I have been self-studying Liptser and Shiryaev (2001), and came up with a little twist on this example related to the optimal filtering of Brownian motion (Page 371, example 1).
There is a Brownian motion with unknown drift $\mu$ and known variance $\sigma^{2}$.
$$X_{t} = \mu t + \sigma Z_{t} \sim N(\mu t , \sigma^{2}t ) $$
The drift parameter $\mu$ is uncertain $\mu \in \{\mu_{H},\mu_{L}\}$, and the prior $Pr(\mu=\mu_{H}|t=0)$ is given by $q_{0}$.
According to Liptser and Shiryaev (2001) in page 371, the posterior $q_{t}$ satisfies :
$$ dq_{t} = q_{t} (1-q_{t}) \frac{\mu_{H}-\mu_{L}}{\sigma} \frac{dX_{t}-q_{t}\mu_{H}dt-(1-q_{t})\mu_{L}d_{t}}{\sigma}$$
$$ $$
This is the result that I have, and I want to add one more trick as follows.
Let denote the state of the world by $\theta$, which can be either high or low : $\theta \in \{H,L\}$.
If the world is high, $Pr(\mu=\mu_{H}|\theta=H)=p_{H}$, whereas if it's low, $Pr(\mu=\mu_{H}|\theta=L)=p_{L}$ such that $p_{H}>p_{L} $.
In other words, interpreting $X_{t}$ as an outcome of the world, the high state is more likely to give more outcome.
I denote the prior $Pr(\theta=\theta_{H}|t=0)$ by $\pi_{0}$. Then what would be the function that characterizes $d\pi_{t}$ just like $dq_{t}$ above?
These are what I have tried.
By Bayes' rule
$$q_{t} = \frac{q_{0} \cdot Pr(X_{t}=x|\theta_{H})}{q_{0} \cdot Pr(X_{t}=x|\theta_{H}) + (1-q_{0})\cdot Pr(X_{t}=x|\theta_{L})}$$
which is equivalent to :
$$\frac{q_{t}}{1-q_{t}} = \frac{q_{0}}{1-q_{0}} \frac{Pr(X_{t}=x|\theta_{H})}{Pr(X_{t}=x|\theta_{L})} $$
where
$$ Pr(X_{t}=x|\theta_{H}) = Pr(X_{t}=x|\mu=\mu_{H}) \cdot p_{H} + Pr(X_{t}=x|\mu=\mu_{L})\cdot (1-p_{H}) $$
and analogously,
$$ Pr(X_{t}=x|\theta_{L}) = Pr(X_{t}=x|\mu=\mu_{H}) \cdot p_{L} + Pr(X_{t}=x|\mu=\mu_{L})\cdot (1-p_{L}) $$
As far as I know, the denominator and the numerator of the likelihood ratio are weighted sums of two independent normal random variables, hence both are normal random variables.
My guess is that, the likelihood ratio can be approximated to another normal random variable (according to this reference https://en.wikipedia.org/wiki/Ratio_distribution), thus $\pi_{t} / (1-\pi_{t})$ becomes a normal random variable.
I would like to know if this approach seems correct (under approximation) and, if so, how I could retrieve the differential equation $d \pi_{t}$.