Polynomial regression

A linear regression in the coefficients with a polynomial mean function $\mathbb{E}(Y\mid t)={\sum }_{j=0}^d{\beta }_j{t}^j$. Successive degrees define nested models that can be compared by an F-test.

With independent prior distributions $\beta \sim N(\mu ,\Sigma )$ and $s\sim \mathrm{IG}(a,b)$ in a normal distribution regression model, the conditional posterior distribution of $\beta$ has covariance matrix $V=(X^{T}X/s+{\Sigma }^{-1}{)}^{-1}$ and mean $V(X^{T}y/s+{\Sigma }^{-1}\mu )$. The conditional posterior distribution of $s$ is inverse-gamma distribution $\mathrm{IG}(a+n/2,b+\Vert y-X\beta {\Vert }^2/2)$. The shape contains no half-dimension contribution from $\beta$, since its prior distribution is independent of $s$. These conditionals give a two-block Gibbs sampler.

A polynomial regression of degree two is linear in its unknown regression coefficients despite being curved in the predictor. Its design matrix has columns $1,x,x^2$; three distinct predictor values give full column matrix rank.