$$\begin{aligned}
\mathbf{E}\left[g\left(\tau\right)|\mathcal{F}_{t}^{P}\right] & =g\left(\tau\right)\mathbb{I}_{\left\{ \tau\leq t\right\} }+\frac{{\displaystyle \int_{\left(t,+\infty\right)}}g\left(r\right)\varphi_{t}\left(r,\beta_{t}\right)dF(r)}{{\displaystyle \int_{\left(t,+\infty\right)}}\varphi_{t}\left(v,\beta_{t}\right)dF(v)}\,\mathbb{I}_{\left\{ t<\tau\right\}}\,. \label{eq:condexptau2}\end{aligned}$$

If the function $g$ is bounded then the statement immediately follows by an application of the monotone class theorem to simple functions where it is possible to use Theorem \[lem:PRE-theo-4.3\]. In the general case $g$ has to be approximated pointwise by bounded functions and by passing to the limit.

We point out that the function $\phi_{t}$ defined by $$\phi_{t}\left(r,x\right):=\frac{\varphi_{t}\left(r,x\right)}{{\displaystyle \int_{\left(t,+\infty\right)}\varphi_{t}\left(v,x\right)dF\left(v\right)}}, \; \left(r,t\right)\in\left(0,+\infty\right)\times\mathbb{R}_+,\; x\in\mathbb{R}\,, \label{eq:densitybeta2}$$ is, for $t<r$, the a posteriori density function of $\tau$ on $\{\tau>t\}$ based on the observation $\beta_t=x$ (see Corollary \[cor:The-conditional-density\]). Then relation (\[eq:condexptau-1\]), representing the a posteriori distribution of $\tau$ based on the observation of $\mathcal{F}_t^P$, can be rewritten as $$\mathbf{P}\left(\tau\leq u|\mathcal{F}_{t}^{P}\right)=\mathbb{I}_{\left\{ \tau\leq t\right\}} +{\displaystyle \int_{\left(t,u\right]}}\phi_{t}\left(r,\beta_{t}\right)dF\left(r\right)\mathbb{I}_{\left\{ t<\tau\right\} },\;\mathbf{P}\textrm{-a.s.},$$ while (\[eq:condexptau2\]) is equal to the expression $$\mathbf{E}\left[g\left(\tau\right)|\mathcal{F}_{t}^{P}\right]=g\left(\tau\right)\mathbb{I}_{\left\{ \tau\leq t\right\} }+{\displaystyle \int_{\left(t,+\infty\right)}g\left(r\right)}\,\phi_{t}\left(r,\beta_{t}\right)dF\left(r\right)\mathbb{I}_{\left\{ t<\tau\right\} },\;\mathbf{P}\textrm{-a.s.}$$ Here it is possible to see how the Bayesian estimate of $\tau$ given above provides a better knowledge on the default time $\tau$ through the observation of the information process $\beta$ at time $t$.

Extensions of the Bayes Estimates
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In this section we shall deal with an extension of the Bayes estimates of $\tau$ provided in Section \[sec:Conditional-Expectations\]. Roughly speaking we shall 