How do you compute fisher information matrix?

[Kabibi]

https://github.com/wiseodd/natural-gradients/blob/7d51f19d315cb393a52a371162aaf2e27d20dbaa/numpy/toy/full_fisher.py#L47

I have no idea how do you compute fisher information matrix. Specifically, I don't know how do you compute p(x|θ) without using a prior in your data. Can you explain? Thanks.

[wiseodd]

The codes in this repo are actually using the empirical Fisher matrix, so take them with a grain of salt.

To compute the true Fisher matrix, you need to do compute the expectation w.r.t. p(y | x, \theta). One way to do that is to approximate it with MC integration: Do forward propagation to get the predictive distribution given x, then sample y ~ p(y | x, \theta) and do backprop and compute the outer product of the gradient. Do this many times and average, and you get the (approximation of) the true Fisher matrix.

[tranvansang]

I also have the same question, and I am trying to figure it out (but still stuck).

Even if the implementation calculates the empirical FIM, I cannot understand where the y - y*y term comes from in the formula of grad_loglik_z. In my below theoretical implication, the term (y-y*y) should not be there, but if I remove it, the program won't work (nan returned).

F = cov(grad_loglik_W)
grad_loglik_W = d_loglik / d_W = d_loglik_z * X_train
grad_loglik_z = grad_loglik_y * dy_dz  # dy_dz is dz in the program
grad_loglik_y := derivative of log likelihood with respect to y

If the output y follows a normal distribution, d_loglik_y will be y - t_train. However, in the code, it is (y - t_train) / (y - y * y). I understand the sign, so no problem, but cannot figure out why y - y * y is there.

Just posting here if luckily, someone can explain it, while I am also trying to do it myself.

============

I nearly figure out the answer.

The distribution of y is a binomial distribution, not a normal distribution. The formula turns to be a classical form of the fisher scoring applied in GLM training procedure.

[wiseodd]

Yes, this particular code is for (binary) classification with Bernoulli likelihood. See https://github.com/wiseodd/natural-gradients/blob/7d51f19d315cb393a52a371162aaf2e27d20dbaa/numpy/toy/full_fisher.py#L25-L26