A Laplace Mixture Representation of the Horseshoe and Some Implications

Maximum a posteriori estimation in graphical models using local linear approximation

“…Given $$$ \left(n,q\right) $$$ and the true precision matrix $$$ {\Omega}_0 $$$, we generate 50 data sets from $$$ \mathcal{N}\left(0,{\Omega}_0^{-1}\right) $$$ and estimate the precision matrix $$$ \hat{\Omega} $$$ using the methods mentioned above. Tuning parameters for graphical lasso and LLA‐based methods are chosen by fivefold cross validation; see Sagar and Bhadra ((2022), Appendix) for other ways of tuning the global scale parameter $$$ \tau $$$. The stopping criterion for LLA‐based methods is set as $$$ {\left\Vert {\Omega}^{\left(t+1\right)}-{\Omega}^{(t)}\right\Vert}_2<1{0}^{-3} $$$, and we draw a total of 6000 samples from the posterior (with 1000 burn‐in samples), for the GHS estimate.…”

“…Proposition 1 Sagar and Bhadra (2022). The marginal horseshoe density for a scalar random variable admits the following representation as a Laplace mixture:…”

“…The ratio of integrals in G ðtÞ ij in (4) can be computed in a computationally efficient manner as simple Riemann sums using the rational approximation of the Dawson function (Lether, 1997) and having the values of D þ ðuÞ pre-computed and stored on a large enough and fine grid of u; see Sagar and Bhadra ((2022), Appendix) for more details. Under the same prior, one can further establish G ðtÞ ij !…”

“…Note that pen 0 ðjxjÞ is completely monotone (Sagar & Bhadra, 2022) and hence non-negative. Thus, the right-hand side of the above bound will suffice to bound jpen 0 ðjxjÞj.…”

“…Given ðn, qÞ and the true precision matrix Ω 0 , we generate 50 data sets from N ð0, Ω À1 0 Þ and estimate the precision matrix Ω using the methods mentioned above. Tuning parameters for graphical lasso and LLA-based methods are chosen by fivefold cross validation; see Sagar and Bhadra ((2022), Appendix) for other ways of tuning the global scale parameter τ. The stopping criterion for LLA-based methods is set as jjΩ ðtþ1Þ À Ω ðtÞ jj 2 < 10 À3 , and we draw a total of 6000 samples from the posterior (with 1000 burn-in samples), for the GHS estimate.…”