Questions regarding the code for Metropolis-adjusted Langevin

[Wenlin-Chen]

Hi, thanks for the repo! I have three questions regarding the code for Metropolis-adjusted Langevin:

1. When a proposal is rejected, why is the gradient not reset (i.e., self.grad[1].data = self.grad[0].data)?
   

   Langevin-dynamics/langevin_sampling/samplers.py

   Lines 212 to 214 in 495de3a

   	else:
   	self.x[1].data = self.x[0].data
   	self.P[1].data = self.P[0].data

2. When computing the density of the proposal distribution, why isP[1] used for both directions? Shouldn't it be P[idx^1]?
   

   Langevin-dynamics/langevin_sampling/samplers.py

   Lines 228 to 233 in 495de3a

   	return (-(.25 / self.lr_fn(self.counter)) *
   	(self.x[idx] - self.x[idx ^ 1] -
   	self.lr_fn(self.counter) * self.grad[idx ^ 1] / self.P[1]) *
   	self.P[1]
   	@ (self.x[idx] - self.x[idx ^ 1] -
   	self.lr_fn(self.counter) * self.grad[idx ^ 1] / self.P[1]))

3. Why are the rejected samples discarded? Shouldn't MH keep both accepted and rejected samples in the list?
   

   Langevin-dynamics/langevin_sampling/samplers.py

   Line 198 in 495de3a

   while not accepted:

[alisiahkoohi]

1. grad[0] represents the gradient in the last accepted sample, and the gradient in the unaccepted sample is not required anymore.
2. What is meant by "direction"? P[1] (preconditioner) acts as the (squared) covariance of the additive noise distribution at the proposed sample and needed for computing the acceptance ratio.
3. The rejected samples themselves are not needed but the duplicates of the accepted samples per rejection might be saved. Feel free to submit a pull request with pointers to references in the comments.