The statify team focuses on statistics.
Statistics can be defined as a science of variation where the main question is how to acquire knowledge in the face of variation.
In the past, statistics were seen as an opportunity to play in various backyards. Today, the statistician sees his own backyard invaded by data scientists, machine learners and other computer scientists of all kinds. Everyone wants to do data analysis and some (but not all) do it very well.
Generally, data analysis algorithms and associated network architectures are empirically validated using domain-specific datasets and data challenges. While winning such challenges is certainly rewarding, statistical validation lies on more fundamentally grounded bases and raises interesting theoretical, algorithmic and practical insights.
Statistical questions can be converted to probability questions by the use of probability models. Once certain assumptions about the mechanisms generating the data are made, statistical questions can be answered using probability theory. However, the proper formulation and checking of these probability models is just as important, or even more important, than the subsequent analysis of the problem using these models. The first question is then how to formulate and evaluate probabilistic models for the problem at hand. The second question is how to obtain answers after a certain model has been assumed. This latter task can be more a matter of applied probability theory, and in practice, contains optimization and numerical analysis.

The statify team aims at bringing strengths, at a time when the number of solicitations received by statisticians increases considerably because of the successive waves of big data, data science and deep learning. The difficulty is to back up our approaches with reliable mathematics while what we have is often only empirical observations that we are not able to explain. Guiding data analysis with statistical justification is a challenge in itself.
statify has the ambition to play a role in this task and to provide answers to questions about the appropriate usage of statistics.

Often statistical assumptions do not hold. Under what conditions then can we use statistical methods to obtain reliable knowledge? These conditions are rarely the natural state of complex systems. The central motivation of statify is to establish the conditions under which statistical assumptions and associated inference procedures approximately hold and become reliable.

However, as George Box said "Statisticians and artists both suffer from being too easily in love with their models". To moderate this risk, we choose to develop, in the team, expertise from different statistical domains to offer different solutions to attack a variety of problems. This is possible because these domains share the same mathematical food chain, from probability and measure theory to statistical modeling, inference and data analysis.

Our goal is to exploit methodological resources from statistics and machine learning to develop models that handle variability and that scale to high dimensional data while maintaining our ability to assess their correctness, typically the uncertainty associated with the provided solutions. To reach this goal, the team offers a unique range of expertise in statistics, combining probabilistic graphical models and mixture models to analyze structured data, Bayesian analysis to model knowledge and regularize ill-posed problems, non-parametric statistics, risk modeling and extreme value theory to face the lack, or impossibility, of precise modeling information and data. In the team, this expertise is organized to target five key challenges:

The first two challenges address sources of complexity coming from data, namely, the fact that observations can be:
1) high dimensional, collected from multiple sensors in varying conditions i.e. multimodal and heterogeneous and 2)
inter-dependent with a known structure between variables or with unknown interactions to be discovered.
The other three challenges focus on providing reliable and interpretable models:
3) making the Bayesian approach scalable to handle large and complex data;
4) quantifying the information processing properties of machine learning methods and 5) allowing to draw reliable conclusions from datasets that are too small or not large enough to be used for training machine/deep learning methods.

These challenges rely on our four research axes:

In terms of applied work, we will target high-impact applications in neuroimaging, environmental and earth sciences.

In a first approach, we consider statistical parametric models,

These models are interesting in that they may point out hidden
variables responsible for most of the observed variability and so
that the observed variables are conditionally independent.
Their estimation is often difficult due to the missing data. The
Expectation-Maximization (EM) algorithm is a general and now
standard approach to maximization of the likelihood in missing
data problems. It provides parameter estimation but also values
for missing data.

Mixture models correspond to independent

Graphical modelling provides a diagrammatic representation of the dependency structure of a joint probability distribution, in the form of a network or graph depicting the local relations among variables. The graph can have directed or undirected links or edges between the nodes, which represent the individual variables. Associated with the graph are various Markov properties that specify how the graph encodes conditional independence assumptions.

It is the conditional independence assumptions that give graphical models their fundamental modular structure, enabling computation of globally interesting quantities from local specifications. In this way graphical models form an essential basis for our methodologies based on structures.

The graphs can be either
directed, e.g. Bayesian Networks, or undirected, e.g. Markov Random Fields.
The specificity of Markovian models is that the dependencies
between the nodes are limited to the nearest neighbor nodes. The
neighborhood definition can vary and be adapted to the problem of
interest. When parts of the variables (nodes) are not observed or missing,
we
refer to these models as Hidden Markov Models (HMM).
Hidden Markov chains or hidden Markov fields correspond to cases where the

Hidden Markov models are very useful in modelling spatial dependencies but these dependencies and the possible existence of hidden variables are also responsible for a typically large amount of computation. It follows that the statistical analysis may not be straightforward. Typical issues are related to the neighborhood structure to be chosen when not dictated by the context and the possible high dimensionality of the observations. This also requires a good understanding of the role of each parameter and methods to tune them depending on the goal in mind. Regarding estimation algorithms, they correspond to an energy minimization problem which is NP-hard and usually performed through approximation. We focus on a certain type of methods based on variational approximations and propose effective algorithms which show good performance in practice and for which we also study theoretical properties. We also propose some tools for model selection. Eventually we investigate ways to extend the standard Hidden Markov Field model to increase its modelling power.

We also consider methods which do not assume a parametric model.
The approaches are non-parametric in the sense that they do not
require the assumption of a prior model on the unknown quantities.
This property is important since, for image applications for
instance, it is very difficult to introduce sufficiently general
parametric models because of the wide variety of image contents.
Projection methods are then a way to decompose the unknown
quantity on a set of functions (e.g. wavelets). Kernel
methods which rely on smoothing the data using a set of kernels
(usually probability distributions) are other examples.
Relationships exist between these methods and learning techniques
using Support Vector Machine (SVM) as this appears in the context
of level-sets estimation (see section 3.3.2). Such
non-parametric methods have become the cornerstone when dealing
with functional data 87. This is the case, for
instance, when observations are curves. They enable us to model the
data without a discretization step. More generally, these
techniques are of great use for dimension reduction purposes
(section 3.3.3). They enable reduction of the dimension of the
functional or multivariate data without assumptions on the
observations distribution. Semi-parametric methods refer to
methods that include both parametric and non-parametric aspects.
Examples include the Sliced Inverse Regression (SIR) method 89
which combines non-parametric regression
techniques
with parametric dimension reduction aspects. This is also the case
in extreme value analysis86, which is based
on the modelling of distribution tails (see section 3.3.1).
It differs from traditional statistics which focuses on the central
part of distributions, i.e. on the most probable events.
Extreme value theory shows that distribution tails can be
modelled by both a functional part and a real parameter, the
extreme value index.

Extreme value theory is a branch of statistics dealing with the extreme
deviations from the bulk of probability distributions.
More specifically, it focuses on the limiting distributions for the
minimum or the maximum of a large collection of random observations
from the same arbitrary distribution.
Let i.e.

To estimate such quantiles therefore requires dedicated
methods to
extrapolate information beyond the observed values of

where both the extreme-value index i.e. such that

for all

More generally, the problems that we address are part of the risk assessment theory. For instance, in reliability, the distributions of interest are included in a semi-parametric family whose tails are decreasing exponentially fast. These so-called Weibull-tail distributions 8 are defined by their survival distribution function:

Gaussian, gamma, exponential and Weibull distributions, among others,
are included in this family. An important part of our work consists
in establishing links between models (2) and (4)
in order to propose new estimation methods.
We also consider the case where the observations were recorded with a covariate information 9. In this case, the
extreme-value index and the

Level sets estimation is a
recurrent problem in statistics which is linked to outlier
detection. In biology, one is interested in estimating reference
curves, that is to say curves which bound

Our work on high dimensional data requires that we face the curse of dimensionality phenomenon. Indeed, the modelling of high dimensional data requires complex models and thus the estimation of high number of parameters compared to the sample size. In this framework, dimension reduction methods aim at replacing the original variables by a small number of linear combinations with as small as a possible loss of information. Principal Component Analysis (PCA) is the most widely used method to reduce dimension in data. However, standard linear PCA can be quite inefficient on image data where even simple image distortions can lead to highly non-linear data. Two directions are investigated. First, non-linear PCAs can be proposed, leading to semi-parametric dimension reduction methods 88. Another field of investigation is to take into account the application goal in the dimension reduction step. One of our approaches is therefore to develop new Gaussian models of high dimensional data for parametric inference 1. Such models can then be used in a Mixtures or Markov framework for classification purposes. Another approach consists in combining dimension reduction, regularization techniques, and regression techniques to improve the Sliced Inverse Regression method 89.

As regards applications, several areas of image analysis can be
covered using the tools developed in the team. More specifically,
in collaboration with team perception, we address various issues
in computer vision involving Bayesian modelling and probabilistic
clustering techniques. Other applications in medical imaging are
natural. We work more specifically on MRI and functional MRI data, in collaboration
with the Grenoble Institute of Neuroscience (GIN). We also consider other
statistical 2D fields coming from other domains such as remote
sensing, in collaboration with the Institut de Planétologie et d'Astrophysique de
Grenoble (IPAG) and the Centre National d'Etudes Spatiales (CNES). In this context, we worked
on hyperspectral and/or multitemporal images. In the context of the "pole
de competivité" project I-VP, we worked of images of PC Boards.

A third domain of applications concerns biology and medicine. We considered the use of mixture models to identify biomakers. We also investigated statistical tools for the analysis of fluorescence signals in molecular biology. Applications in neurosciences are also considered. In the environmental domain, we considered the modelling of high-impact weather events and the use of hyperspectral data as a new tool for quantitative ecology.

The footprint of our research activities has not been assessed yet. Most of the team members have validated the "charte d'éco-resposnsabilité" written by a working group from Laboratoire Jean Kuntzmann, which should have practical implications in the near future.

A lot of our developments are motivated by and target applications in medicine and environmental sciences. As such they have a social impact with a better handling and treatment of patients, in particular with brain diseases or disorders. On the environmental side, our work has an impact on geoscience-related decision making with e.g. extreme events risk analysis, planetary science studies and tools to assess biodiversity markers. However, how to truly measure and report this impact in practice is another question we have not really addressed yet.

A recent work has been accepted in Journal of Machine Learning Research, which is one of the top journals in the domain 13.

M. Vladimirova received the best poster award for her work
at the BAyesian Young Statisticians Meeting in Université de Montréal, Canada, in June 22-23, 2022.

H. Lbath received the student award for her work on Clusted-based inter-regional correlation estimation presented at 2022 IMS International Conference on Statistics and Data Science (ICSDS) December 13-16, 2022, Florence, Italy, 54.

Joint work with: Hien Nguyen, University of Queensland, Brisbane Australia, Gersende Fort, IMT Toulouse and Olivier Cappé ENS Paris.

Modern statistical and machine learning settings often involve high data volume and data streaming, which require the development of online estimation algorithms. The online Expectation-Maximization (EM) algorithm extends the popular EM algorithm to this setting, via a stochastic approximation approach. We show that an online version of the Minorization-Maximization (MM) algorithm, which includes the online EM algorithm as a special case, can also be constructed in a similar manner. We demonstrate our approach via an application to the logistic regression problem and compare it to existing methods 36.

Joint work with: Hien Nguyen, University of Queensland, Brisbane Australia.

Due to the changing nature of data, online and mini-batch variants of EM and EM-like algorithms have become increasingly popular. The consistency of the estimator sequences that are produced by these EM variants often rely on an assumption regarding the continuous differentiability of a parameter update function. In many cases, the parameter update function is often not in closed form and may only be defined implicitly, which makes the verification of the continuous differentiability property difficult. We demonstrate how a global implicit function theorem can be used to verify such properties in the cases of finite mixtures of distributions in the exponential family and more generally when the component specific distribution admits a data augmentation scheme in the exponential family. We demonstrate the use of such a theorem in the case of mixtures of beta distributions, gamma distributions, fully-visible Boltzmann machines and Student distributions. Via numerical simulations, we provide empirical evidence towards the consistency of the online EM algorithm parameter estimates in such cases 27.

Joint work with: Hien Nguyen, University of Queensland, Brisbane Australia.

We study the large sample behaviors of approximate Bayesian computation (ABC) posterior measures in situations when the data generating process is dependent on non-identifiable parameters. In particular, we establish the concentration of posterior measures on sets of arbitrarily small size that contain the equivalence set of the data generative parameter, when the sample size tends to infinity. Our theory also makes weak assumptions regarding the measurement of discrepancy between the data set and simulations, and in particular, does not require the use of summary statistics and is applicable to a broad class of kernelized ABC algorithms. We provide useful illustrations and demonstrations of our theory in practice, and offer a comprehensive assessment of the nature in which our findings complement other results in the literature.

Joint work with: Hien Nguyen, University of Queensland, Australia and Michel Dojat Grenoble Institute of Neurosciences.

A popular way to approach clustering tasks is via a parametric mixture model. The vast majority of the work on such mixtures has been based on Gaussian mixture models. However, in some applications the tails of Gaussian distributions are shorter than appropriate or parameter estimations are affected by atypical observations (outliers). To address this issue, mixtures of so-called multiple scale Student distributions have been proposed and used for clustering. In contrast to the Gaussian case, no closed-form solution exists for such mixtures but tractability is maintained via the use of the expectation-maximisation (EM) algorithm. However, such mixtures require more parameters than the standard Student or Gaussian mixtures and the EM algorithm used to estimate the mixture parameters involves more complex numerical optimizations. Consequently, when the number of samples to be clustered becomes large, applying EM on the whole data set (Batch EM) may become costly both in terms of time and memory requirements. A natural approach to bypass this issue is to consider an online version of the algorithm, that can incorporate the samples incrementally or in mini batches. In this work, we proposed to design a tractable online EM for mixtures of multiple scale Student distributions in order to use it then to detect subtle brain anomalies from MR brain scans for patients suffering from Parkinson disease. The application to Parkinson disease will be carried out jointly with Grenoble Institute of Neuroscience.

Joint work with: Faicel Chamroukhi, University of Caen and Hien Nguyen, University of Queensland, Brisbane, Australia.

Mixture of experts (MoE) are a popular class of statistical and machine learning models that have gained attention over the years due to their flexibility and efficiency. In this work, we consider Gaussian-gated localized MoE (GLoME) and block-diagonal covariance localized MoE (BLoME) regression models to present nonlinear relationships in heterogeneous data with potential hidden graph-structured interactions between high-dimensional predictors. These models pose difficult statistical estimation and model selection questions, both from a computational and theoretical perspective. This paper 84 is devoted to the study of the problem of model selection among a collection of GLoME or BLoME models characterized by the number of mixture components, the complexity of Gaussian mean experts, and the hidden block-diagonal structures of the covariance matrices, in a penalized maximum likelihood estimation framework. In particular, we establish non-asymptotic risk bounds that take the form of weak oracle inequalities, provided that lower bounds for the penalties hold. The good empirical behavior of our models is then demonstrated on synthetic and real datasets.

Joint work with: Guillaume Kon Kam King (INRAE).

Bayesian nonparametric mixture models are common for modeling complex data. While these models are well-suited for density estimation, their application for clustering has some limitations. Recent results proved posterior inconsistency of the number of clusters when the true number of clusters is finite for the Dirichlet process and Pitman–Yor process mixture models. In 68, we extend these results to additional Bayesian nonparametric priors such as Gibbs-type processes and finite-dimensional representations thereof. The latter include the Dirichlet multinomial process, the recently proposed Pitman–Yor, and normalized generalized gamma multinomial processes. We show that mixture models based on these processes are also inconsistent in the number of clusters and discuss possible solutions. Notably, we show that a post-processing algorithm introduced for the Dirichlet process can be extended to more general models and provides a consistent method to estimate the number of components.

Joint work with: Mathieu Fauvel, INRAE

Recent satellite missions have led to a huge amount of earth observation data, most of them being freely available. In such a context, satellite image time series have been used to study land use and land cover information. However, optical time series, like Sentinel-2 or Landsat ones, are provided with an irregular time sampling for different spatial locations, and images may contain clouds and shadows. Thus, pre-processing techniques are usually required to properly classify such data. The proposed approach is able to deal with irregular temporal sampling and missing data directly in the classification process. It is based on Gaussian processes and allows to perform jointly the classification of the pixel labels as well as the reconstruction of the pixel time series. The method complexity scales linearly with the number of pixels, making it amenable in large scale scenarios. Experimental classification and reconstruction results show that the method does not compete yet with state of the art classifiers but yields reconstructions that are robust with respect to the presence of undetected clouds or shadows and does not require any temporal preprocessing 16.

Joint work with: M. Allouche and E. Gobet (CMAP, Ecole Poytechnique)

Feedforward neural networks based on Rectified linear units (ReLU) cannot efficiently approximate quantile functions which are not bounded, especially in the case of heavy-tailed distributions. We thus propose a new parametrization for the generator of a Generative adversarial network (GAN) adapted to this framework, basing on extreme-value theory. We provide an analysis of the uniform error between the extreme quantile and its GAN approximation. It appears that the rate of convergence of the error is mainly driven by the second-order parameter of the data distribution. The above results are illustrated on simulated data and real financial data 13.

A similar investigation has been conducted to simulate fractional Brownian motion with ReLU neural networks 12.

In 69, we propose new parametrizations for neural networks in order to estimate extreme quantiles in both non-conditional and conditional heavy-tailed settings. All proposed neural network estimators feature a bias correction based on an extension of the usual second-order condition to an arbitrary order. The convergence rate of the uniform error between extreme log-quantiles and their neural network approximation is established. The finite sample performances of the non-conditional neural network estimator are compared to other bias-reduced extreme-value competitors on simulated data. It is shown that our method outperforms them in difficult heavy-tailed situations where other estimators almost all fail. Finally, the conditional neural network estimators are implemented to investigate the behaviour of extreme rainfalls as functions of their geographical location in the southern part of France. The results are submitted for publication.

Joint work with: J. El-Methni (Univ. Paris Cité), G. Stupfler (Univ. Angers) and A. Usseglio-Carleve (Univ. Avignon).

One of the most popular risk measures is the Value-at-Risk (VaR) introduced in the 1990's.
In statistical terms,
the VaR at level

The Value-at-Risk however suffers from several weaknesses. First, it provides us only with a pointwise information:
VaR(

Risk measures of a financial position are, from an empirical point of view, mainly based on quantiles. Replacing quantiles with their least squares analogues, called expectiles, has recently received increasing attention 3. The novel expectile-based risk measures satisfy all coherence requirements. Currently available estimators of extreme expectiles are typically biased and hence may show poor finite-sample performance even in fairly large samples. In 22, we focus on the construction of bias-reduced extreme expectile estimators for heavy-tailed distributions. The rationale for our construction hinges on a careful investigation of the asymptotic proportionality relationship between extreme expectiles and their quantile counterparts, as well as of the extrapolation formula motivated by the heavy-tailed context. We accurately quantify and estimate the bias incurred by the use of these relationships when constructing extreme expectile estimators. This motivates the introduction of a class of bias-reduced estimators whose asymptotic properties are rigorously shown, and whose finite-sample properties are assessed on a simulation study and three samples of real data from economics, insurance and finance.

G. Stupfler (Univ. Angers) and A. Usseglio-Carleve (Univ. Avignon).

As explained in Paragraph 8.2.2, expectiles have recently started to be considered as serious candidates to become standard tools in actuarial and financial risk management. However, expectiles and their sample versions do not benefit from a simple explicit form, making their analysis significantly harder than that of quantiles and order statistics. This difficulty is compounded when one wishes to integrate auxiliary information about the phenomenon of interest through a finite-dimensional covariate, in which case the problem becomes the estimation of conditional expectiles.

We exploit the fact that the expectiles of a distribution

Joint work with: H. Nguyen, Universityof Queensland, Brisbane, Australia and A. Usseglio-Carleve (Univ. Avignon).

Expectiles form a family of risk measures that have recently gained interest over the more common value-at-risk or return levels, primarily due to their capability to be determined by the probabilities of tail values and magnitudes of realisations at once. However, a prevalent and ongoing challenge of expectile inference is the problem of uncertainty quantification, which is especially critical in sensitive applications, such as in medical, environmental or engineering tasks. In 14, we address this issue by developing a novel distribution, termed the multivariate expectilebased distribution (MED), that possesses an expectile as a closed-form parameter. Desirable properties of the distribution, such as log-concavity, make it an excellent fitting distribution in multivariate applications. Maximum likelihood estimation and Bayesian inference algorithms are described. Simulated examples and applications to expectile and mode estimation illustrate the usefulness of the MED for uncertainty quantification.

Joint work with: G. Enjolras (CERAG).

In the context of the PhD thesis of Meryem Bousebata, we propose a new approach, called Extreme-PLS, for dimension reduction in regression and adapted to distribution tails. The objective is to find linear combinations of predictors that best explain the extreme values of the response variable in a non-linear inverse regression model. The asymptotic normality of the Extreme-PLS estimator is established in the single-index framework and under mild assumptions. The performance of the method is assessed on simulated data. A statistical analysis of French farm income data, considering extreme cereal yields, is provided as an illustration 15.

Joint work with: A. Dutfoy (EDF R&D).

Combining extreme value theory with Bayesian methods offers several advantages, such as a quantification of uncertainty on parameter estimation or the ability to study irregular models that cannot be handled by frequentist statistics. However, it comes with many options that are left to the user concerning model building, computational algorithms, and even inference itself. Among them, the parameterization of the model induces a geometry that can alter the efficiency of computational algorithms, in addition to making calculations involved. In 75, we focus on the Poisson process characterization of extremes and outline two key benefits of an orthogonal parameterization addressing both issues. First, several diagnostics show that Markov chain Monte Carlo convergence is improved compared with the original parameterization. Second, orthogonalization also helps deriving Jeffreys and penalized complexity priors, and establishing posterior propriety. The analysis is supported by simulations, and our framework is then applied to extreme level estimation on river flow data. The results are submitted for publication.

Joint work with: A. Dutfoy (EDF R&D).

Diagnosing convergence of Markov chain Monte Carlo (MCMC) is crucial in Bayesian analysis.
Among the most popular methods, the potential scale reduction factor (commonly named

Joint work with: H. Lorenzo and J. Saracco (Inria Bordeaux Sud-Ouest).

Since its introduction in the early 90's, the Sliced Inverse Regression (SIR) methodology has evolved adapting to increasingly complex data sets in contexts combining linear dimension reduction with non linear regression. The assumption of dependence of the response variable with respect to only a few linear combinations of the covariates makes it appealing for many computational and real data application aspects. In 19, we propose an overview of the most active research directions in SIR modeling from multivariate regression models to regularization and variable selection.

Joint work with: Wilfried Thuiller, LECA - Laboratoire d'Ecologie Alpine.

We investigate modelling species distributions over space and time which is one of the major research topics in both ecology and conservation biology. Joint Species Distribution models (JSDMs) have recently been introduced as a tool to better model community data, by inferring a residual covariance matrix between species, after accounting for species' response to the environment. However, these models are computationally demanding, even when latent factors, a common tool for dimension reduction, are used. To address this issue, previous research proposed to use a Dirichlet process, a Bayesian nonparametric prior, to further reduce model dimension by clustering species in the residual covariance matrix. Here, we built on this approach to include a prior knowledge on the potential number of clusters, and instead used a Pitman-Yor process to address some critical limitations of the Dirichlet process. We therefore propose a framework that includes prior knowledge in the residual covariance matrix, providing a tool to analyze clusters of species that share the same residual associations with respect to other species. We applied our methodology to a case study of plant communities in a protected area of the French Alps (the Bauges Regional Park), and demonstrated that our extensions improve dimension reduction and reveal additional information from the residual covariance matrix, notably showing how the estimated clusters are compatible with plant traits, endorsing their importance in shaping communities. A book chapter describing latent factor models as a tool for dimension reduction in joint species distribution models is aslo available.

Joint work with: Thibaud Rahier from Criteo and Sylvain Marié from Schneider.

Learning the structure of Bayesian networks from data is a NP-Hard problem that involves optimization over a super-exponential sized space. Still, in many real-life datasets a number of the arcs contained in the final structure correspond to strongly related pairs of variables and can be identified efficiently with information-theoretic metrics. In this work, we propose a meta-algorithm to accelerate any existing Bayesian network structure learning method. It contains an additional arc pre-screening step allowing to narrow the structure learning task down to a subset of the original variables, thus reducing the overall problem size. We conduct extensive experiments on both public benchmarks and private industrial datasets, showing that this approach enables a significant decrease in computational time and graph complexity for little to no decrease in performance score 37.

Joint work with: Senan Doyle, Alan Tucholka from Pixyl and Michel Dojat from Grenoble Institute of Neurosciences.

The full acceptance of Deep Learning (DL) models in the clinical field is rather low with respect to the quantity of high-performing solutions reported in the literature. Particularly, end users are reluctant to rely on the rough predictions of DL models. Uncertainty quantification methods have been proposed in the literature as a potential response to reduce the rough decision provided by the DL black box and thus increase the interpretability and the acceptability of the result by the final user. In this review, we propose an overview of the existing methods to quantify uncertainty associated to DL predictions. We focus on applications to medical image analysis, which present specific challenges due to the high dimensionality of images and their quality variability, as well as constraints associated to real-life clinical routine. We then discuss the evaluation protocols to validate the relevance of uncertainty estimates. Finally, we highlight the open challenges of uncertainty quantification in the medical field 72.

Joint work with: Senan Doyle, Alan Tucholka from Pixyl and Michel Dojat from Grenoble Institute of Neurosciences.

Deep neural networks have become the gold-standard approach for the automated segmentation of 3D medical images. Their full acceptance by clinicians remains however hampered by the lack of intelligible uncertainty assessment of the provided results. Most approaches to quantify their uncertainty, such as the popular Monte Carlo dropout, restrict to some measure of uncertainty in prediction at the voxel level. In addition not to be clearly related to genuine medical uncertainty, this is not clinically satisfying as most objects of interest (e.g. brain lesions) are made of groups of voxels whose overall relevance may not simply reduce to the sum or mean of their individual uncertainties. In this work, we propose to go beyond voxel-wise assessment using an innovative Graph Neural Network approach, trained from the outputs of a Monte Carlo dropout model. This network allows the fusion of three estimators of voxel uncertainty: entropy, variance, and model's confidence; and can be applied to any lesion, regardless of its shape or size. We demonstrate the superiority of our approach for uncertainty estimate on a task of Multiple Sclerosis lesions segmentation 35.

Joint work with: Senan Doyle, Alan Tucholka from Pixyl and Michel Dojat from Grenoble Institute of Neurosciences.

Deep Learning (DL) techniques have become the gold standard for biomedical image segmentation. Although extensively used, they tend to be considered as black-boxes, preventing their full acceptance in clinical routine. This opacity is partly due to the inability of neural networks to express the uncertainty in their predictions. In recent years, tremendous work has been carried out to alleviate this limitation and develop models that know when they don't know 1. In this work, we propose an in-depth evaluation of 3 state-of-the-art approaches to quantify uncertainty attached to DL predictions : Monte- Carlo Dropout, Deep Ensemble, and Heteroscedastic network. We performed a multi-scale analysis of these techniques by evaluating uncertainty estimates at the voxel, lesion and image levels. We illustrate this comparison on an automatic segmentation task to detect White-Matter Hyperintensities (WMH) from T2-weighted FLAIR MRI sequences of multiple sclerosis patients 56.

Joint work with: Senan Doyle, Alan Tucholka from Pixyl and Michel Dojat from Grenoble Institute of Neurosciences.

Deep Learning models are easily disturbed by variations in the input images that were not seen during training, resulting in unpredictable behaviours. Such Out-of-Distribution (OOD) images represent a significant challenge in the context of medical image analysis, where the range of possible abnormalities is extremely wide, including artifacts, unseen pathologies, or different imaging protocols. In this work, we evaluate various uncertainty frameworks to detect OOD inputs in the context of Multiple Sclerosis lesions segmentation. By implementing a comprehensive evaluation scheme including 14 sources of OOD of various nature and strength, we show that methods relying on the predictive uncertainty of binary segmentation models often fails in detecting outlying inputs. On the contrary, learning to segment anatomical labels alongside lesions highly improves the ability to detect OOD inputs.

Joint work with: Michel Dojat from Grenoble Institute of Neurosciences, Carole Lartizien, Nicolas Pinon, Robin Trombetta from Creatis.

Neural network-based anomaly detection remains challenging in clinical applications with little or no supervised information and subtle anomalies such as hardly visible brain lesions. Among unsupervised methods, patch-based auto-encoders with their efficient representation power provided by their latent space, have shown good results for visible lesion detection. However, the commonly used reconstruction error criterion may limit their performance when facing less obvious lesions. In this work, we design two alternative detection criteria. They are derived from multivariate analysis and can more directly capture information from latent space representations.
Their performance compares favorably with two additional supervised learning methods, on a difficult de novo Parkinson Disease (PD) classification task.

Joint work with: Hien Nguyen from University of Queensland Brisbane Australia and Fatoumata Dama, Laboratoire des Sciences du Numérique de Nantes, France

Hidden Markov random fields (HMRFs) have been widely used in image segmentation and more generally, for clustering of data indexed by graphs. Dependent hidden variables (states) represent the cluster identities and determine their interpretations. Dependencies between state variables are induced by the notion of neighborhood in the graph. A difficult and crucial problem in HMRFs is the identification of the number of possible states

We proposed an application of the discrete-data model to a new risk mapping model for traffic accidents in the region of Victoria, Australia 17. The partition into regions using labels yielded by HMRFs was interpreted using covariates, which showed a good discrimination with regard to labels.

As a perspective, Bayesian nonparametric models for hidden Markov random fields could be extended to non-Poissonian models (particularly to account for zero-inflated and over-/under-dispersed cases of application) and to regression models.Further perspectives of this work include the improvement of the convergence in the VBEM algorithm, since however the KL divergence between the posterior distribution and its approximation converges, the sequence of optimizing parameters is shown to diverge in our current approach.

Joint work with: Anne Guérin-Dugué (GIPSA-lab)
and Benoit Lemaire (Laboratoire de Psychologie et Neurocognition)

In the last years, GIPSA-lab has developed computational models of information search in web-like materials, using data from both eye-tracking and electroencephalograms (EEGs). These data were obtained from experiments, in which subjects had to decide whether a text was related or not to a target topic presented to them beforehand. In such tasks, reading process and decision making are closely related. Statistical analysis of such data aims at deciphering underlying dependency structures in these processes. Hidden Markov models (HMMs) have been used on eye-movement series to infer phases in the reading process that can be interpreted as strategies or steps in the cognitive processes leading to decision. In HMMs, each phase is associated with a state of the Markov chain. The states are observed indirectly though eye-movements. Our approach was inspired by Simola et al. (2008) 91, but we used hidden semi-Markov models for better characterization of phase length distributions (Olivier et al., 2021) 29. The estimated HMM highlighted contrasted reading strategies, with both individual and document-related variability.
New results were obtained in the standalone analysis of the eye-movements: 1) a statistical comparison between the effects of three types of texts was performed, considering texts either closely related, moderately related or unrelated to the target topic; 2) a characterization of the effects of the distance to trigger words on transition probabilities and 3) highlighting a predominant intra-individual variability in scanpaths.

As a perspective of this work, the segmentation induced by our eye-movement model could be used to obtain a statistical characterization of functional brain connectivity through simultaneous EEG recordings. This should lead to some integrated models coupling EEG and eye movements within one single HMM for better identification of strategies.

The results of this study have been partially included in a dissemination work at Inria Interstice journal. 85.

Joint work with: Grégoire Vincent, IRD, AMAP, Montpellier, France.

Covering just 7% of the Earth’s land surface, tropical forests play a disproportionate role in the biosphere: they store about 25% of the terrestrial carbon and contribute to over a third of the global terrestrial productivity. They also recycle about a third of the precipitations through evapotranspiration and thus contribute to generate and maintain a humid climate regionally, with positive effects also extending well beyond the tropics. However, the seasonal variability in fluxes between tropical rainforests and atmosphere is still poorly understood. Better understanding the processes underlying flux seasonality in tropical forests is thus critical to improve our predictive ability on global biogeochemical cycles. Leaf area, one key variable controlling water efflux and carbon influx, is poorly characterized. To monitor evolutions of biomass, leaf area density (LAD) or gas exchange, aerial and terrestrial laser scanner (LiDAR) measurements have been frequently used.

The principle is, for different LiDAR shoots assumed as independent, to measure the portions of beam lengths between successive hits. Possible censoring comes from beams not being intercepted within a given voxel. Current approaches aim at connecting LAD to the distribution of beam lengths through some statistical model. Such a simplified model does not currently take into account several effects that may impact either LAD or beam lengths: heterogeneity and dependencies in the vegetation properties in different voxels on the one hand and the nature of hit material on the other hand: wood vs. leaves, leading to biases or deteriorated uncertainty in estimation.

This collaboration, supported by Y. Bai's PhD work, aims at developping machine learning methods to address these issues.Current work is now focusing on 1) Applying 3D point convolution neural network to discriminate wood from leaves 2) Modeling the effect of underdetection of vegetal elements due to gradual loss of laser power through different censoring assumptions and validating these assumptions on simulated and real data sets.

The study of feature propagation at initialization in neural networks lies at the root of numerous initialization
designs. An assumption very commonly made in the field states that the pre-activations are Gaussian.
Although this convenient Gaussian hypothesis can be justified when the number of neurons per layer tends to infinity, it is challenged by both theoretical and experimental works for finite-width neural networks.
In 79, our major contribution of this work is to construct a family of pairs of activation functions and initialization distributions that ensure
that the pre-activations remain Gaussian throughout the network's depth, even in narrow neural networks.
In the process, we discover a set of constraints that a neural network should fulfill to ensure Gaussian
pre-activations.
Additionally, we provide a critical review of the claims of the Edge of Chaos line of works and build an exact Edge of Chaos analysis.
We also propose a unified view on pre-activations propagation, encompassing the framework of several well-known
initialization procedures.
Finally, our work provides a principled framework for answering the much-debated question: is it desirable to initialize the training of
a neural network whose pre-activations are ensured to be Gaussian?

We also investigate the cold posterior effect through the lens of PAC-Bayes generalization bounds. We argue that in the non-asymptotic setting, when the number of training samples is (relatively) small, discussions of the cold posterior effect should take into account that approximate Bayesian inference does not readily provide guarantees of performance on out-of-sample data. Instead, out-of-sample error is better described through a generalization bound. In this context, we explore the connections of the ELBO objective from variational inference and the PAC-Bayes objectives. We note that, while the ELBO and PAC-Bayes objectives are similar, the latter objectives naturally contain a temperature parameter

Joint work with: Michel Dojat from GIN, Univ. Grenoble Alpes

In recent accepted publication, we worked on the notion of graph comparisons. Node role explainability in complex networks is very difficult, yet is crucial in different application domains such as social science, neurosciences or computer science. Many efforts have been made on the quantification of hubs revealing particular nodes in a network using a given structural property. Yet, in several applications, when multiple instances of networks are available and several structural properties appear to be relevant, the identification of node roles remains largely unexplored. Inspired by the node automorphically equivalence relation, we define an equivalence relation on graph's nodes associated with any collection of nodal statistics (i.e. any functions on the node set). This allows us to define new graph global measures, the power coefficient and the orthogonality score to evaluate the collection parsimony and heterogeneity of a given nodal statistics collection. In addition, we introduce a new method based on structural patterns to compare graphs that have the same vertices set. This methods assigns a value to a node to determine its role distinctiveness in a graph family. Extensive numerical results of our method are conducted on both generative graph models and real data concerning human brain functional connectivity. The differences in nodal statistics are shown to be dependent on the underlying graph structure. Comparisons between generative models and real networks combining two different nodal statistics reveal the complexity of human brain functional connectivity with differences at both global and nodal levels. Using a group of 200 healthy controls connectivity networks, our method is able to compute high correspondence scores among the whole population, to detect homotopy, and finally to quantify differences between comatose patients and healthy controls.

Joint work with: Alex Petersen, Brigham Young University, US and Wendy Meiring, University Santa Barbara California, US

A novel non-parametric estimator of the correlation between regions, or groups of arbitrarily dependent variables, is proposed in the presence of noise. The challenge resides in the fact that both noise and low intra-regional correlation lead to inconsistent inter-regional correlation estimation using classical approaches. While some existing methods handle one of these issues or the other, none tackle both at the same time. To address this problem, we propose a trade-off between two approaches: correlating regional averages, which is not robust to low average intra-regional correlation, and averaging pairwise inter-regional correlations, which is not robust to high noise. To that end, we project the data onto a space where the Euclidean distance can be used as a proxy for the sample correlation. We then leverage hierarchical clustering to gather together highly correlated variables within each region prior to averaging. We prove our estimator is consistent for an appropriate cut-off height of the dendogram. We also empirically show our approach surpasses popular estimators in terms of quality and provide illustrations on real-world datasets that further demonstrate its usefulness. 54

Bayesian Optimal Experimental Design is a technique that allows practitioners to efficiently make use of limited experimental resources. In the sequential case however, we often have to run from start computationally intensive inference techniques such as Markov Chain Monte Carlo or Variational Inference to obtain the current estimate of the posterior. In this study, we show how we can leverage Sequential Monte Carlo Samplers in order to simultaneously estimate the best possible design and the current posterior. This iterative procedure make use of past information in order to avoid to have to run an inference algorithm from start at each iteration. We demonstrate its applicability in a source location inverse problem.

A wide class of problems can be formulated as inverse problems where the goal is to find parameter values that best explain some observed measures. Typical constraints in practice are that relationships between parameters and observations are highly nonlinear, with high-dimensional observations and multi-dimensional correlated parameters. To handle these constraints, we consider probabilistic mixtures of locally linear models, which can be seen as particular instances of mixtures of experts (MoE). We have shown in previous studies that such models had a good approximation ability provided the number of experts was large enough. This contribution is to propose a general scheme to design a tractable Bayesian nonparametric (BNP) MoE model to avoid any commitment to an arbitrary number of experts. A tractable estimation algorithm is designed using a variational approximation and theoretical properties are derived on the predictive distribution and the number of components. Illustrations on simulated and real data show good results in terms of selection and computing time compared to more traditional model selection procedures.

Joint work with: Thomas Moreau and Alexandre Gramfort from Inria Saclay and Gilles Louppe from Université de Liège

Inferring the parameters of a stochastic model based on experimental observations is central to the scientific method. A particularly challenging setting is when the model is strongly indeterminate, i.e. when distinct sets of parameters yield identical observations. This arises in many practical situations, such as when inferring the distance and power of a radio source (is the source close and weak or far and strong?) or when estimating the amplifier gain and underlying brain activity of an electrophysiological experiment. In a recent work, we have proposed the hierarchical neural posterior estimation (HNPE), a novel method for cracking such indeterminacy by exploiting additional information conveyed by an auxiliary set of observations sharing global parameters. This method extends recent developments in simulation-based inference (SBI) based on normalizing flows to Bayesian hierarchical models. We validated HNPE quantitatively on a motivating example amenable to analytical solutions and then applied it to invert a well known non-linear model from computational neuroscience, using both simulated and real EEG data.

Joint work with: Nicholas Tolley and Stephanie Jones from Brown University, Alexandre Gramfort from Inria Saclay

The Human Neocortical Neurosolver (HNN) is a framework whose foundation is a cortical column model with cell and circuit level detail designed to connect macroscale signals to meso/microcircuit level phenomena. We apply this model to study the cellular and circuit mechanisms of beta generation using local field potential (LFP) recordings from the non-human primate (NHP) motor cortex. To characterize beta producing mechanisms, we employ simulation based inference (SBI) in the HNN modeling tool. This framework leverages machine learning techniques and neural density estimators to characterize the relationship between a large space of model parameters and simulation output. In this setting, Bayesian inference can be applied to models with intractable likelihood functions (Gonçalves 2020, Papamakarios 2021). The main goal of this project is to provide a set of guidelines for scientists that wish to apply simulation-based inference to their neuroscience studies with a large-scale simulator such as HNN. This involves developing new methods for extracting summary features, checking the quality of the posterior approximation, etc. This work is mostly carried out by the Ph.D. student Nicholas Tolley from Brown University.

LANDER stands for "Latent Analysis, Adversarial Networks, and DimEnsionality Reduction" and is an associate team with researchers in Australia that has started in 2019. Details can be found at Lander website.

During the year, a number of journal publications have been finalized on subjects directly linked to the project, mixture of expert models and variants related to high dimensional regression, distributions and properties for non-Gaussian data, clustering and classification, Approximate Bayesian Computation (ABC), Bayesian non parametrics and Markov random fields, Expectiles, etc. The corresponding publications and working papers are listed on the website.

All this work served as a basis for a new submission to the Australian Research Council (ARC), which has been recently granted.

MSTGA and AIGM INRAE (French National Institute for Agricultural Research) networks: F. Forbes and J.B Durand are members of the INRAE network called AIGM (ex MSTGA) network since 2006, website, on Algorithmic issues for Inference in Graphical Models.
It is funded by INRAE MIA and RNSC/ISC Paris. This network gathers researchers from different disciplines.
Statify co-organized and hosted 2 of the network meetings in 2008 and 2015 in Grenoble.

S. Achard and J-B. Durand published an illustration of the process of statistical modeling, in the Inria journal for popularization of science (in French). The topic was illustrated through the analysis of eye movements to infer cognitive processes.