For a continuous (forward) transition density $p(x_{t+1}|x_t)$, is there a general rule how to compute the backward transition density $p(x_t|x_{t+1})$ ? I am working on a discrete time grid $t=1,2,...$.
I thought it may work with Bayes' rule $\quad$ $p(x_t|x_{t+1}) = \frac{p(x_{t+1}|x_t) p(x_t)}{p(x_{t+1})} , \quad\quad$ however do not see how to treat the marginal densities $p(x_t)$ and $p(x_{t+1})$.
My three concrete cases are the ones having as forward transition density:
a) a normal density, $p(x_{t+1}|p(x_t) = n(x_{t+1}; x_t+\mu, \sigma^2)$
b) a mixture of two normal densities, $p(x_{t+1}|p(x_t) = a_1 n(x_{t+1}; x_t+\mu_1, \sigma_1^2) + a_2 n(x_{t+1}; x_t+\mu_2, \sigma_2^2)$
c) a mixture of two exponentially damped standard normal cdfs at a linear function of the difference, $\quad p(x_{t+1}|p(x_t) = a_1 e^{-b_1 x_t} N(c_1(x_{t+1}-x_t)+d_1) + a_2 e^{-b_2 x_t} N(c_2(x_{t+1}-x_t)+d_2) $
where $n(x;\mu, \sigma^2)$ denotes the normal density at $x$ with mean $\mu$ and variance $\sigma^2$ and $N(x)$ is the cumulative standard normal distribution at $x$.
Thanks for any hints / thoughts!