HiddenMarkovModel | Python implementation of Hidden Markov Model | Natural Language Processing library

The TFP HiddenMarkovModel implements message passing algorithms for chain-structured graphs, so it can't natively handle the graph in which the Cs are additional latent variables. I can think of a few approaches:

1. Fold the Cs into the hidden state H, blowing up the state size. (that is, if H took values in 1, ..., N and C took values in 1, ..., M, the new combined state would take values in 1, ..., NM).

2. Model the chain conditioned on values for the Cs that are set by some approximate inference algorithm. For example, if the Cs are continuous, you could fit them using gradient-based VI or MCMC:

HMMs work well with temporal data, but it may be a suboptimal fit for this problem.

As you've identified, the observations {x, y, timestamp} are temporal in nature. As a result, it is best cast as the emissions of the HMM, while reserving the digits as states of the HMM.

• Explicitly, if numbers (0 to 9) are encoded as hidden states, then for a 100 x 100 "image", the emission can be one of 10000 possible pixels coordinates.
• The model predicts a digit state at every timestamp (on-line). The output is a non-unique pixel location. This is cumbersome but not impossible to encode (you'd just have a huge emission matrix).
• The initial state probabilities of which digit to start can be a uniform distribution (1/10). More cleverly, you can invoke Benford's law to approximate the frequency of digits appearing in text and distribute your starting probability accordingly.
• State transition and emission probabilities are tricky. One strategy is to train your HMM using the Baum-Welch (a variation of Expectation-Maximization) algorithm to iteratively and agnostically estimate parameters for your transition and emission matrices. The training data would be known digits with pixel locations registered across time.

Casting the problem the other way is less natural because of this lack of temporal fit, but not impossible.

• You can also make 10000 possible states aligning to the pixels, while having 10 emissions (0-9).
• However, most commonly used algorithms for HMMs have run times quadratically related to number of states ( ie. Viterbi algorithm for the most likely valid hidden states runs O(n_emissions * n_states^2)). You are incentivized to keep the number of hidden states low.

Unsolicited suggestions

Maybe what you might be looking for are Kalman filters, which may be a more elegant way to develop this on-line digit recognition (outside of CNNs which appear to the most effective) using this time-series format.

You may also want to look at structured perceptrons, if your emissions are multivariate (ie. x, y) and independent. Here, I believe x, y coordinates should be correlated and should be respected as such.