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I am interested in ways to reduce the width/size of a credible interval in a Bayesian regression. Suppose you have a simple Bayesian linear regression $y \sim \mathcal{N}(\mu, \sigma)$, formulated with:

  • Independent variables $x_1, x_2, x_3$ with priors on the intercept $\beta_0$ and coefficients $\beta_1, \beta_2, \beta_3$ with $\mu = \beta_0 +\beta_1 x_1 + \beta_2x_2 + \beta_3 x_3$
  • Prior on the standard deviation $\sigma$

As below:

enter image description here

When you make predictions on the dependent variable $y$ you find that the credible interval for each prediction is wider than you would expect (generated by sampling from $\mathcal{N}(\mu, \sigma)$ for a given set of independent variables).

My question is, what methods are appropriate for narrowing the credible interval. For example, are any of the below inappropriate and any I am missing?

  • Strong Informative Prior on the Standard Deviation (if this aligns with your belief)
  • Increase sample size (if consistent effect in data to reduce uncertainty)
  • Change type of interval of confidence level (ideally fixed)
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  • $\begingroup$ Why exactly it is "wider than you would expect"? $\endgroup$
    – Tim
    Sep 9 at 13:46

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