Say we have a Bayesian polynomial regression like the following.
$$y_i \sim N(\mu_i, \sigma^2)$$ $$\mu_i = \beta_0 + \beta_1 x_i + \beta_2 x_i^2 + \beta_3 x_i^3 $$
where $x_i$ is some mean centred predictor variable.
I'm interested in what kind of prior to specify that is reasonably uninformative. While I generally find a uniform prior to be adequate for the linear coefficient $\beta_1$, I imagine uniform priors on the quadratic $\beta_2$ and cubic $\beta_3$ coefficients might be problematic because small increases can have very large effect on predictions. Thus, I was thinking that perhaps some sort of
highly skewed distribution that makes values closer to zero more likely might be more appropriate.
- Is there a standard choice of noninformative priors for quadratic and cubic parameters in polynomial regression?
- Or alternatively, is there a good default strategy for choosing priors for such parameters (e.g., perhaps based on features of x, y or their relationship) ?