I am trying to perform the symbolic integration of the following expression $$\int \prod_{i=1}^n p(x_i | z) p(z) dz$$ where $$p(x_i|z) = \frac{\exp\left(\tfrac{-(x-z)^2}{2\sigma_i^2}\right)}{\sqrt{2 \pi \sigma_i^2}}$$ and $p(z)$ is also normal. I did search the site and I think that answers to n-fold symbolic integral in Mathematica might be applicable but I really couldn't understand what was going on. Any clues are welcome, Thanks.
UPDATE: I tried to manually compute the integral for product of 3 terms by the following expression $$\int \frac{\sqrt{2 \pi }}{e^{z^2} \left(\sqrt{2 \pi \sigma _1^2} e^{\frac{\left(x_1-z\right){}^2}{2 \sigma _1^2}}\right) \left(\sqrt{2 \pi \sigma _3^2} e^{\frac{\left(x_3-z\right){}^2}{2 \sigma _3^2}}\right) \left(\sqrt{2 \pi \sigma _2^2} e^{\frac{\left(x_2-z\right){}^2}{2 \sigma _2^2}}\right)} \, dz$$
And Mathematica produced a pretty bad expression: $$\tiny\frac{\sigma _1 \sigma _2 \sigma _3 \exp \left(-\frac{\left(\left(2 \sigma _3^2+1\right) \sigma _2^2+\sigma _3^2\right) x_1^2-2 \sigma _2^2 x_3 x_1+\left(\left(2 \sigma _2^2+1\right) \sigma _1^2+\sigma _2^2\right) x_3^2-2 x_2 \left(\sigma _1^2 x_3+\sigma _3^2 x_1\right)+\left(\left(2 \sigma _3^2+1\right) \sigma _1^2+\sigma _3^2\right) x_2^2}{2 \left(\left(\left(2 \sigma _3^2+1\right) \sigma _2^2+\sigma _3^2\right) \sigma _1^2+\sigma _2^2 \sigma _3^2\right)}\right) \text{erf}\left(\frac{\sigma _1^2 \left(\sigma _2^2 \left(-x_3+2 \sigma _3^2 z+z\right)+\sigma _3^2 \left(z-x_2\right)\right)+\sigma _2^2 \sigma _3^2 \left(z-x_1\right)}{\sqrt{2} \sigma _1 \sigma _2 \sigma _3 \sqrt{\left(\left(2 \sigma _3^2+1\right) \sigma _2^2+\sigma _3^2\right) \sigma _1^2+\sigma _2^2 \sigma _3^2}}\right)}{2 \sqrt{2 \pi } \sqrt{\sigma _1^2} \sqrt{\sigma _2^2} \sqrt{\sigma _3^2} \sqrt{\left(\left(2 \sigma _3^2+1\right) \sigma _2^2+\sigma _3^2\right) \sigma _1^2+\sigma _2^2 \sigma _3^2}}$$
multiple-integral
tag is for? Your expression appears to be the integral over a single variable $z$. $\endgroup$