Fig. 1.


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Illustration of conditional (pure Gibbs; orange) and marginal sampling (blue) algorithms for a highly correlated (Pearson’s correlation coefficient of ρ = 0.99) two-dimensional Gaussian distribution (black contours). The initial position (βinit, ainit) is indicated by a purple dot. Left: comparison of un-normalized conditional P(β ∣ ainit) distribution evaluated at the initial position and the corresponding marginal P(β) distribution; note that the latter is much wider than the former. Assuming β samples drawn at the 10th percentile, the graphs along the vertical lines represent un-normalized conditial distributions P(a ∣ β) evaluated at the β values drawn with the conditional (orange) and marginal (blue) distributions of β; note that the marginal sampling case results in a much longer step length between the initial and final sample values. Right: samples of a standard Gibbs sampling chain (orange) using conditional sampling for both a and β, and a sampling chain (blue) using marginal sampling for β and conditional sampling for a. Both cases show the first 100 samples initialized from the purple point.

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