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Consider a set $U$. My signal is a piece-wise constant "function" $Sig: t \mapsto s$, i.e. the signal at time $t$ equals to some subset $s \subset U$. One can see $Sig(t)$ as a stochastic process.

For a given sequence of time points $\{t_i\}$, we have sequence of partial observations $\{ Ob(t_i) \}$. Each observation $Ob(t_i)$ is not a noised value of $Sig(t_t)$, but rather a "random" part of the "real" signal. I mean:

$$ Ob(t_i) \subset Sig(t_i) \,.$$

(For example, $Sig(t_i)$ can be a graph, and $Ob(t_i)$ will result in a random subgraph of $Sig(t_i)$).

Is there a theory that describes such partially observed stochastic processes? In particular, is there a method to infer (estimate) the parameters of the underlying stochastic process $Sig(t)$ using only a given sequence of observations $\{Ob(t_i)\}$?

If you want to add some addition properties (constraints) to $Sig(t)$ or to $Ob(t)$, you are welcome.

Update:

For example, a very simple problem from this class is considered here: https://stats.stackexchange.com/questions/83998/if-maria-performs-more-observations-per-unit-of-time-than-maximilien-how-can-he

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kerzol
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    Maybe you could try to dial it down a specific application, because with this general description is too mathematical to be able to point out a particular model or theory that would work. but you could look into hidden markov models, a good way to estimate the true states of a hidden markov model based on observations is the Viterbi Algorithm – bone Jan 31 '14 at 19:25
  • I second @bone's suggestion: have a look at hidden Markov models and some of their extensions. – Peter K. Jan 31 '14 at 22:27
  • @bone and Peter K., Thank you! I should certainly think about the Markov chains, but the problem is that we can easily have an infinite number of hidden states... – kerzol Jan 31 '14 at 22:52
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    That's why I added "and extensions. See, for example, http://books.nips.cc/papers/files/nips14/AA01.pdf – Peter K. Jan 31 '14 at 23:18

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