Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 36 / 4 / f / Solution

The PDF output places $-1$ inside the 95% credible interval for the time slope, whose endpoints are approximately $-1.329$ and $-0.7955$, and places $1$ inside the baseline slope interval, approximately $0.7915$ to $1.231$. Relative to their posterior standard deviations, the proposed fixed values are onlyfrom the posterior means. Thus the simpler choice $\beta =-1,\gamma =1$ is compatible with the fitted coefficient summaries and has a natural proportional-decay interpretation. It is a reasonable constrained model to compare, not proof of the exact equalities: separate marginal intervals do not establish a joint constraint, and a continuous posterior distribution assigns a specified point probability zero.

The reported Monte Carlo errors measure numerical uncertainty in the estimated posterior means, not uncertainty about the coefficients themselves. In particular, the baseline slope's Monte Carlo error is about 9% of its posterior standard deviation, appreciably larger than the corresponding proportion for the time slope. Longer or better-mixed chains may be needed for precise summaries.