Reducing Size of Credible Interval Bayesian Regression Pymc3

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:

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)
• Why exactly it is "wider than you would expect"?
– Tim
Sep 9 at 13:46