The Statistics for Structures Seminar is an informal seminar focusing on the recent development of statistics on structural
data. The talks concern methodological, theoretical, and applied findings. The focus of the seminar is both on discussing
the existing literature and presenting new results in the topic.

For more information please contact Botond Szabó (b.t.szabo@math.leidenuniv.nl) or Moritz Schauer (schauermr@math.leidenuniv.nl).

##
Mark van de Wiel (VUMC): Empirical Bayes for p>n: use of auxiliary information to improve prediction and variable selection

Leiden, December 1, 2017

Empirical Bayes (EB) is a versatile approach to 'learn from a lot' in two ways: first, from a large number of variables
and second, from a potentially large amount of prior information, e.g. stored in public repositories. I will present applications
of a variety of EB methods to several prediction methods, with examples on ridge regression and Bayesian models with a
spike-and-slab prior. Both (marginal) likelihood and moment-based EB methods will be discussed. I consider a simple empirical
Bayes estimator in a linear model setting to study the relation between the quality of an empirical Bayes estimator and
p.
I argue that EB is particularly useful when the prior contains multiple parameters, modeling a priori information on
variables, termed 'co-data'. This will be illustrated with an application to cancer genomics. Finally, some ideas on how
to include prior structural information in a ridge setting will be shortly discussed.

##
Ioan Gabriel Bucur (RU): Robust Causal Estimation in the Large-Sample Limit without Strict Faithfulness

Leiden, November 3, 2017

Causal effect estimation from observational data is an important and much studied research topic. The instrumental variable
(IV) and local causal discovery (LCD) patterns are canonical examples of settings where a closed-form expression exists
for the causal effect of one variable on another, given the presence of a third variable. Both rely on faithfulness
to infer that the latter only influences the target effect via the cause variable. In reality, it is likely that this
assumption only holds approximately and that there will be at least some form of weak interaction. This brings about
the paradoxical situation that, in the large-sample limit, no predictions are made, as detecting the weak edge invalidates
the setting. We introduce an alternative approach by replacing strict faithfulness with a prior that reflects the existence
of many 'weak' (irrelevant) and 'strong' interactions. We obtain a posterior distribution over the target causal effect
estimator which shows that, in many cases, we can still make good estimates. We demonstrate the approach in an application
on a simple linear-Gaussian setting, using the MultiNest sampling algorithm, and compare it with established techniques
to show our method is robust even when strict faithfulness is violated. This is joint work with Tom Claassen and Tom
Heskes

##
Felix Lucka (CWI & UCL): Sparse Bayesian Inference and Uncertainty Quantification for Inverse
Imaging Problems

Leiden, October 20, 2017

During the last two decades, sparsity has emerged as a key concept to
solve linear and non-linear ill-posed inverse problems, in particular
for severely ill-posed problems and applications with incomplete,
sub-sampled data. At the same time, there is a growing demand to obtain
quantitative instead of just qualitative inverse results together with a
systematic assessment of their uncertainties (Uncertainty
quantification, UQ). Bayesian inference seems like a suitable framework
to combine sparsity and UQ but its application to large-scale inverse
problems resulting from fine discretizations of PDE models leads to
severe computational and conceptional challenges.
In this talk, we will focus on two different Bayesian approaches to
model sparsity as a-priori information: Via convex, but non-smooth prior
energies such as total variation and Besov space priors and via
non-convex but smooth priors arising from hierarchical Bayesian
modeling. To illustrate our findings, we will rely on experimental data
from challenging biomedical imaging applications such as EEG/MEG source
localization and limited-angle CT.
We want to share the experiences, results we obtained and the open
questions we face from our perspective as researchers coming from a
background in biomedical imaging rather than in statistics and hope to
stimulate a fruitful discussion for both sides.

We develop a sparse Bayesian Simultaneous Equations Models (SEMs) approach to network reconstruction which incorporates prior
knowledge. We use an extended version of the horseshoe prior for the regressions parameters where on one hand priors
have been assigned to hyperparameters and on the other hand the hyperparameters have been estimated by Empirical
Bayes (EB). We use fast variational Bayes method for posterior densities approximation and compare its accuracy
with that of MCMC strategy. Compared to their ridge counterpart, both models perform well in sparse situations,
specially, the EB approach seems very promising. In a simulation study we show that accurate prior data can greatly
improve the reconstruction of the network, but need not harm the reconstruction if wrong.

##
Name of Speaker: Title of the Talk

Location

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occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

* Academic year 2016-2017*

##
Nurzhan Nurushev: Oracle uncertainty quantification for biclustering model

Leiden, Friday, May 12, 2017

We study the problem of inference on the unknown parameter in the biclustering model by using the penalization method which
originates from the empirical Bayes approach. The underlying biclustering structure is that the high-dimensional
parameter consists of a few blocks of equal coordinates. The main inference problem is the uncertainty quantification
(i.e., construction of a conference set for the unknown parameter), but on the way we solve the estimation problem
as well. We pursue a novel local approach in that the procedure quality is characterized by a local quantity, the
oracle rate, which is the best trade-off between the approximation error by a biclustering structure and the best
performance for that approximating biclustering structure. The approach is also robust in that the additive errors
in the model are not assumed to be independent (in fact, in general dependent) with some known distribution, but
only satisfying certain mild exchangeable exponential moment conditions. We introduce the excessive bias restriction
(EBR) under which we establish the local (oracle) confidence optimality of the proposed confidence ball. Adaptive
minimax results (for the graphon estimation and posterior contraction problems) follow from our local results.
The results for the stochastic block model follow, with implications for network modeling. [Joint work with E.
Belitser.]

##
Peter Bloem: Network motif detection at scale

Leiden, April 7, 2017

Network motif analysis is a form of pattern mining on graphs. It searches for subgraphs that are unexpectedly frequent with
respect to a null model. To compute the expected frequency of the subgraph, the search for motifs is normally repeated
on as many as 1 000 random graphs sampled from the null model. This is an expensive operation that currently limits
motif analysis to graphs of around 10 000 links. Using the minimum description length principle, we have developed
an approximation that avoids the graph samples and computes motif significance efficiently, allowing us to perform
motif detection on graphs with billions of links, using commodity hardware.

We consider the problem of change-point detection in multivariate time-series. The multivariate distribution of the observations
is supposed to follow a graphical model, whose graph and parameters are affected by abrupt changes throughout time.
We demonstrate that it is possible to perform exact Bayesian inference whenever one considers a simple class of
undirected graphs called spanning trees as possible structures. We are then able to integrate on the graph and
segmentation spaces at the same time by combining classical dynamic programming with algebraic results pertaining
to spanning trees. In particular, we show that quantities such as posterior distributions for change-points or
posterior edge probabilities over time can efficiently be obtained. We illustrate our results on both synthetic
and experimental data arising from biology and neuroscience.

##
Jarno Hartog: Nonparametric Bayesian label prediction on a graph

Leiden, March 17, 2017

I will present an implementation of a nonparametric Bayesian approach to solving binary classification problems on graphs.
I consider a hierarchical Bayesian approach with a randomly scaled Gaussian prior.

##
Guus Regts: Approximation algorithms for graph polynomials and partition functions

The correlation decay method, pioneered by Weitz in 2006, is a method that yields efficient (polynomial time) deterministic
approximation algorithms for computing partition functions of several statistical models. While the method yields
deterministic algorithms it has a probabilistic flavour. In this talk I will sketch how this method works for the
hardcore model, i.e., for counting independent sets in bounded degree graphs. After that I will discuss a different
method pioneerd by Barvinok based on Taylor approximations of the logarithm of the partition function and on the
location of zeros of the partition function. I will explain how this approach can give polynomial time approximation
algorithms for computing several partition functions on bounded degree graphs.

This is based on joint
work with Viresh Patel (UvA)

##
Marco Grzegorczyk: Bayesian inference of semi-mechanistic network models

A topical and challenging problem for statistics and machine learning is to infer the structure of complex systems of interacting
units.In many scientific disciplines such systems are represented by interaction networks described by systems
of differential equations. My presentation is about a novel semi-mechanistic Bayesian modelling approach for infering
the structures and parameters of these interaction networks from data. The inference approach is based on gradient
matching and a non-linear Bayesian regression model. My real.-world applications stem from the topical field of
computational systems biology, where researchers aim to reconstruct the structure of biopathways or regulatory
networks from postgenomic data. My focus is on investigating to which extent certain factors influence the network
reconstruction accuracy. To this end, I compare not only (i) different methods for model selection, including various
Bayesian information criteria and marginal likelihood approximation methods, but also (ii) different ways to approximate
the gradients of the observed time series. Finally, I cross-compare the performance of the new method with a set
of state-of-the art network reconstruction networks, such as Bayesian networks. Within the comparative evaluation
studies I employ ANOVA schemes to disambiguate to which extents confounding factors impact on the network reconstruction
accuracies.

##
Joris Mooij: Automating Causal Discovery and Prediction

The discovery of causal relationships from experimental data and the construction of causal theories to describe phenomena
are fundamental pillars of the scientific method. How to reason effectively with causal models, how to learn these
from data, and how to obtain causal predictions has been traditionally considered to be outside of the realm of
statistics. Therefore, most empirical scientists still perform these tasks informally, without the help of mathematical
tools and algorithms. This traditional informal way of causal inference does not scale, and this is becoming a
serious bottleneck in the analysis of the outcomes of large-scale experiments nowadays. In this talk I will describe
formal causal reasoning methods and algorithms that can help to automate the process of scientific discovery from
data.

## Stephanie van der Pas: Bayesian community detection

In the stochastic block model, nodes in a graph are partitioned into classes ('communities') and it is assumed that the probability
of the presence of an edge between two nodes solely depends on their class labels. We are interested in recovering
the class labels, and employ the Bayesian posterior mode for this purpose. We present results on weak consistency
(where the fraction of misclassified nodes converges to zero) and strong consistency (where the number of misclassified
nodes converges to zero) of the posterior mode , in the 'dense' regime where the probability of an edge occurring
between two nodes remains bounded away from zero, and in the 'sparse' regime where this probability does go to
zero as the number of nodes increases.

*Academic year 2015-2016*

Consider the problem of recovering the support of a sparse signal, that is we are given an unknown s-sparse vector $x$, whose
non-zero elements are $\mu>0$ and we are tasked with recovering the support of $x$. Suppose each coordinate of $x$
is measured independently with additive standard normal noise. In case the support can be any s-sparse set, we know
that $\mu$ needs to scale as $\sqrt{\log n}$ for us to be able to reliably recover the support. However, in some
practical settings the support set has a certain structure. For instance in gene-expression studies the signal support
can be viewed a submatrix of the gene-expression matrix, or when searching for network anomalies the support set
can be viewed as a star in the network graph. In such cases we might be able to recover the support of weaker signals.
This question has been recently addressed by various authors. Now consider a setting where instead of measuring every
coordinate of $x$ the same way, we can collect observations sequentially using our knowledge accumulated from previous
observations. This setup is usually referred to as ``active learning" or ``adaptive sensing". We aim to characterize
the difficulty of accurately recovering structured support sets using adaptive sensing, and also provide near optimal
procedures for support recovery. In particular we are interested in the gains adaptive sensing provides over non-adaptive
sensing in these situations. We consider two measurement models, namely coordinate-wise observations and compressive
sensing. Our results show that adaptive sensing strategies can improve on non-adaptive ones both by better mitigating
the effect of measurement noise, and capitalizing on structural information to a larger extent.
## Moritz Schauer: Working with graphs in Julia

Julia is an emerging technical programming language, which has some properties which make it especially interesting for the
implementation of Bayesian methods. In this talk I give an introduction into the graph-related functionality Julia
provides. After a demonstration how to create and display graphs in Julia using the package Graphs.jl, I show how
to perform Bayesian inference on a "smooth" function defined on a graph in Julia.

## Fengnan Gao: On the Estimation of the Preferential Attachment Network Model

The preferential attachment (PA) network is a popular way of modeling the social networks, the collaboration networks and
etc. The PA network model is an evolving network where new nodes keep coming in. When a new node comes in, it establishes
only one connection with an existing node. The random choice on the existing node is via a multinormial distribution
with probability weights based on a preferential function $f$ on the degrees. f is assumed apriori nondecreasing,
which means the nodes with high degrees are more likely to get new connections, i.e. "the rich get richer". We
proposed an estimator on f, that maps the natural numbers to the positive real line. We show, with techniques from
branching process, our estimator is consistent. If $f$ is affine, meaning $f(k) = k + delta$, it is well known
that such a model leads to a power-law degree distribution. We proposed a maximum likelihood estimator for delta
and establish a central limit result on the MLE of delta.

In statistics we often want to discover (sometimes impose) structure on observed data, and dimension plays a crucial role
in this task. The setting that I will consider in this talk is the following: some high-dimensional data has been
collected but it (potentially) lives in some lower dimensional space (this lower dimension is called the intrinsic
dimension of the dataset); the objective is to estimate the intrinsic dimension of the high-dimensional dataset.

Why would we want to to this? Dimensionality reduction techniques (e.g., PCA, manifold learning) usually rely on
knowledge about intrinsic dimension. Knowledge about dimension is also important to try to avoid the curse of dimensionality.
From a computational perspective, the dimension of a dataset has impact in terms of the amount of space needed
to store data (compressibility). The speed of algorithms is also commonly affected by the dimension of input data.
One can also envision situations where we have access to some regression data, but the design points are unknown
(this occurs, for example, in graphon estimation problems); the dimension of the design space has a large impact
on the rate with which the regression function can be recuperated.

Our approach relies on having access to
a certain graph: each vertex represents an obser- vation, and there is an edge between two vertices if the corresponding
observations are close in some metric. We model this graph as a random connection model (a model from continuum
percolation), and use this to propose estimators for the intrinsic dimension based on the dou- bling property of
the Lebesgue measure. I will give some conditions under which the dimension can be estimated consistently, and
some bounds on the probability of correctly recuperating an integer dimension. I will also show some numerical
results and compare our estimators with some competing approaches from the literature.

This is joint work with
Michel Mandjes.

## Rui Castro (TU/e): Distribution-Free Detection of Structured Anomalies: Permutation and Rank-Based Scans

The scan statistic is by far the most popular method for anomaly detection, being popular in syndromic surveillance, signal
and image processing and target detection based on sensor networks, among other applications. The use of scan statistics
in such settings yields an hypothesis testing procedure, where the null hypothesis corresponds to the absence of
anomalous behavior. If the null distribution is known calibration of such tests is relatively easy, as it can be
done by Monte-Carlo simulation. However, when the null distribution is unknown the story is less straightforward.
We investigate two procedures: (i) calibration by permutation and (ii) a rank-based scan test, which is distribution-free
and less sensitive to outliers. A further advantage of the rank-scan test is that it requires only a one-time calibration
for a given data size making it computationally much more appealing than the permutation-based test. In both cases,
we quantify the performance loss with respect to an oracle scan test that knows the null distribution. We show
that using one of these calibration procedures results in only a very small loss of power in the context of a natural
exponential family. This includes for instance the classical normal location model, popular in signal processing,
and the Poisson model, popular in syndromic surveillance. Numerical experiments further support our theory and
results (joint work with Ery Arias-Castro, Meng Wang (UCSD) and Ervin Tánczos (TU/e)).

## Koen van Oosten: Achieving Optimal Misclassification Proportion in Stochastic Block Model

Community detection is a fundamental statistical problem in network data analysis. Many algorithms have been proposed to
tackle this problem. Most of these algorithms are not guaranteed to achieve the statistical optimality of the problem,
while procedures that achieve information theoretic limits for general parameter spaces are not computationally
tractable. In my talk I present a computationally feasible two-stage method that achieves optimal statistical performance
in misclassification proportion for stochastic block model under weak regularity conditions. This two-stage procedure
consists of a refinement stage motivated by penalized local maximum likelihood estimation. This stage can take
a wide range of weakly consistent community detection procedures as initializer, to which it applies and outputs
a community assignment that achieves optimal misclassification proportion with high probability.

## Wessel van Wieringen: A tale of two networks: two GGMs and their differences

The two-sample problem is addressed from the perspective of Gaussian graphical models (GGMs), in exploratory and confirmatory
fashion. The former amounts to the estimation of a precision matrix for each group. First, this is done group-wise
by means of penalized maximum likelihood with an algebraically proper l2-penalty, for which an analytic expression
of the estimator and its properties are derived. To link the groups the ridge penalty is then augmented with an
fused term, which penalizes the difference between the group precisions. The confirmatory part concentrates on
the situation in which partial correlations are systematically smaller/larger (in an absolute sense) in one of
the groups. Data in both groups again are assumed to follow a GGM but now their partial correlations are proportional,
differing by a multiplier (common to all partial correlations). The multiplier reflects the overall strength of
the conditional dependencies. As before model parameters are estimated by means of penalized maximum likelihood,
now using a ridge-like penalty. A permutation scheme to test for the multiplier differing from zero is proposed.
A re-analysis of publicly available gene expression data on the Hedgehog pathway in normal and cancer prostate
tissue combines both strategies to show its activation in the disease group.

## Pariya Behrouzi: Detecting Epistatic Selection in the Genome of RILs via a latent Gaussian Copula Graphical Model

Recombinant Inbred Lines (RILs) derived from divergent parental lines can display extensive segregation distortion and long-range
linkage disequilibrium (LD) between distant loci on same or different chromosomes. These genomic signatures are
consistent with epistatic selection having acted on entire networks of interacting parental alleles during inbreeding.
The reconstruction of these interaction networks from observations of pair-wise marker-marker correlations or pair-wise
genotype frequency distortions is challenging as multiple testing approaches are under-powered and true long-range
LD is difficult to distinguish from drift, particularly in small RIL panels. Here we develop an efficient method
for reconstructing an underlying network of genomic signatures of high-dimensional epistatic selection from multi-locus
genotype data. The network captures the conditionally dependent short- and long-range LD structure of RIL genomes
and thus reveals aberrant marker-marker associations that are due to epistatic selection rather than gametic linkage.
The network estimation relies on penalized Gaussian copula graphical models, which accounts for the large number
of markers p and the small number of individuals n. We overcome the p >> n problem by using a penalized maximum
likelihood technique that imposes an l1 penalty on the precision matrix of the latent process inside the EM estimation.
A multi-core implementation of our algorithm makes it feasible to estimate the graph in high-dimensions (max markers
approximately 3000). We demonstrate the efficiency of the proposed method on simulated datasets as well as on genotyping
data in A.thaliana and Maize.

* Academic year 2014-2015*

Network analysis is becoming one of the most active research areas in statistics. Significant advances have been made recently
on developing theories, methodologies and algorithms for analyzing networks. However, there has been little fundamental
study on optimal estimation. In this paper, we establish optimal rate of convergence for graphon estimation. Link:
http://arxiv.org/abs/1410.5837

I will review different algorithms for community detection described in: NEWMAN, M. E. J. (2004). Detecting community structure
in networks. Eur. Phys. J. B 38 321-330. NEWMAN, M. E. J. (2006). Finding community structure in networks using
the eigenvectors of matrices. Phys. Rev. E (3) 74 036104, 19. MR2282139 MR2282139 (2007j:82115)

In this short talk I look at the connections between Gauss-Markov process priors on a line and Gaussian Markov Random fields
on a tree via the midpoint displacement procedure. The Markov-property of the prior corresponds to a sparsity constraint
for the prior precision on the tree which allows to solve the Gaussian inverse problem under quasi-linear time
and space constraints using a divide and conquer algorithm. This leads to the notion of computationally desirable
sparsity properties connecting Gramian matrix stemming from an Gaussian inverse problem and the prior precision
matrix.

## Johannes Schmiedt-Hieber: High-dimensional covariance estimation

I am going to talk about the paper: Ravikumar, Pradeep, Martin J. Wainwright, Garvesh Raskutti and Bin Yu High-dimensional
covariance estimation by minimizing l1-penalized log-determinant divergence, EJS, 2011

## Bartek Knapik: Point process modelling for directed interaction networks

I will present the paper: Perry, Patrick O.; Wolfe, Patrick J. Point process modelling for directed interaction networks.
J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 (2013), no. 5, 821-849

## Fengnan Gao: A quick survey in random graph models

We will review several important random graph models, their definitions and important results on them. The models include
Erdős–Rényi model, configuration model and preferential attachment model. We will focus on preferential attachment
model. Most of the presentation is based on Remco van der Hofstad's lecture notes http://www.win.tue.nl/~rhofstad/NotesRGCN.pdf

I will review the paper: Antoine Channarond, Jean-Jacques Daudin, and Stéphane Robin. Classification and estimation in the
Stochastic Blockmodel based on the empirical degrees. Electron. J. Statist. Volume 6 (2012), 2574-2601. Link: http://projecteuclid.org/euclid.ejs/1357913089

## Kolyan Ray: Estimating Sparse Precision Matrix

I will present the paper: Cai, Tony, Weidong Liu and Harrison H. Zhou. Estimating Sparse Precision Matrix: Optimal Rates
of Convergence and Adaptive Estimation Link: http://arxiv.org/abs/1212.2882

Recently we developed a Bayesian structural equation model (SEM) framework with shrinkage priors for undirected network reconstruction.
It was shown that Bayesian SEM in combination with variational Bayes is particularly attractive as it performs
well, is computationally very fast and a flexible framework. A posteriori variable selection is feasible in our
Bayesian SEM and so is the use of shrinkage priors. These shrinkage priors depend on all regression equations
allowing borrowing of information across equations and improve inference when the number of features is large.
An empirical Bayes procedure is used to estimate our hyperparameters. We also showed in simulations that our
approach can outperform popular (sparse) methods. Here, we focus on addressing the problem of incorporating external
data and/or prior information into network inference. In many settings information regarding network connectivity
is often available. It is then natural to take such information into account during network reconstruction. Based
on Bayesian SEM we propose a new model that focuses on the use of external data. It performs better than that
of our Bayesian SEM when the external information is relevant, and as good when it is not.

I will discuss the basic kernel approach to regression in the context of graphs and present an number of methods to construct
such kernels as in section 8.4 of Statistical Analysis of Network Data by Eric D. Kolaczyk.