Suppose that we have the following probability distributuions
- $q$ is the probability that gives positive measure in any state $\omega\in\Omega$ where $\Omega$ is a fixed and finite state space.
- $p$ is the probability that gives a new information $t\in T$ given the state that is $p:\Omega\to\Delta(T)$ where $T$ is also fixed and finite.
- $\mu$ is the probability that given $\omega\in\Omega$ and $t\in T$, then the element $m\in M$ is realized wit probability $\mu$, where $M$ is a (discrete) random variable. In other words $\mu(m|t,\omega)\to\Delta(M) $.
- $l$ is a transition probability that maps the measurable space of $M$, to a measurable space of $R$, such that $l:S\times\Omega\times M\to\Delta(R)$ and hence we can define the following probability function
$$\nu(t,m,r)=\sum_{\omega\in\Omega}q(\omega)p(t|\omega)\mu(m|t,\omega) l(r|t,\omega,\mu)$$
Suppose now that the four bullets above can generate a procedure that is reapeted, in other words keepin the fist bullet fixed, the otehr steps are repeated again as it follows
- at any stage of the repeated procedure denoted as $k>1$ then $p^{k+1}:T^k\times R\to\Delta(T^{k+1})$ where $p^1:\Omega\to\Delta(T)$.
- $\mu^{k+1}$ is the probability that given $\omega\in\Omega$ and $t^{k+1}\in\Delta(T^{k+1})$, then the element $m^{k+1}\in M^{k+1}$ is realiazed with probability $\mu^{k+1}(m^{k+1}|t^{k+1},\omega,\mu^k)$.
- $l^{k+1}$ is a transition probability that maps the measurable space of $M$, to a measurable space of $R$, such that $l^{K+1}:S^{k+1}\Omega\times M^{K+1}\times R^{k}\to\Delta(R^{k+1})$ and hence we can define the following probability function for the stage $k$ in time
$$\nu^{k+1}(t,m,r)=\sum_{\omega\in\Omega}q(\omega)p(t^{k+1}|\omega,r^{k})\mu^{k+1}(m^{k+1}|t^{k+1},\omega,\mu^k) l(r^{k+1}|s^{k+1},\omega,\mu^{k+1},r^{k})$$
$\textbf{Question:}$ Are the probability functions $\nu$ and $\nu^{k+1}$ defined well above? Yes or no and why? Is there any other way to write the probability function in simpler way?