Conjugate priors for exponential-type processes (Q1186642)

Let \(X(t)\), \(t\in T\), be a stochastic process defined on a probability space \((\Omega,{\mathcal F},P_ \theta)\) with values in the measurable space \((R^ k,{\mathcal B}(R^ k))\) where \(T=[0,\infty)\) or \(T=\{0,1,2,\dots\}\) and \(\theta\) is a parameter with values in an open set \(\Theta\subset R^ n\). Let \(P_{\theta,t}\) denote the restriction of \(P_ \theta\) to the \(\sigma\)-algebra \({\mathcal F}_ t=\sigma\{X(s): s\leq t\}\). Suppose that for each \(t\) the family \(P_{\theta,t}\), \(\theta\in\Theta\), is dominated by the measure \(\mu_ t\) being the restriction of a probability measure \(\mu\) to \({\mathcal F}_ t\), and \[ dP_{\theta,t}/d\mu_ t=\exp\left\{\sum^ n_{i=1}\theta_ i Z_ i(t)+\Phi(\theta)S(t)\right\},\tag{1} \] where \(\Phi(\theta)\) is a continuously differentiable function and \((Z(t),S(t))\), \(t\in T\), is a stochastic process adapted to the filtration \({\mathcal F}_ t\), \(t\in T\). The authors characterize the family of proper priors on \(\Theta\) which are conjugate to the curved exponential family defined by (1). They also show that for this family of priors the posterior mean value of the parameter function \(-(\partial/\partial\theta)\Phi(\theta)\) given \((Z(t),S(t))\) has the form \((Z(t)+r)/(S(t)+\alpha)\) where \((r,\alpha)\) are parameters of the prior distribution.