Spatial Science and Urban Systems, UC Irvine Spatial Science and Urban Systems, UC Irvine

Fast and Unbiased Log-Determinants for Large Spatial Autoregressive Models

Current

Two line charts of bias in the posterior mean of rho against the true value of rho, negative values at left and positive at right. Six log-determinant methods lie on zero through the interior; near rho of 0.9 and above, the Barry–Pace line climbs to about 1.4 posterior standard deviations, and two others rise modestly.
Bias in the posterior mean of ρ, in posterior standard deviations, by true ρ, for a spatial autoregressive model of all 85,230 U.S. census tracts. Factorize-and-interpolate and Pace–LeSage lie on zero throughout; Barry–Pace lifts off near either boundary, and Smirnov–Anselin and grid-and-spline follow, more modestly, at ±0.95.

Fitting a spatial autoregressive model means evaluating the log-determinant of its Jacobian over and over, thousands of times in a Bayesian sampling run, and that cost has long limited the sample sizes the models reach. The paper introduces two methods. The first treats the log-determinant as an analytic function of the autoregressive parameter and interpolates it with the AAA algorithm; it suits most spatial weights matrices, directed or undirected. The second is matrix-free, built on stochastic Chebyshev traces, for weights that are dense or have hub-and-spoke structure. Both stay unbiased as the autoregressive parameter approaches the edge of its stability region, where the most scalable methods in current use do not.

With them, a Bayesian spatial autoregressive model of all 85,230 U.S. census tracts needs about a second of Jacobian computation per chain of 2,000 draws, where the next-best unbiased method takes several minutes. The methods ship in neighbayes.