next up previous
Next: Challenges for Future Research Up: State Space Models Previous: State Space Models

Hybrids of Discrete and Continuous State

 

One disadvantage of the discrete representation of the state is that it is an inefficient representation in comparison to a distributed representation with multiple state variables. When the state variable can take n values, only bits of information about the past of the observed sequence are carried by its value. For example, if instead n binary variables were used, exponentially more values would be available. In general such models would be very expensive to maintain, but so-called factorial HMMs [&make_named_href('', "node31.html#Zoubin-nips8","[125]")] have been proposed with such properties. On the other hand, models with a continuous-valued state have been typically restricted to a linear-Gaussian model, again for reasons of computational tractability. Some particular exceptions with nonlinear and non-Gaussian models are also explored in [&make_named_href('', "node31.html#kitagawa87","[126]"), &make_named_href('', "node31.html#kitagawa96","[127]"), &make_named_href('', "node31.html#kitagawa+gersch96","[47]")]. For example, the above models avoid the Gaussian assumptions with non-parametric (e.g., piecewise-linear) density functions [&make_named_href('', "node31.html#kitagawa87","[126]")], or with a Monte-Carlo approximation [&make_named_href('', "node31.html#kitagawa96","[127]")]. One objective is to model both the abrupt and gradual changes in time series. With a similar objective, several researchers have in fact proposed hybrids of state space models and discrete-state HMMs (or IOHMMs), also known as state space models with switching, or jump-linear systems. See [&make_named_href('', "node31.html#Bar-Shalom93","[122]")] and [&make_named_href('', "node31.html#Zoubin96b","[14]")] for a review of such models. The hybrid state can be represented with a distinct vector variable for each value of the discrete variable.

Many early models assume that some of the parameters of the distribution are known a-priori, and others [&make_named_href('', "node31.html#Shumway91","[13]")] approximate the EM algorithm with a heuristic, because the E-step would require exponential computations. Others [&make_named_href('', "node31.html#Carter94","[128]"), &make_named_href('', "node31.html#Athaide95","[129]")] used expensive Monte-Carlo simulations to address this problem. Instead, in [&make_named_href('', "node31.html#Zoubin96b","[14]")], a function that is a lower bound on the log likelihood is maximized with a tractable algorithm. This paper uses the idea of variational approximation that has already been proposed in [&make_named_href('', "node31.html#Saul96","[130]")] for other intractable models. A simpler version of this idea used in physics is the mean-field approximation [&make_named_href('', "node31.html#Parisi88","[131]")] for statistical mechanics systems.


next up previous
Next: Challenges for Future Research Up: State Space Models Previous: State Space Models

Yoshua Bengio
Tue Oct 7 08:34:36 EDT 1997