Add support for parallel hmm sampler

[calebweinreb]

Currently there are associative scan implementations for hmm smoothing and filtering, but not for sampling.

[AdrienCorenflos]

The HMM sampling can't really be done using prefix-sum. You would need to use a divide and conquer approach as in Section 3.2 in my paper https://arxiv.org/abs/2303.00301 (Note that it's for Gaussians but the general principle applies).

On Tue, 5 Sept 2023, 22:31 Caleb Weinreb, @.***> wrote:

Currently there are associative scan implementations for hmm smoothing and filtering, but not for sampling.

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[slinderman]

@AdrienCorenflos, I think @calebweinreb has actually come up with a clever way to implement the HMM (and LGSSM) sampling with associative_scan! Curious what you think of his implementation in this PR https://github.com/probml/dynamax/pull/342. You can see the corresponding parallel LGSSM sampler here.

[AdrienCorenflos]

The LGSSM is unsurprising (it's just a cumulative sum over pre-generated Gaussians). In fact I think I had it implemented some time ago.

I'm actually surprised by the HMM case though. I had given it some thought and decided it was not possible because I didn't see how selecting indices could be associative. I need to check how that is done here. Is there a small proof for it somewhere?

On Wed, 6 Sept 2023, 00:06 Scott Linderman, @.***> wrote:

@AdrienCorenflos https://github.com/AdrienCorenflos, I think @calebweinreb https://github.com/calebweinreb has actually come up with a clever way to implement the HMM (and LGSSM) sampling with associative_scan! Curious what you think of his implementation in this PR #342 https://github.com/probml/dynamax/pull/342. You can see the corresponding parallel LGSSM sampler here https://github.com/probml/dynamax/blob/main/dynamax/linear_gaussian_ssm/parallel_inference.py#L356 .

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[calebweinreb]

Hi Adrien,

The way I think about the algorithm, each associative scan element is a function $E_{st}: [K] \to [K]$ where $E_{st}(i) \mapsto z_s \sim P(z_s \mid x_{1:t-1}, z_t=i)$. The functions can be thought of as draws from a random process. The operator is function composition and it's associative because function composition is associative.

[slinderman]

I think the other necessary ingredient to the proof is that the function composition $E_{su} = E_{st} \circ E_{tu}$ defined by $E_{su}(i) = E_{st}(E_{tu}(i))$ ensures that $E_{su}(i) \mapsto z_s \sim P(z_s \mid x_{1:u-1}, z_u=i)$. Moreover, $(E_{su}(i), E_{tu}(i))$ are jointly distributed as $P(z_s, z_t \mid x_{1:u-1}, z_u=i)$. These follow from the fact that $z_s$ is conditionally independent of $x_{t:u-1}$ given $z_t$ in an HMM.

Finally, note that the base cases $E_{s,s+1}$ are straightforward to sample given the filtering distributions. The final message is defined as $E_{T,T+1} \triangleq E_T \mapsto z_T \sim p(z_T \mid x_{1:T})$.