uses the same format as the previous figure -- the only difference is that the target motion has been rectified so that it is (approximately) hemi-sinusoidal. The thing to note here is that the improved accuracy of the pursuit previously apparent at the onset of the second cycle of motion has now disappeared -- because active inference does not have access to the immediately preceding trajectory. This failure of an anticipatory improvement in tracking is contrary to empirical predictions. \item \odot
the generative model has been equipped with a second hierarchical level that contains hidden states, modelling latent periodic behaviour of the (hidden) causes of target motion. With this addition, the improvement in pursuit accuracy apparent at the onset of the second cycle of motion is reinstated. This is because the model has an internal representation of latent causes of target motion that can be called upon even when these causes are not expressed explicitly in the target trajectory.

Smooth Pursuit with oclusion

Finally, the generative model is equipped with a hierarchical structure, so that it can recognise and remember unseen (occluded) trajectories and emit anticipatory responses.

This figure uses the same format as the previous figure -- the only difference is that the target motion has been rectified so that it is (approximately) hemi-sinusoidal. The thing to note here is that the improved accuracy of the pursuit previously apparent at the onset of the second cycle of motion has now disappeared -- because active inference does not have access to the immediately preceding trajectory. This failure of an anticipatory improvement in tracking is contrary to empirical predictions. \item \odot
the generative model has been equipped with a second hierarchical level that contains hidden states, modelling latent periodic behaviour of the (hidden) causes of target motion. With this addition, the improvement in pursuit accuracy apparent at the onset of the second cycle of motion is reinstated. This is because the model has an internal representation of latent causes of target motion that can be called upon even when these causes are not expressed explicitly in the target trajectory.

This figure uses the same format as the previous figure -- the only difference is that the target motion has been rectified so that it is (approximately) hemi-sinusoidal. The thing to note here is that the improved accuracy of the pursuit previously apparent at the onset of the second cycle of motion has now disappeared -- because active inference does not have access to the immediately preceding trajectory. This failure of an anticipatory improvement in tracking is contrary to empirical predictions. \item \odot
the generative model has been equipped with a second hierarchical level that contains hidden states, modelling latent periodic behaviour of the (hidden) causes of target motion. With this addition, the improvement in pursuit accuracy apparent at the onset of the second cycle of motion is reinstated. This is because the model has an internal representation of latent causes of target motion that can be called upon even when these causes are not expressed explicitly in the target trajectory.

Outline

Problem statement

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Practical example: How to decode neural activity?

Active inference, EMs & oculomotor delays

Take-home message

Probabilities, Bayes and the Free-energy principle

Laurent Udo Perrinet, INT

Acknowledgements:

Laurent Pezard & Demian Battaglia, INS
Anna Montagnini, Nicole Malfait & Frédéric Chavane, INT
Stéphanie Ouine, PhD program

PhD program in Neuroscience, Marseille
March 27th, 2018
ⓦ https://invibe.net/LaurentPerrinet/Presentations/2018-03-26_cours-NeuroComp_FEP
Presentation made with slides.py

Probabilities, Bayes and the Free-energy principle

Laurent Udo Perrinet, INT

Acknowledgements:

PhD program in Neuroscience, Marseille March 27th, 2018 ⓦ https://invibe.net/LaurentPerrinet/Presentations/2018-03-26_cours-NeuroComp_FEPPresentation made with slides.py

Probabilities, Bayes and the Free-energy principle

Laurent Udo Perrinet, INT

Acknowledgements:

PhD program in Neuroscience, Marseille March 27th, 2018 ⓦ https://invibe.net/LaurentPerrinet/Presentations/2018-03-26_cours-NeuroComp_FEPPresentation made with slides.py

Problem statement

Summary: the encoding / decoding problem

Summary: the encoding / decoding problem

Examples of Bayesian mechanisms in perception

Examples of Bayesian mechanisms in perception

Examples of Bayesian mechanisms in perception

Examples of Bayesian mechanisms in perception

Examples of Bayesian mechanisms in perception

Outline

Problem statement

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Practical example: How to decode neural activity?

Active inference, EMs & oculomotor delays

Take-home message

Probabilities and Bayesian inference

Probabilities and Bayesian inference

Probabilities and Bayesian inference

Probabilities and Bayesian inference

Probabilities and Bayesian inference

Probabilities and Bayesian inference

Probabilities and Bayesian inference

Outline

Problem statement

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Practical example: How to decode neural activity?

Active inference, EMs & oculomotor delays

Take-home message

Variational Inference and the Free-energy principle

Variational Inference and the Free-energy principle

Variational Inference and the Free-energy principle

Variational Inference and the Free-energy principle

Variational Inference and the Free-energy principle

Outline

Problem statement

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Practical example: How to decode neural activity?

Active inference, EMs & oculomotor delays

Take-home message

Practical example: How to decode neural activity?

Practical example: How to decode neural activity?

Practical example: How to decode neural activity?

Practical example: How to decode neural activity?

Practical example: How to decode neural activity?

Optimal representation of sensory information

Optimal representation of sensory information

Outline

Problem statement

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Practical example: How to decode neural activity?

Active inference, EMs & oculomotor delays

Take-home message

Outline

Problem statement

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Practical example: How to decode neural activity?

Active inference, EMs & oculomotor delays

Take-home message

Probabilities, Bayes and the Free-energy principle

Laurent Udo Perrinet, INT

Acknowledgements:

PhD program in Neuroscience, Marseille March 27th, 2018 ⓦ https://invibe.net/LaurentPerrinet/Presentations/2018-03-26_cours-NeuroComp_FEPPresentation made with slides.py

PhD program in Neuroscience, Marseille
March 27th, 2018
ⓦ https://invibe.net/LaurentPerrinet/Presentations/2018-03-26_cours-NeuroComp_FEP
Presentation made with slides.py

PhD program in Neuroscience, Marseille
March 27th, 2018
ⓦ https://invibe.net/LaurentPerrinet/Presentations/2018-03-26_cours-NeuroComp_FEP
Presentation made with slides.py

PhD program in Neuroscience, Marseille
March 27th, 2018
ⓦ https://invibe.net/LaurentPerrinet/Presentations/2018-03-26_cours-NeuroComp_FEP
Presentation made with slides.py