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SUMMARY:On L2 posterior contraction rates in Bayesian nonparametric regres
 sion models - Paul Rosa (University of Cambridge)
DTSTART:20250828T103000Z
DTEND:20250828T110000Z
UID:TALK234514@talks.cam.ac.uk
DESCRIPTION:The nonparametric regression model with normal errors has been
  extensively studied\, both from the frequentist and Bayesian viewpoint. A
  central result in Bayesian nonparametrics is that under assumptions on th
 e prior\, the data-generating distribution (assuming a true frequentist mo
 del) and a semi-metric d(.\,.) on the space of regression functions that s
 atisfy the so called testing condition\, the posterior contracts around th
 e true distribution with respect to d(.\,.)\, and the rate of contraction 
 can be estimated. In the regression setting\, the semi-metric d(.\,.) is o
 ften taken to be the Hellinger distance or the empirical L2 norm (i.e.\, t
 he L2 norm with respect to the empirical distribution of the design) in th
 e present regression context. However\, extending contraction rates to the
  "integrated" L2 norm usually requires more work\, and has previously been
  done for instance under sufficient smoothness or boundedness assumptions\
 , which may not necessarily hold. In this work we show that\, for priors b
 ased on truncated random basis expansions and in the random design setting
 \, a high probability two sided inequality between the empirical L2 norm a
 nd the integrated L2 norm holds in appropriate spaces of functions of low 
 frequencies\, under mild assumptions on the underlying basis (which can be
  for instance a Fourier\, wavelet or Laplace eigenfunction basis)\, allowi
 ng us to directly deduce an L2 contraction rate from an empirical L2 one w
 ithout further assumption on the true regression function. Time allowing w
 e will also discuss extensions to Gaussian process priors and semi supervi
 sed learning on graphs.
LOCATION:Seminar Room 1\, Newton Institute
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