Batch posterior vs Recursive posterior

Question

Consider for a unknown parameter $\Theta\in\mathbb{R}^n$ the following posterior density conditioned on a dataset $d=\{y_1, y_2\}$ of two generic measures $y_1,y_2 \in \mathbb{R}^p$ \begin{equation} p_{\Theta|D}(\theta) \triangleq p_{\Theta|Y_1, Y_2}(\theta| y_1,y_2) \qquad \forall \theta \in \mathbb{R}^n \end{equation} thanks to the Bayes theorem such posterior can be written as \begin{equation} p_{\Theta|D}(\theta)=\frac{\ell_D (\theta) \, p_\Theta (\theta)}{\int \ell_D (t) \, p_\Theta (t) \text{ d}t} \qquad \forall \theta\in\mathbb{R}^n \tag{1} \end{equation} where $p_\Theta (\theta)$ is a prior density and $\ell_D (\theta)$ is the joint likelihood of the two measurements $y_1,y_2$ \begin{equation} \ell_D (\theta) \triangleq p_{D|\Theta}(d|\theta) \triangleq p_{Y_1, Y_2 |\Theta}(y_1, y_2|\theta) \qquad \forall \theta\in\mathbb{R}^n \end{equation} suppose that the two measures are conditionally independent in order to simplify the likelihood as follows \begin{equation} \ell_D (\theta) = \underbrace{p_{Y_1|\Theta}(y_1|\theta)}_{\triangleq \ell_1(\theta)}\,\underbrace{p_{Y_2|\Theta}(y_1|\theta)}_{\triangleq \ell_2(\theta)} \qquad \forall \theta\in\mathbb{R}^n \end{equation} in this case the posterior $(1)$ can be written \begin{equation} p_{\Theta|D}(\theta)=\frac{\ell_1 (\theta) \, \ell_2 (\theta) \, p_\Theta (\theta)}{\int \ell_1 (t)\,\ell_2 (t) \, p_\Theta (t) \text{ d}t} \qquad \forall \theta\in\mathbb{R}^n \tag{2} \end{equation}

Now consider the following recursion:

1. compute the "partial" posterior \begin{equation}p_{\Theta|Y_1}(\theta)\triangleq \frac{\ell_1 (\theta)\,p_\Theta(\theta)}{\int \ell_1 (t)\,p_\Theta(t) \text{ d}t} \qquad \forall \theta \in \mathbb{R}^n\end{equation}
2. compute the "full" posterior \begin{equation}p'_{\Theta|D}(\theta)\triangleq \frac{\ell_2 (\theta)\,p_{\Theta|Y_1}(\theta)}{\int \ell_2 (t)\,p_{\Theta|Y_1}(t) \text{ d}t} \qquad \forall \theta \in \mathbb{R}^n \tag{3}\end{equation}

My question is, the "batch" posterior $p_{\Theta|D}$ expressed in $(2)$ is equivalent to the "recursive" posterior $p'_{\Theta|D}$ expressed in $(3)$?

I have te suspect that the answer is yes because, if we consider the non-normalized posteriors then we have for $(2)$: \begin{equation}P_{\Theta|D}(\theta)\triangleq \ell_2(\theta)\,\ell_1(\theta)\,p_\Theta(\theta) \qquad \forall \theta \in \mathbb{R}^n\end{equation} while for $(3)$: \begin{equation}P'_{\Theta|D}(\theta)\triangleq \ell_2(\theta)\,P_{\Theta|Y_1}(\theta)=\ell_2(\theta)\, \ell_1(\theta)\,p_\Theta(\theta) \qquad \forall \theta \in \mathbb{R}^n\end{equation} where $P_{\Theta|Y_1}(\theta)\triangleq \ell_1(\theta)\,p_\Theta(\theta)$. Clearly, $P'_{\Theta|D}=P_{\Theta|D}$. How can we deal with the normalizing factors?

Answer 1

In (3), \begin{align}p'_{\Theta|D}(\theta) &= \frac{\ell_2 (\theta)\,p_{\Theta|Y_1}(\theta)}{\int \ell_2 (t)\,p_{\Theta|Y_1}(t) \text{ d}t}\\ &= \frac{\ell_2 (\theta)\,\frac{\ell_1 (\theta)\,p_{\Theta}(\theta)}{\int \ell_1 (\tau)\,p_{\Theta}(\tau) \text{ d}\tau}}{\int \ell_2 (t)\,\frac{\ell_1 (t)\,p_{\Theta}(t)}{\int \ell_1 (\tau)\,p_{\Theta}(\tau) \text{ d}\tau} \text{ d}t}\\ &=\frac{\ell_2 (\theta)\,\ell_1 (\theta)\,p_{\Theta}(\theta)}{\int \ell_2 (t)\,{\ell_1 (t)\,p_{\Theta}(t)} \text{ d}t}\\ &=p_{\Theta|D}(\theta)\end{align}