A Bayesian probability density for an interpolating function is developed, and its desirable properties and practical potential are demonstrated. This density has an often needed but previously unachieved property, here called cardinal interpolation, which ensures extrapolation to the density of the least-squares linear model. In particular, the mean of the cardinal interpolation density is a smooth function that intersects given (x, y) points and which extrapolates to their least-squares line, and the variance of this density is a smooth function that is zero at the point x values, that increases with distance from the nearest point x value, and that extrapolates to the well-known quadratic variance function for the least-squares line. The new cardinal interpolation density is developed for Gaussian radial basis interpolators using fully Bayesian methods that optimize interpolator smoothness. This optimization determines the basis function widths and yields an interpolating density that is non-Gaussian except for large magnitude x and which is therefore not the outcome of a Gaussian process. Further, new development shows that the salient property of extrapolation to the density of the least- squares linear model can be achieved for more general approximating (not just interpolating) functions.