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can you help find the maximum (analytically) of the following posterior pdf?

$p(\theta|x) = \frac{\alpha}{\sqrt{2\pi}}e^{-\frac{1}{2}(\theta-x)^2} + \frac{1-\alpha}{\sqrt{2\pi}}e^{-\frac{1}{2}(\theta+x)^2}$

In particular, I am interested in values $\alpha=\frac{1}{2}$ and $\alpha=\frac{3}{4}$. If I draw $p(\theta|x)$ it is easy to find the maximum at $\theta=\pm x$ for $\alpha=\frac{1}{2}$, and $\theta=x$ for $\alpha=\frac{3}{4}$, but I am not able to show it analytically.

Thank you!

st7488
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  • Seems like a mixture model. Also $\hat\theta = x \implies \hat\alpha = 1 $ not 0.5. Consider applying the mixture model technique invoking a variable Z where by $Z=1\implies x\sim f_1$ – Onyambu Mar 10 '23 at 10:39
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    This is a mixture of two normal distributions with means at $\pm{x}$ and variances of 1. There doesn't appear to be a general closed-form solution for the mode(s). Here is one paper: https://mat.polsl.pl/sjpam/zeszyty/z6/Silesian_J_Pure_Appl_Math_v6_i1_str_059-068.pdf – jblood94 Mar 10 '23 at 14:48
  • Welcome to CV! Re "easy to find the maximum:" I don't think so, because when $x$ is sufficiently large, your putative maximum is a local minimum. See https://stats.stackexchange.com/a/416216/919 for illustrations. – whuber Mar 10 '23 at 17:28

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