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PRODID:-//Columba Systems Ltd//NONSGML CPNG/SpringViewer/ICal Output/3.3-
M3//EN
VERSION:2.0
CALSCALE:GREGORIAN
METHOD:PUBLISH
BEGIN:VEVENT
DTSTAMP:20191022T134424Z
DTSTART:20191029T120000Z
DTEND:20191029T130000Z
SUMMARY:Natalia Bochkina - Bayesian Inverse Problems with Heterogeneous V
ariance
UID:{http://www.columbasystems.com/customers/uom/gpp/eventid/}ojm-k0mg1r7
c-5qzspc
DESCRIPTION:***Please note unusual location***\n\nJoin us for this resear
ch seminar\, part of the SQUIDS (Statistics\, quantification of uncertai
nty\, inverse problems and data science) seminar series.\n\nAbstract: We
consider inverse problems in Hilbert spaces contaminated by Gaussian no
ise\, and use a Bayesian approach to find its regularised smooth solutio
n. We consider the so called conjugate diagonal setting where the covari
ance operators of the noise and of the prior are diagonalisable in the o
rthogonal bases associated with the forward operator of the inverse prob
lem. Firstly\, we derive the minimax rate of convergence in such problem
s with known covariance operator of the noise\, showing that in the case
of heterogeneous variance the ill posed inverse problem can become self
regularised in some cases when the eigenvalues of the variance operator
decay to zero\, achieving parametric rate of convergence\; as far as we
are aware\, this is a striking novel result that have not been observed
before in nonparametric problems. Secondly\, we give a general expressi
on of the rate of contraction of the posterior distribution in case of k
nown noise covariance operator in case the noise level is small\, for a
given prior distribution. We also investigate when this contraction rate
coincides with the optimal rate in the minimax sense which is typically
used as a benchmark for studying the posterior contraction rates. We ap
ply our results to known variance operators with polynomially decreasing
or increasing eigenvalues as an example. We also discuss when the plug
in estimator of the eigenvalues of the covariance operator of the noise
does not affect the rate of the contraction of the posterior distributio
n of the signal. We show that plugging in the maximum marginal likelihoo
d estimator of the prior scaling parameter leads to the optimal posterio
r contraction rate\, adaptively. Effect of the choice of the prior param
eters on the contraction in such models is illustrated on simulated data
with Volterra operator. This is joint work with Jenovah Rodrigues.
STATUS:TENTATIVE
TRANSP:TRANSPARENT
CLASS:PUBLIC
LOCATION:G.108\, Alan Turing Building\, Manchester
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