Quadratic/cubic regression: scaling vs. using different priors

So I’ve run into a problem while constructing logistic models in brms for
a quadratic model: i.e.

y ~ x + I(x^2)

and a cubic model: i.e.

y ~ x + I(x^2) + I(x^3)

So far I’ve been scaling x to have a mean of 0 and a SD of 0.5 and using a Student prior - student_t(7, 0, 1) . The aim is to have some form of regularisation and to calculate BFs. However, the problem is that the beta values for the quadratic/cubic is naturally more extreme than the linear one. Should I therefore scale the quadratic term seperately? Or should I just use different priors? If the latter, what would be a student_t(7, 0, 1) equivalent for quadratic and cubic betas? Does it even make sense to have variants of x in model that have different scales?

If any one has an accessible text how to construct priors with this in mind, please let me know because currently I’ve hit a wall.

Yes, you can scale the different terms separately and use different priors.

I’m not sure what you mean by Student-t equivalent for quadratic and cubic betas. I’m guessing beta is the coefficient name and you want to know how to scale coefficients for x, x^2, x^3. What are you thinking about in terms of equivalence here?

One thing you can do is just standardize the x, x^2, x^3 predictors. lt’s a linear transform, so you don’t lose any power and you can always scale the answer back if you want. A similar alternative is to inversely scale the parameters by sd(x), sd(x^2), sd(x^3).

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