_sample_from_history output size

[Romain3Ch216]

Hi everyone,

Thank you for the Batch BALD implementation.

I am trying to understand it and I am confused with the output size of _sample_from_history. I don't get why we reshape choices with dimensions (N x K x S) into expanded_choices_N_K_K_S with dimensions (N x K x K x S).

Then the dimension of the output samples_M_K is ( (KxS) x K ) whereas I expected it to be (S x K) since S is the number of draws.

Thank you for your help,
Romain

[Romain3Ch216]

I will rephrase my question.

If I understand it right, the function _sample_from_history samples probabilities of p(y_1, ..., y_(n-1)) among the c^(n-1) values (y_1, ..., y_(n-1)) can take, where c is the number of classes. However, I don't understand how the sampling strategy works.

_sample_from_history takes as argument probs, the probabilities of previously selected points and num_draws, the number of draws to sample.

If we consider the following example:

probs = tensor([[0.2, 0.3, 0.4], 
                [0.8, 0.7, 0.6]])

We have 2 classes and 3 MC samples.

Let's take num_draws = 5 :

choices = tensor([[[1, 1, 1, 0, 1],
                   [1, 0, 0, 1, 0],
                   [0, 0, 1, 0, 0]]])

where the ith row of choices is 5 independant draws of the bernoulli distribution defined by the ith column of probs.

Then, gather_expand is called which yields a new tensor probs where:

probs[:,:,:,0] = tensor([[[0.8, 0.8, 0.2], 
                          [0.7, 0.7, 0.3],
                          [0.6, 0.6, 0.4]]])

probs[:,:,:,1] = tensor([[[0.8, 0.2, 0.2], 
                          [0.7, 0.3, 0.3],
                          [0.6, 0.4, 0.4]]])

etc.

We can see that the rows of probs[:,:,:,i] are defined by the index of the ith column of choices. I don't understand why we would do that since only the kth element of this column is related to the kth MC sample we drew.

I hope my question is clear.
Thanks,
Romain

[Dref360]

Hello! Thanks for the question.

Let batch N = 2, MC-Iteration K=20 and num_sample S = 100.
The result of this function would be a Tensor with shape [K*S, K].
I see now that in the documentation we say that it returns [num_draws, K]. We should be more clear.

choices = self.draw_choices(...) # Has shape [B, K, S]
probs = gather_expand(...) # Has shape [N,K,K,S]
entropies = ... # Has shape [K*S, K]

If I understand your question properly, we are not indexing by the MC samples (K), but by the draw (S).

Yes there is a typo in the documentation here # [batch, draw, iterations] should be # [batch, iterations, draw]

We should make this heuristic clearer. In the next release, we will be refactoring this part of the code so I'll try to improve the documentation. Also, Andreas Kirsch has recently released a new version of their algorithm here: https://github.com/BlackHC/batchbald_redux

Did I answer your question correctly?

Cheers,
Fred

[Romain3Ch216]

Thanks for the reply. Does N = B ? What I don't understand is why we expand probs to a new dim of size K that leads to the size K*S of the first dim of entropies. It seems like an arbitrary choice.

I also don't get why this variable is named entropies whereas it contains probability values, or maybe I understood it wrong ?

Thanks a lot,
Romain

[Dref360]

1. Yes this is arbitrary, we follow the same behavior as the official implementation. Maybe it changed in the new one? PRs are welcome!
2. We use that variable later to compute the mutual entropy. Could be better named too. I'll keep that in mind.
   Thank you!

[Romain3Ch216]

Ok thank you very much!