My primary field of interest has been nonparametric
Bayesian Statistics for a long time. My thesis
work concerned the asymptotic behaviour of posterior
distributions under model misspecification.
Subsequently my efforts were focussed on the semiparametric
BernsteinVon Mises theorem which formulates an
asymptotic relation at the inferential level between the
Bayesian marginal posterior and Frequentist sampling
distributions for bestregular estimators for the
parameter of interest.
Further efforts have focused on criteria for
posterior consistency: whereas Schwartz'
theorem requires priors that charge KullbackLeibler
balls, adaptations of Schwartz' theorem exchange model
properties for flexibility with regard to the choice of
the priors, enabling priors that charge Hellinger balls,
uniformnorm balls and other types of neighbourhoods. To
enable the central theorems of posterior convergence to
situations in which the data is not i.i.d., we have to go
beyond Schwartz's theorem. A fully general
version of Schwartz's frequentist posterior consistency
theorem has been found in The frequentist
validity of Bayesian limits, as well as frequentist
theorems for posterior convergence at a rate,
for consistent testing with posterior odds and
model selection, and for conversion of
credible sets to confidence sets.
Those posterior convergence theorems have been applied in
a wellknown problem from network science,
in Recovery, detection and confidence sets of
communities in a sparse stochastic block model (with
J. van Waaij). Most striking is the occurrence of three
distinct regimes of edgesparsity, in
which estimation of communities behaves quite differently,
corresponding to the three phases of the ErdösRényi
random graph.
The analysis of the stochastic block model has also led
to a novel approach to frequentist uncertainty
quantification with finite amounts of data: when a
posterior distribution is available, credible sets are
easily found. If the posterior satisfies a concentration
inequality, credible sets can be enlarged to form
confidence sets, with full control over the confidence
level in terms of the credible level and enlargement
radius. In the paper Confidence sets in a sparse
stochastic block model with two communities of unknown
sizes, J. van Waaij and I have applied this new
(nonasymptotic) methodology to the stochastic block
model, to find confidence sets for the community
structure.
In finance, the paper Clearing prices in call
auctions (with M. Derksen) developes a stochastic
model for the pricing and trade in assets based on an equilibrium
equation between ask and bidsides of the
market. Applied to closing auctions for Eurostoxx 50
shares, the model predicts the closing price through a
distribution, which matches the distribution of the real
closing price to very good degree, as assessed through
QQplots and KolmogorovSmirnov statistics. Other
properties include an efficient limit when liquidity is
large and the emergence of resistance levels on the price
axis. The model is also used for the analysis and
explanation of the heavytailedness of auction return
distributions, in the paper Heavy tailed distributions
in closing auctions.
Most recently, my research has concentrated on existence
theorems for random inverse limit measures,
like the Dirichlet and Polya tree families of
prior/posterior measures of Bayesian nonparametrics (see
presentations in Bremen
and the KdV
Institute). The goal is to lend mathematical rigor
to the definition of scalar quantum field
theories in d Euclidean dimensions. The path
integral of the free theory can be viewed as an
inverse limit probability measure, and
martingale limits to prove the existence and
renormalization of RadonNikodym densities,
whose logarithms describe effective interaction
terms in the Euclidean action. (A PhD project
in this direction is to start in Sep 2024, see https://vacatures.uva.nl/UvA/job/PhDcandidateRandomhistogramlimitsandexactquantumfieldtheories/789992902/.)
