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Korteweg-de Vries Institute for Mathematics


My primary field of interest has been non-parametric Bayesian Statistics for a long time. My thesis work concerned the asymptotic behaviour of posterior distributions under model mis-specification. Subsequently my efforts were focussed on the semi-parametric Bernstein-Von Mises theorem which formulates an asymptotic relation at the inferential level between the Bayesian marginal posterior and Frequentist sampling distributions for best-regular estimators for the parameter of interest.

Further efforts have focused on criteria for posterior consistency: whereas Schwartz' theorem requires priors that charge Kullback-Leibler balls, adaptations of Schwartz' theorem exchange model properties for flexibility with regard to the choice of the priors, enabling priors that charge Hellinger balls, uniform-norm 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 well-known 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 edge-sparsity, in which estimation of communities behaves quite differently, corresponding to the three phases of the Erdös-Rényi 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 (non-asymptotic) 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 bid-sides 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 QQ-plots and Kolmogorov-Smirnov 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 heavy-tailedness 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 non-parametrics (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 Radon-Nikodym 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/PhD-candidate-Random-histogram-limits-and-exact-quantum-field-theories/789992902/.)


Book project (work in progress)

B. Kleijn, The frequentist theory of Bayesian statistics
Springer Verlag, New York (expected 2023)


Ph.D. Theses

B. Kleijn, New couplings in N=2 supergravity
Supervision: prof. B. de Wit
Institute for Theoretical Physics, Utrecht University (1998)

B. Kleijn, Bayesian asymptotics under misspecification
Supervision: prof. A. van der Vaart
Mathematics Department, Free Univesity Amsterdam (2003)


Research statement and List of publications

B. Kleijn, Curriculum Vitae (Feb 2022)

B. Kleijn, Research statement (Feb 2022)

B. Kleijn, List of Publications (Feb 2022)