Hard attention loss function

I am referring to paper: Show, Attend and Tell: Neural Image Caption Generation with Visual Attention (page 4). I wished to know, why we look to maximize the lower bound of the log likelihood probability $$log p(y|a)$$ and not itself. I agree that we sample images for the $$t_{th}$$ word stochastically, but I don't get the complete reasoning as to why we do this...

Because $$\log p(y|a) = \log \sum_s p(s|a)p(y|s,a)$$ is computationally infeasible to compute. The hard attention attention $$s$$ is a discrete random variable which can take on $$T^L$$ values, where $$T$$ is the number of words and $$L$$ is the number possible of attention locations. Therefore the sum cannot be realistically computed, and variational lower bound is used instead.
• @kg__ in the soft attention case the $s$ are a deterministic function of $a$, and not a random variable conditioned on $a$, so there is no need to marginalize it out, so there is no intractable sum. – shimao Jan 16 at 21:19