Does Bayesian model averaging improve polynomial extrapolations? Two toy problems as tests

We assess the accuracy of Bayesian polynomial extrapolations from small parameter values, x, to large values of x. We consider a set of polynomials of fixed order, intended as a proxy for a fixed-order effective field theory (EFT) description of data. We employ Bayesian model averaging (BMA) to combine results from different order polynomials (EFT orders). Our study considers two 'toy problems' where the underlying function used to generate data sets is known. We use Bayesian parameter estimation to extract the polynomial coefficients that describe these data at low x. A 'naturalness' prior is imposed on the coefficients, so that they are $\mathcal{O}(1)$. We BMA different polynomial degrees by weighting each according to its Bayesian evidence and compare the predictive performance of this BMA with that of the individual polynomials. The credibility intervals on the BMA forecast have the stated coverage properties more consistently than does the highest evidence polynomial, though BMA does not necessarily outperform every polynomial.