Is it possible to define a non-homogenous Poisson process (NHPP) with an arrival rate $\lambda(t)=\exp(W_t)$ given by the exponential of a Wiener process $W_t$ with variance $u$ satisfying $W_0=0$?
If yes, how is it possible to do Bayesian inference on data on events up to time $T$? That is, what is the posterior over $W_T$ (and hence $\lambda(T)$) and perhaps $u$ after observing data on time of events up to $T$ (namely, $T_1,T_2,...,T_n<T$). Is there a class of priors on $u$ such that this has an analytic solution? (It's also OK to assume $u$ is known.)
More generally, how does one model an NHPP where the rate follows a stochastic process, and how does one do Bayesian inference on the most recent arrival rate and process parameters?
I know how to do Bayesian inference on a homogenous PP with arrival rate Gamma distributed, for example: https://bookdown.org/kevin_davisross/bayesian-reasoning-and-methods/poisson.html