Hidden Markov Model with Multiple Agents

EDIT: The author clarified that they are interested in the situation where only the aggregate number of good products at each time is known (rather than the output of each individual machine). This answer does not address that case, and assumes that a separate time series is available for each machine.

You can run a state decoding algorithm (e.g., Viterbi or forward-backward) separately on each machine, if they are independent of each other. You can also decide which of the parameters are shared between machines and which are not. For example, if you have $M$ machines, consider the following scenarios:

1. All machines have the same success probabilities $p_1$ and $p_2$ in the two states, and the same transition probabilities between the states (stored as a 2-by-2 transition matrix $\Gamma$). Probably reasonable if the machines have the same configurations.

2. All machines have the same probabilities $p_1$ and $p_2$, but different transition probabilities. This model would require machine-specific transition matrices $\Gamma_1, \Gamma_2, \dots, \Gamma_M$. This might be useful if some machines spend more time in the "bad" state than others.

3. All machines have the same transition probabilities, but they have different success probabilities $(p_{11}, p_{21}), \dots, (p_{1M}, p_{2M})$. This represents the case where the "bad" state doesn't correspond to exactly the same behaviour for the different machines.

You could even have both transition matrices and success probabilities be machine-dependent, at which point you would be using a different hidden Markov model for each machine.

Let's say that you have $M$ time series of length $n$, where $z_{ij}$ is the $j$-th observation for the $i$-th machine ($z_{ij}$ is either "good product" or "bad product"). The most common state decoding algorithms take an inputs a time series of observations, and model parameters. So, for each machine $m = 1, 2, \dots, M$, you can run the decoding algorithm on the corresponding observations $(z_{m1}, z_{m2}, \dots, z_{mn})$, given the corresponding (possibly machine-dependent) parameters $\Gamma_m$ and $(p_{1m}, p_{2m})$.

References on state decoding:

• Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2), 257-286. See "Solution to Problem 2" section.

• Zucchini, W., MacDonald, I. L., & Langrock, R. (2017). Hidden Markov Models for Time Series: An Introduction Using R. Second Edition. CRC Press. See Section 5.4.