In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required.
In random signal theory, this distribution is typically a stochastic process, e.g. a Gaussian or a uniform process.
What do such distributions become in deterministic signal theory?, that is the question.
To make it simple, consider a discrete-time real deterministic signal
$ s\left( {1} \right),s\left( {2} \right),...,s\left( {M} \right) $
For instance, they may be samples from a continuous-time real deterministic signal.
By the standard definition of a discrete-time deterministic dynamical system, there exists:
- a phase space $\Gamma$, e.g. $\Gamma \subset \mathbb{R} {^d}$
- an initial condition $ z\left( 1 \right)\in \Gamma $
- a state-space equation $f:\Gamma \to \Gamma $ having $ z\left( 1 \right)$ in its domain of definition such as $z\left( {m + 1} \right) = f\left[ {z\left( m \right)} \right]$
- an output or observation equation $g:\Gamma \to \mathbb{R}$ such as $s\left( m \right) = g\left[ {z\left( m \right)} \right]$
Hence, by definition we have
$\left[ {s\left( 1 \right),s\left( 2 \right),...,s\left( M \right)} \right] = \left\{ {g\left[ {z\left( 1 \right)} \right],g\left[ {f\left( {z\left( 1 \right)} \right)} \right],...,g\left[ {{f^{M - 1}}\left( {z\left( 1 \right)} \right)} \right]} \right\}$
or, in probabilistic notations
$p\left[ {\left. {s\left( 1 \right),s\left( 2 \right),...,s\left( M \right)} \right|z\left( 1 \right),f,g,\Gamma ,d} \right] = \prod\limits_{m = 1}^M {\delta \left\{ {g\left[ {{f^{m - 1}}\left( {z\left( 1 \right)} \right)} \right] - s\left( m \right)} \right\}} $
Therefore, by total probability and the product rule, the marginal joint prior probability distribution for a discrete-time deterministic signal conditional on phase space $\Gamma$ and its dimension $d$ formally/symbolically writes
$p\left[ {\left. {s\left( 1 \right),s\left( 2 \right),...,s\left( M \right)} \right|\Gamma ,d} \right] = \int\limits_{{\mathbb{R}^\Gamma }} {{\text{D}}g\int\limits_{{\Gamma ^\Gamma }} {{\text{D}}f\int\limits_\Gamma {{{\text{d}}^d}z\left( 1 \right)\prod\limits_{m = 1}^M {\delta \left\{ {g\left[ {{f^{m - 1}}\left( {z\left( 1 \right)} \right)} \right] - s\left( m \right)} \right\}p\left( {z\left( 1 \right),f,g} \right)} } } } $
Should phase space $\Gamma$ and its dimension $d$ be also unknown *a priori*, they should be marginalized as well so that the most general marginal prior probability distribution for a deterministic signal I'm interested in formally/symbolically writes
$p\left[ {s\left( 1 \right),s\left( 2 \right),...,s\left( M \right)} \right] = \sum\limits_{d = 2}^{ + \infty } {\int\limits_{\wp \left( {{\mathbb{R}^d}} \right)} {{\text{D}}\Gamma \int\limits_{{\mathbb{R}^\Gamma }} {{\text{D}}g\int\limits_{{\Gamma ^\Gamma }} {{\text{D}}f\int\limits_\Gamma {{{\text{d}}^d}z\left( 1 \right)\prod\limits_{m = 1}^M {\delta \left\{ {g\left[ {{f^{m - 1}}\left( {z\left( 1 \right)} \right)} \right] - s\left( m \right)} \right\}p\left( {z\left( 1 \right),f,g,\Gamma ,d} \right)} } } } } } $
where ${\wp \left( {{\mathbb{R}^d}} \right)}$ stands for the powerset of ${{\mathbb{R}^d}}$.
Dirac's $\delta$ distributions are certainly welcome to "digest" those very high dimensional integrals. However, we may also be interested in probability distributions like
$p\left[ {s\left( 1 \right),s\left( 2 \right),...,s\left( M \right)} \right] \propto \sum\limits_{d = 2}^{ + \infty } {\int\limits_{\wp \left( {{\mathbb{R}^d}} \right)} {{\text{D}}\Gamma \int\limits_{{\mathbb{R}^\Gamma }} {{\text{D}}g\int\limits_{{\Gamma ^\Gamma }} {{\text{D}}f\int\limits_\Gamma {{{\text{d}}^d}z\left( 1 \right)\int\limits_{{\mathbb{R}^ + }} {{\text{d}}\sigma {\sigma ^{ - M}}{e^{ - \sum\limits_{m = 1}^M {\frac{{{{\left\{ {g\left[ {{f^{m - 1}}\left( {z\left( 1 \right)} \right)} \right] - s\left( m \right)} \right\}}^2}}}{{2{\sigma ^2}}}} }}p\left( {\sigma ,z\left( 1 \right),f,g,\Gamma ,d} \right)} } } } } } $
Please, what can you say about those important probability distributions beyond the fact that they should not be invariant by permutation of the time points, i.e. not De Finetti-exchangeable?
What can you say about such strange looking functional integrals (for the state-space and output equations $f$ and $g$) and even set-theoretic integrals (for phase space $\Gamma$) over sets having cardinal at least ${\beth_2}$? Are they already well-known in some branch of mathematics I do not know yet or are they only abstract nonsense?
More generally, I'd like to learn more about functional integrals in probability theory. Any pointer would be highly appreciated. Thanks.
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