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Random projections for Bayesian regression
This article deals with random projections applied as a data reduction
technique for Bayesian regression analysis. We show sufficient conditions under
which the entire -dimensional distribution is approximately preserved under
random projections by reducing the number of data points from to in the case . Under mild
assumptions, we prove that evaluating a Gaussian likelihood function based on
the projected data instead of the original data yields a
-approximation in terms of the Wasserstein
distance. Our main result shows that the posterior distribution of Bayesian
linear regression is approximated up to a small error depending on only an
-fraction of its defining parameters. This holds when using
arbitrary Gaussian priors or the degenerate case of uniform distributions over
for . Our empirical evaluations involve different
simulated settings of Bayesian linear regression. Our experiments underline
that the proposed method is able to recover the regression model up to small
error while considerably reducing the total running time
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