How to get conditional from mixed normals (Bayes rule)?

Question

I need help with something I am getting confused on (this is not a homework problem). If you have a mixture of 2 normal distributions, how do I use Bayes rule with another normal to get a conditional distribution?

Here are some assumptions. Let's say there are data points $i$ from group $j=1$ where $$Q_{ij} = A_{j} + \upsilon_{ij}, \quad \upsilon_{ij} \sim \mathcal{N}(0, \sigma^{2}_{q})$$ $$S_{ij} \mid Q_{ij} = Q_{ij} + \varepsilon_{ij}, \quad \varepsilon_{ij} \sim \mathcal{N}(0, \sigma^{2}_{\varepsilon})\text{, where }\upsilon_{ij} \perp \!\!\!\!\!\perp \varepsilon_{ij} \label{subeqn:qualitycondscore}$$ So, based on Bayes Rule, it should follow that : $$Q_{ij} \mid S_{ij}\sim \mathcal{N}(\gamma S + (1- \gamma) A_{j} , \gamma \sigma^{2}_{\varepsilon})\text{, where }\gamma =\frac{\sigma^{2}_{q}}{\sigma^{2}_{q}+\sigma^{2}_{\varepsilon}}$$

Now, say there is another group $j=2$ with same variance $\sigma^{2}_{q}$ but different mean where $A_{1} \neq A_{2}$. Say that group $j=2$ is pooled together with $j=1$. I know that if group $j=1$ represents a proportion $p$ of the total, then the combined group is a mixture of normals with a pdf of $$f(Q) = p\mathcal{N}(A_{1}, \sigma^{2}_{q}) + (1-p)\mathcal{N}(A_{2}, \sigma^{2}_{q})$$.

Let's say that the assumptions about $S_{ij} \mid Q_{ij} = Q_{ij} + \varepsilon_{ij}, \quad \varepsilon_{ij} \sim \mathcal{N}(0, \sigma^{2}_{\varepsilon})$ still hold with the new distribution. Given that $Q$ is distributed with a mixture of normals, how do I now get the new pdf of $Q_{ij} \mid S_{ij}$, $E(Q_{ij} \mid S_{ij})$, and variance of $Q_{ij} \mid S_{ij}$?

Answer 1

My advice would be to write the PDF of each variable. Bayes rule in that case would be $$f_{Q_{ij}|S_{ij}=s}(q) = \frac{f_{S_{ij}|Q_{ij}=q}(s)\cdot f_{Q_{ij}}(q)}{f_{S_{ij}}(s)}=\frac{f_{S_{ij}|Q_{ij}=q}(s)\cdot f_{Q_{ij}}(q)}{\int_{\mathbb R} f_{S_{ij}|Q_{ij}=q}(s)\cdot f_{Q_{ij}}(q) \,dq},$$ and you know that $Q_{ij}\sim N(A_{ij},\sigma^2_q)$ and that $S_{ij|Q_{ij}=q}\sim N(q,\sigma^2_{\varepsilon})$: I mean, you can actually write the densities, as in $$f_{Q_{ij}}(q)=\frac1{\sqrt{2\pi \sigma^2_q}}\cdot e^{-\tfrac1{2\sigma^2_q}(q-A_{ij})^2}$$ and so on.

Once you have $f_{Q_{i1}|S_{i1}=s}$ and $f_{Q_{i2}|S_{i2}=s}$, you'll have that $$f_{Q_i}=p\cdot f_{Q_{i1}}+(1-p)\cdot f_{Q_{i2}},$$ and you can condition on any event...

...but from the beginning your use of $i,j$ saying that $j$ is $1$ and then is $2$ was a little complicated for me to understand; at this point, I think I can no longer follow you... mind rethinking that?