Description
Question 1. [50 points] Assume that a neural population computes weighted linear
combinations of its input x, characterized by a system of equations Ax = b. Here A is the
transfer function and b is the output vector.
A single output measurement is recorded, given by
1 0 −1 2
2 1 −1 5
3 3 0 9
x1
x2
x3
x4
=
1
4
9
.
Answer the questions below. Derive the results by hand first and then confirm them on the
computer.
a) Find all solutions xn to Ax = 0.
b) Find a particular solution xp to Ax = b.
c) Find all solutions to Ax = b.
e) Find the pseudo-inverse of A.
e) Find the sparsest solution to Ax = b (the solution with the least number of non-zero
entries).
f) Find the least-norm solution to Ax = b (the solution with the minimum Euclidean norm).
Question 2. [50 points]
‘Reverse inference’ is a common, albeit poorly exercised method in neuroscience. It refers
to the practice of inferring that a cognitive process is engaged on the basis of activation
in some brain area. For example, if Broca’s area was found to be activated in some task,
researchers might infer that the subjects were using language. After a comprehensive search
of the literature, we find that Broca’s area was reported to be activated in 103 out of 869
fMRI tasks involving engagement of language, but this area was also active in 199 out of
2353 tasks not involving language.
a) Assume that the conditional probability of activation given language and activation given
no language, each follow a Bernoulli distribution (i.e., active with some probability p, or
not with probability 1 − p). Compute the likelihoods of observed frequencies of activation
in literature, as functions of the possible values of their respective Bernoulli probability parameters p = xl and p = xnl. Compute these functions at the values x =[0:.001:1] and plot
them as separate bar charts.
b) Find the values of xl and xnl that maximize their respective discretized likelihood functions.
c) Using the likelihood functions computed for discrete x, compute and plot the discrete posterior distributions P(X|data) and the associated cumulative distributions P(X ≤ x|data)
for both processes (language and no language cases). To do this, assume a uniform prior
P(x) ∝ 1 and note that it will be necessary to compute (rather than ignore) the normalizing
constant for Bayes’ rule. Use the cumulative distributions to compute (discrete approximations to) upper and lower 95% confidence bounds on each proportion (xl,nl).
d) Consider the joint posterior distribution P(Xl
, Xnl|data) over xl and xnl, the Bernoulli
probability parameters for the language and non-language contrasts. Given that these two
frequencies are independent, the (discrete) joint distribution is given by the outer product of
the two marginals. Plot it (with imagesc). Compute (by summing the appropriate entries
in the joint distribution) the posterior probabilities that P(Xl > Xnl|data) and conversely
that P(Xl ≤ Xnl|data).
e) Using the estimates from part b as the relevant conditional probabilities, and assuming
the prior that a contrast engages language, P(language) = 0.5, compute the probability
P(language|activation) that observing activation in this area implies engagement of language processes. Is the critique on ‘reverse inference’ correct? How confident should you be
in implicating language if you observe activity in Broca’s area?
