Likelihood Functions

June 6, 2026 · View on GitHub

GraphLD provides Gaussian likelihood helpers for precision-premultiplied GWAS summary statistics under an infinitesimal model.

The model is:

β ~ N(0, D)

z|β ~ N(n^(1/2) R β, R)

where β is the effect-size vector in s.d.-per-s.d. units, D is a diagonal matrix of per-variant heritabilities, z is the GWAS summary statistic vector, R is the LD correlation matrix, and n is the sample size.

The likelihood functions operate on precision-premultiplied summary statistics:

pz = n^(-1/2) R^(-1) z ~ N(0, M), where M = D + n^(-1) R^(-1)

Available functions:

gaussian_likelihood(pz, M): compute the log-likelihood
gaussian_likelihood_gradient(pz, M, del_M_del_a=None, n_samples=10, seed=None, trace_estimator="xdiag"): compute the gradient with respect to either the diagonal of M or parameters a
gaussian_likelihood_hessian(pz, M, del_M_del_a=None, trace_estimator="xdiag"): compute an approximate Hessian with respect to a, or diagonal-only Hessian output when del_M_del_a is omitted

The Hessian approximation is minus the average of the Fisher information matrix and the observed information matrix, and is most useful near the optimum.