Bayes graph model parameter dependencies problem with prior density Suppose I have a graph model

with underlying density (denote $f(\theta_1|\theta_0):=f(\theta_1)$)
$$f(\theta|x)\propto \prod_{i=1}^k f(x_i|\theta_{1:i})f(\theta_i|\theta_{1:i-1}). $$
Suppose markov property such that $x_i$ does not depend on $\theta_{1:i-1}$ if I know $\theta_i$ and $\theta_i$ does not depend on $\theta_{1:i-2}$ if I know $\theta_{i-1}$. Then model can be described by
$$f(\theta|x)\propto \prod_{i=1}^k f(x_i|\theta_{i})f(\theta_i|\theta_{i-1}) .$$
I know densities $f(x_i|\theta_i)$ and $f(\theta_i|\theta_{i-1})$, but I would like to enlist into this model that I know also $f(\theta_i)\;\forall i\in\{1,\dotsc, k\}$. I have prior information about each $\theta$ that I would like to add into model, but still can't find a way how to construct such a model. Does anybody know how to deal with that?

Either $f(\theta_i)$ (which should be denoted $f_i$ to avoid confusing notations) is the marginal of
$$\prod_{i=1}^k f(\theta_i|\theta_{i-1})\tag{1}$$
and in this case there is no clear interest in considering it.
Or $f_i(\theta_i)$ is not the marginal of (1) and in this case the model is incoherent from a probabilistic perspective.
