Probabilities, Bayes and the Free-energy principle

Laurent U Perrinet, INT

Acknowledgements:

Jim Bednar @ Continuum Analytics & University of Edinburgh
Rick Adams and Karl Friston @ UCL - Wellcome Trust Centre for Neuroimaging
Wahiba Taouali, Giacomo Benvenuti, Frédéric Chavane - ANR BalaV1
Mina Aliakbari Khoei, Guillaume Masson and Anna Montagnini - FACETS-ITN Marie Curie Training

Telluride Neuromorphic Workshop - "Compound Eyes and Event-based Vision" - June 28th, 2017 https://laurentperrinet.github.io/talk/2017-06-30-telluride

Outline

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Active inference, EMs & oculomotor delays

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 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 still works!

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

Probabilities and Bayesian inference

Variational Inference and the Free-energy principle

Active inference, EMs & oculomotor delays

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.