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I have a general question concerning probabilistic models. Imagine that I have a system with a high-dimensional state $x$ with a pdf $p(x)$. The state $x$ is time-dependent, and a propagating stochastic model $f(x_t|x_{t-1})$ describes its evolution through time.

In many applications where an approximation of the pdf $p(x)$ by deterministic samples is employed, such as (for example) the particle filter, the propagating model $f(x_t|x_{t-1})$ is taken as a combination of a determinstic propagating function $g(x)$ operating on the determinstic previous state value(s) plus some error $\epsilon$:

$$f(x_t|x_{t-1}) = g(x_{t-1}) + \epsilon$$

This error $\epsilon$ is often assumed to be, for example, a (multi)gaussian with mean $0$ and (co)variance $\sigma$:

$$\epsilon=\mathcal{N}(0,\sigma)$$

Now my question: How much freedom do I have in choosing the error type? Is it, for example, justified to introduce a more complex error type that may restrict the outcomes to a subspace of the state-space (as opposed to, for example, a gaussian error, which could theoretically move anywhere)?

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