Let's assume we have a counting process with Poisson distribution
${P(N_t=k)={(λ_t )^k e^{-λ_t t}}/k!}$
when the rate function , ${λ_t}$, can be described as an exponent of an auto-regressive process, or specifically
${λ_t=exp{(μ+x_t)}}$ when ${x_t=∑_{i=1}^kx_{t-i } a_i+e_i}$
when ${P(e)=N(0,\sigma )}$ and ${A}$ is the AR coefficient matrix.
Further, we know
${P(x_{t} |x_{t-k},A,σ )=(1/{σ√2π}) \cdot exp(-(x_{t}-Ax_{t-k})^2/(2σ^2 ))}$
What is the Likelihood funtion ${P(N_t|A,σ)}$ ?
I've started solving this by the law of total distribution :
${L=P(N│Σ,A)=∑_λP(N│λ,Σ,A ) P(λ│Σ,A)=E[P(N│λ,Σ,A)]}$
but I must have a second opinion This problem is a reduction of the multivariate process I am working with when I need to estimates for ${A}$ and ${\sigma}$