A gravitational-wave search pipeline and its veto-efficiency check extend the…
PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Data analysis and model fitting in the laboratory · Data analysis and model fitting in the laboratory
Error, Chance and the Machine, Ch Tests, intervals and Bayesian inference
Problem
A gravitational-wave search pipeline and its veto-efficiency check extend the boundary form of Wilks' theorem to two parameters and use Bayesian conjugate updating.
((a)) (Boundary Wilks, two amplitudes.) The pipeline now fits two templates, one for each polarisation, with amplitudes A1≥0 and A2≥0, since neither can be negative, and tests the null hypothesis A1=A2=0. Let A^u,j be the unconstrained maximum-likelihood estimates. Suppose that under the null hypothesis Zj=A^u,j/σj for j=1,2 are independent N(0,1) variables, where σj is the known standard error of A^u,j. Each physical estimate is A^j=A^u,j if A^u,j≥0 and A^j=0 otherwise, and correspondingly −2lnΛ is the sum of Zj2 over those j with Zj≥0, and 0 if there are none. Derive the null distribution of −2lnΛ, and use it to find the p-value of an observed −2lnΛ=6.0. Compare it with the χ22 value that Wilks' theorem would give if both null values were interior.
((b)) (Conjugate posterior.) The same pipeline's veto is checked by injecting N=20 simulated signals and counting the k=17 that are correctly flagged. Taking a uniform prior on the veto's true efficiency θ, derive the posterior distribution from Bayes' theorem and identify it as a Beta distribution. A Beta(a,b) density has mean a/(a+b) and variance ab/[(a+b)2(a+b+1)]. Give its mean, and, treating the posterior as approximately Gaussian with the same mean and variance, state an approximate central 68 per cent credible interval for θ.