1 1 2 1 1 2 2 2 1 3 2 3 1 4 2 4 1 2 3 chol V$_{1}$ 1 chol V$_{2}$ 1 chol V$_{2}$ 2 mean 1 mean 2 0.90 0.95 1.00 1.05 1.10 1.15 0.1 0.2 0.3 0.4 0.5 1.0 1.2 1.4 1.6 0.5 0.6 0.7 −0.02 0.00 0.02 0.04 −0.04 0.00 0.04 0.08 −0.10 −0.05 0.00 0.05 0.10 −0.025 0.000 0.025 0.050 0.0 0.1 0.2 0.3 0.4 0.5 −0.2 −0.1 0.0 25 30 0.80 0.85 0.90 −0.02 0.00 0.02 0.04 0.19 0.20 0.21 0.22 1.00 1.01 1.02 1.03 0.008 0.012 0.016 HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours HMC Ours method value Figure 6: Estimated marginal distributions for population-level parameters using our method, and HMC. quantiles. For the elements of D and the Cholesky decomposition of V , our method’s estimated distributions are close to the HMC estimates. For other parameters, the convergence of the es- timates is less clear. However, the parameters for which the densities are not aligned are the same parameters for which there is high autocorrelation, and little movement, in the HMC chains. Thus, we infer that our method compares with HMC well in terms of the marginal densities that it generates, with substantially computational effort. The ability for our method to generate independent samples in parallel already makes it an at- tractive alternative to MCMC. In this section, we present an argument in favor of our method’s scalability. Our criterion for scalability is that the cost of running the algorithm grows close to lin- early in the number of households. Our analysis considers the fundamental computational tasks involved: computing the log posterior, its gradient and its Hessian; computing the Cholesky de- composition of the Hessian; and sampling from an MVN proposal distribution. We will show that 4 Scalability and Sparsity 19