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7:51 PM
@ Liviu. Stochastic processes are just collections of usual (= non-functional) r.v. indexed by the time. Here, I have only one "r.v." $f$ but it is quite unusual because it is defined on a functional space. I'm aware of only two kinds of functional integrals: Wiener measure and Feynman path integral, but they look quite different from the hypothetical functional integrals I'm considering here. I guess you were talking about Wiener measure, isn't it?
I don't think I understand your question. What exactly do you mean by a "probability distribution" and what exactly would it mean for it to be "uniform"? One common interpretation of "uniform" in vector spaces is "translation invariant", but there is no translation invariant probability measure on any vector space, not even on $\mathbb{R}$. Maybe you just want a translation invariant measure, i.e. a Lebesgue measure, but it's well known that there is no Lebesgue measure on infinite-dimensional spaces.
@ Nate. Many thanks for your comments. In some extent, you are the first to answer my question: starting from some practical problems (see the questions), I fell, at least formally, on some nasty functional integrals over sets like ${\mathbb{R}^\mathbb{R}}$ or ${\left( {{\mathbb{R}^d}} \right)^{{\mathbb{R}^d}}}$ or event the set of all subsets of all Cartesian powers of $\mathbb{R}$. So my question was: are those functional probability distributions and integrals well known or crazy? It seems like they are really crazy. But without them it is not possible to solve many interesting problems...
Please, please don't put the question on hold: it's not homework, it's not chat, it's not about MO, these are well defined, serious, important mathematical questions about the existence of some unusual (=functional) mathematical objects arising from a collision between dynamical system theory and Bayesian probability theory. Please refer to both companion questions and please provide me with some tips on how to improve my question. Or just answer it please!
8:10 PM
I'm afraid that I don't know what you mean by the "mean image". I suspect that once you have given a precise statement of what kind of mathematical object you are looking for, and precisely what properties it should satisfy, the answer will be "no such object exists". But for now, the question is still too vague to answer, at least for me.
I'm waiting for somebody to tell me: yes, this is Mister X theory, introduces a long time ago. Your "functional mean image" is in fact know as "sdfjlsjfdl".
I just know about a very little bit about Wiener abstract measure and Feynman path integrals. But in my understanding, they are different kind of functional integrals
9:31 PM
my "functional mean image" arises in a violent collision between dynamical system theory (phase space, initial condition, state-space equation, output equation and Bayesian probability theory
all that comes from an attempt to fix classical cross-correlations functions for deterministic signals
Because crosscorrelation and covariance don't make much sense for deterministic signals: they are invariant by permutation of the time points!
They functional probability distribution I wrote for a deterministic signal should not be invariant by permutation of the time points (= i.i.d. if you are a frequentist or De Finetti-exchangeable otherwise)
$p\left[ {s\left( 1 \right),s\left( 2 \right),...,s\left( M \right)} \right] = \int\limits_{{\mathbb{R}^\Gamma }} {\int\limits_{{\Gamma ^\Gamma }} {\int\limits_\Gamma {{\text{D}}g{\text{D}}f{{\text{d}}^d}{z_1}\prod\limits_{m = 1}^M {\delta \left\{ {g\left[ {{f^{m - 1}}\left( {{z_1}} \right)} \right] - s\left( m \right)} \right\}p\left( {{z_1},f,g} \right)} } } } $
10:49 PM
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Apr '1619
Apr20
Discussion between Fabrice Pautot and…
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