Very interesting insight! But I still have one question here. In your post, you’re only trying to estimate $\theta$ given $\beta$. In reality, we need to estimate both. Can you method be used to estimate both parameters? I know we need to add the prior of $\beta$ to the objective function. Then how should we optimize our objective and get the optimal value of $\theta$ and $\beta$. I’m sorry I cannot figure it out as I don’t have a strong background in optimization.

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