Hidden Markov Model with transition matrix constraints and partial state labels

Misc.
1

Hi all,

I’m trying to fit a special hidden markov model to some sequential data similar to the hmm examples page. In addition, I would like to:

  1. for some of my observations, i have partial labels. Meaning for some observations, i partially know the set of states that generated these observations. For example, sequences[0][9] was drawn from either state 0 or state 2. Is there a way i can incorporate this information in the model?

  2. impose constraints on the transition matrix by not allowing certain state transitions to occur. For example I would like the transition matrix to have a specific structure. In the code below i tried to do this by manually setting certain entries of the transition matrix log probs to -inf (or a small value). This seems to do the trick, but i was wondering if there is a cleaner way to do it?

def hmm(sequences, lengths, args, batch_size=None, include_prior=True):
    num_sequences, max_length, data_dim = map(int, sequences.shape)
    with poutine.mask(mask=include_prior):
        prior_probs = pyro.sample("prior_probs", dist.Dirichlet(0.9 * torch.ones(args.hidden_dim) + 0.1))
        transition_probs = pyro.sample("transition_probs", dist.Dirichlet(9 * torch.eye(args.hidden_dim) + 0.1).to_event(1))
        emission_probs = pyro.sample("emission_probs", dist.Beta(0.1, 0.9).expand([args.hidden_dim, data_dim]).to_event(2))

    with pyro.plate("sequences", num_sequences, batch_size, dim=-1) as batch:
        lengths = lengths[batch]
        y = sequences[batch, : lengths.max()]
        init_logits = prior_probs.log()
        trans_logits = transition_probs.log()
        trans_logits[1, 0] = -1e2 #torch.inf
        trans_logits[2, 0] = -1e2 #-torch.inf
        trans_logits[3, 0] = -1e2#-torch.inf
        obs_dist = dist.Bernoulli(emission_probs).to_event(1)
        pyro.sample("y", dist.DiscreteHMM(init_logits, trans_logits, obs_dist), obs=y)
2

Hi @zade,

  1. To model partial observations, you can use different emission_probs for full vs partial observations. For example, when walking through a city you might see only a north-south avenue sign or only an east-west street sign, so you could let latent state be the city block (an N*M-valued categorical) and let northsouth_probs be an (N*M,M)-shaped matrix and eastwest_probs be an (N*M,M) shaped matrix. Then you could use an HMM to localize which block you’re on based on a sequence of partial observations.

  2. Sure we use structured transition matrices often. I usually use torch.mask with a static mask that’s defined outside of the learning loop. One issue here is that you should re-normalize your trans_logits after zeroing-out some entries, since they will no longer be normalized.

Good luck!

3

Thanks for the response @fritzo

I’ll try out your recommendations, and let you know how it goes.

Z