Bayesian Models in Cognitive Neuroscience: A Tutorial

Abstract

This chapter provides an introduction to Bayesian models and their application in cognitive neuroscience. The central feature of Bayesian models, as opposed to other classes of models, is that Bayesian models represent the beliefs of an observer as probability distributions, allowing them to integrate information while taking its uncertainty into account. In the chapter, we will consider how the probabilistic nature of Bayesian models makes them particularly useful in cognitive neuroscience. We will consider two types of tasks in which we believe a Bayesian approach is useful: optimal integration of evidence from different sources, and the development of beliefs about the environment given limited information (such as during learning). We will develop some detailed examples of Bayesian models to give the reader a taste of how the models are constructed and what insights they may be able to offer about participants’ behavior and brain activity.

Appendices

Appendix A: One-Armed Bandit Model

We can write down the generative model, by which the rewarded action (A or B) is selected as follows:

p(A rewarded on trial i)~ Bernoulli(q_i)

$$ {q_i} = \left\{{\begin{matrix} {{q_{i-1}}}&{if~J = 0} \\ {rand(0,1)}&{if~J = 1}\end{matrix}}\right. $$

… where J is a binary variable which determines whether there was a jump in the value of q between trial i-1 and trial i; J itself is determined by

$$ J\tilde{\ }~{\text{ }}Bernoulli{(}v{)} $$

… where v is the probability of a jump, e.g. if a jump occurs on average every 15 trials, v =${{}^{1}\!/\!{}_{{15}}}.$

Then we can construct a Bayesian computer participant which infers the values of q and v on trial i as follows:

$$ p(q,v\left|{{x_{1:i}}) = p({x_i}\left|{{q_i}}\right.,v)p({q_i},v)}\right. $$

where the prior at trial i, $p({q_i},v),$ is given by

$$ p({{q_i},v})=p({q_i}|{q_{i-1}},v)~p({q_{i-1}},v|{x_{1:i-1}}) $$

and the transition function $p({q_i}\left|{{q_{i-1}},v)}\right.$ is given by

$$p({{q}_{i}}\left| {{q}_{i-1}},v)=(1-v) \right.{{q}_{i-1}}+v\left( \frac{1}{Uniform\left( 0,1 \right)} \right)$$

Exercises

Exercise 1. Look at Fig. 9.5. How do you interpret the shadow on the surface shapes? Most people see the left hand side bumps as convex and the right hand bumps as concaves. Can you explain why that might be, using your Bayesian perspective? Hint: think of the use of priors.

Fig. 9.5

Convex or concave?

Exercise 2. In Fig. 9.4 we saw some interesting behavior by a Bayesian learner. For instance, at point c the model very quickly changed its belief of an environment where left was rewarded into one where right was rewarded. One important goal of model-based cognitive neuroscience is to link this type of changes probability distributions to observed neural phenomena. Can you come up with some phenomena that can be linked with changes in the model’s parameters?

Exercise 3. In this final exercise we will ask you to construct a simple Bayesian model. The solutions include example Matlab code, although they are platform independent. Consider the following set of observations of apple positions x, which Isaac made in his garden:

i	xi
1	63
2	121
3	148
4	114
5	131
6	121
7	90
8	108
9	76
10	126

• 1. Find the mean, E(x), and variance, E(x ² )–E(x) ², of this set of observations using the formulae

$$ E(x) = \frac{1}{n}\sum_i {{x_i}}$$

$$ E({x^2}) = \frac{1}{n}{\sum_i {{x_i}}^2} $$

• 2. If I tell you that these samples were drawn from a normal distribution, x~N(μ, σ ² ) how could you use Bayes’ theorum to find the mean and variance of x? Or more precisely, how could you use Bayes’ theorem to estimate the parameters, μ and σ ², of the normal distribution from which the samples are drawn?

Hint: remember from the text that we can write

$$ p(x\tilde{\ }N(\mu,{\sigma^2})\left|{{x_1}\ldots {x_n}}\right.)\propto p({x_1}\ldots {x_n}\left|{x\tilde{\ }N(\mu,{\sigma^2}))p(x\tilde{\ }N(\mu,{\sigma^2}))}\right. $$

…where the likelihood function, $p({x_i}\left|{x\tilde{\ }N(\mu,\sigma))}\right.$, is given by the standard probability density function for a normal distribution:

$$ p(x) = \frac{1}{{\sqrt {2\pi {\sigma^2}}}}\exp \left({\frac{{-{{(x-\mu)}^2}}}{{2{\sigma^2}}}}\right) $$

…and you can assume:

• 1. The prior probability p(x~N(μ, σ ² )) is equal for all possible values of μ; and σ ²,and

• 2. The observations are independent samples such that p(x_i∩x_j) = p(x_i)p(x_j) for all pairs of samples {x_i, x_j}.

Now use MATLAB to work out the posterior probability for a range of pairs of parameter values μ and σ ², and find the pair with the highest joint posterior probability. This gives a maximum likelihood estimate for μ and σ ².

• 3. Can you adapt this model to process each data point sequentially, so that the posterior after observation i becomes the prior for observation i + 1?

Hint: remember from the text that (assuming the underlying values of μ and σ ² cannot change between observations), we can write:

$$ p(x\tilde{\ }N(\mu,{\sigma^2})\left|{{x_1}\ldots }\right.{x_i})\propto p({x_i}\left|{x\tilde{\ }N(\mu,{\sigma^2})}\right.)p(x\tilde{\ }N(\mu,{\sigma^2})\left|{{x_1}\ldots {x_{i-1}})}\right. $$

… where the prior at trial i, $p(x\tilde{\ }N(\mu,{\sigma^2})\left|{{x_1}}\right.\ldots {x_{i-1}})$ is the posterior from trial i-1.

• 4. If you have done parts 2 and 3 correctly, the final estimates of {μ, σ ² } should be the same whether you process thedata points sequentially, or all at once. Why is this?

Further Reading

• 1. McGrayne [14] provides an historical overview of the development of Bayes’ theorem, its applications, and its gradual acceptance in the scientific community;

• 2. Daniel Wolpert’s TED talk (available at http://www.ted.com/talks/daniel_wolpert_the_real_reason_for_brains.html) provides a nice introduction in to consequences of noise in neural systems and the Bayesian way of dealing with it;

• 3. O’Reilly [15] discusses Bayesian approaches to dealing with changes in the environment and how different types of uncertainty are incorporated into Bayesian models and dealt with in the brain.

• 4. Nate Silver’s book The signal and the noise [23] contains some nice example about how humans make predictions and establish beliefs. Silver advocates a Bayesian approach to dealing with uncertainty. It served him very well in the 2012 USA presidential elections, when he correctly predicted for each of the 50 states whether they would be carried by Obama or Romney.

• 5. David MacKay’s book Information theory, inference, and learning algorithms [12] is a much more advanced treatment of many of the principle of Bayesian thinking. It is available for free at http://www.inference.phy.cam.ac.uk/itprnn/book.html.