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Given $$\begin {align} \beta&\sim\mathcal N(1.5,1)\\ y&\sim\mathcal N(\beta x,5.5) \end{align} $$I have 10,000 samples from the posterior $\beta_S\sim P(\beta|x)$

Is it enough to get the prediction posterior samples by computing $\hat \mu_s=\beta_s x$ for $s=1\cdots10,000$?

And getting a Single sample with dimension $(10000,1)$ from $ \mathcal N(\hat\mu,5.5) $?


Or do I still need to take N samples of the above and get a sample dimension $(N,10000,1)$?

I need the prediction distribution samples and I don't know how to properly get the posterior predictive samples.

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You want the posterior predictive for $y$, which will be given as $p(y^{rep}|y)=\int p(y^{rep}|\beta,y)p(\beta|y)=\int p(y^{rep}|\beta)p(\beta|y)$. This suggests the following procedure for generating samples from the posterior predictive.

  1. Draw from your posterior of $\beta_s\sim p(\beta|y)$

  2. From each of your sampled $\beta_s$ draw $y|\beta_s\sim N(\beta_s x,5.5)$.

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  • $\begingroup$ In step 2, how many samples $y|\beta_s$ for a single sampled $\beta_s$? Is one sample sufficient or do I take M samples and get the mean $\endgroup$
    – wd violet
    Nov 10, 2023 at 3:14
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    $\begingroup$ One sample is sufficient in step 2 for each $\beta_s$. $\endgroup$ Nov 10, 2023 at 3:15
  • $\begingroup$ Thank you for your clarification! $\endgroup$
    – wd violet
    Nov 10, 2023 at 3:15

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