Bastien X Nony
email: bastien.nony@cerfacs.fr

Mélanie C Rochoux
email: melanie.rochoux@cerfacs.fr

Thomas Jaravel
email: thomas.jaravel@cerfacs.fr

Didier Lucor
email: didier.lucor@lisn.upsaclay.fr

REDUCED-ORDER MODEL FOR MICROSCALE ATMOSPHERIC DISPERSION COMBINING MULTI-FIDELITY LES AND RANS DATA

Keywords: Multi-fidelity, Co-kriging, Proper Orthogonal Decomposition, Convolutional Autoencoder, Atmospheric Boundary-Layer Flows, Computational Fluid Dynamics

Because of the complex turbulent flow dynamics caused by interactions between the atmospheric surface layer and the urban topography, forecasting microscale pollutant concentration fields is crucial for monitoring plume dispersion in urban regions. To capture these dynamics, large-eddy simulation (LES) is regarded as a high-fidelity numerical approach, however, it lacks real-time capabilities and remains costly because highly dimensional. Reynoldsaveraged Navier-Stokes (RANS) approach is a cruder path to modelling these phenomena based on a statistical description of turbulent phenomena, leading to averaged transport equations. It is much less computationally expensive but also less accurate in representing interactions between the atmospheric boundary-layer flow and the buildings in urban areas. Designing an efficient reduced-order model (ROM) with a level of precision similar to the LES approach and accounting for atmospheric and point-source emission uncertainties is of primary importance. In this paper, we propose a novel multi-fidelity ROM, which combines two levels of data fidelity -LES and RANS -and we evaluate it in the context of atmospheric flow dispersion. RANS data are obtained by injecting detailed flow information from LES into a lower-fidelity tracer transport equation, in the RANS formalism, using physics-constrained machine learning. There are two steps in our approach: i) dimension reduction is performed using a convolutional autoencoder trained using pre-training, and ii) mapping the uncertain parameters to the autoencoder latent space is obtained using co-kriging and an autoregressive model. We consider a case of a simplified two-dimensional flow configuration around a surface-mounted obstacle, where the main physical quantity of interest is the time-average tracer concentration field and its variability with respect to uncertain atmospheric forcing and emission source location. We show that the multi-fidelity approach achieves increased performance compared to the LES and RANS single-fidelity approaches at equivalent computational budget. This is a promising approach to designing an efficient ROM applicable to realistic field-scale atmospheric dispersion cases.

INTRODUCTION

Accidental short-term pollutant emissions occurring in urban areas or industrial sites can significantly degrade air quality and pose a risk to public health (e.g. 2011 Fukushima power plant explosion - [START_REF] Tsuruta | First retrieval of hourly atmospheric radionuclides just after the Fukushima accident by analyzing filter-tapes of operational air pollution monitoring stations[END_REF]2019/2020Australian bushfires -Graham et al., 2021). The ability to accurately predict pollutant dispersion in urban areas is essential for making informed decisions in emergency situations [START_REF] Da Silva | The impact of urban block typology on pollutant dispersion[END_REF][START_REF] Mendil | Hazardous atmospheric dispersion in urban areas: A deep learning approach for emergency pollution forecast[END_REF]. Unfortunately, urban areas, due to highly specific topographies and complex interactions with meteorology, present unique challenges when attempting to predict the range of possible scenarios for nearsource air pollutant dispersion [START_REF] Philips | Large-eddy simulation of passive scalar dispersion in an urban-like canopy[END_REF][START_REF] Dauxois | Confronting grand challenges in environmental fluid mechanics[END_REF].

Computational fluid dynamics has emerged as a powerful tool to accurately model the complex microscale atmospheric dispersion processes in urban environments [START_REF] Tominaga | CFD simulation of near-field pollutant dispersion in the urban environment: A review of current modeling techniques[END_REF][START_REF] Dauxois | Confronting grand challenges in environmental fluid mechanics[END_REF], using both Reynolds-averaged Navier-Stokes (RANS) approaches [START_REF] Milliez | Numerical simulations of pollutant dispersion in an idealized urban area, for different meteorological conditions[END_REF][START_REF] García-Sanchez | Quantifying inflow uncertainties in rans simulations of urban pollutant dispersion[END_REF] and large-eddy simulations (LES) [START_REF] Philips | Large-eddy simulation of passive scalar dispersion in an urban-like canopy[END_REF][START_REF] Vervecken | Dynamic dose assessment by Large Eddy Simulation of the near-range atmospheric dispersion[END_REF][START_REF] García-Sanchez | Predictive large eddy simulations for urban flows: Challenges and opportunities[END_REF][START_REF] Grylls | Evaluation of an operational air quality model using large-eddy simulation[END_REF]. LES is of high interest due to its greater accuracy to represent flow/obstacle interactions and its ability to provide not only mean statistics but also higher-order statistics. However, this comes with a high computational cost, making it challenging to use in uncertainty quantification applications to account for uncertainties associated with large-scale atmospheric conditions (Sousa and Gorlé, 2019) and the lack of information about the emission source in an accidental context [START_REF] Mons | Data assimilation-based reconstruction of urban pollutant release characteristics[END_REF]. To address these uncertainties, a multi-query ensemble must be built but requires overwhelming computational resources.

In this context, reduced-order models (ROM) aim to accurately capture the behaviour of high-dimensional systems while reducing computational costs. Recent studies have explored the use of machine-learning algorithms for atmospheric dispersion and natural convection emulation [START_REF] Margheri | A hybrid anchored-ANOVA-POD/Kriging method for uncertainty quantification in unsteady high-fidelity CFD simulations[END_REF][START_REF] García-Sanchez | Quantifying inflow uncertainties in rans simulations of urban pollutant dispersion[END_REF][START_REF] Lamberti | A multi-fidelity machine learning framework to predict wind loads on buildings[END_REF][START_REF] Lucor | Simple computational strategies for more effective physics-informed neural networks modeling of turbulent natural convection[END_REF], which is a particularly challenging problem for ROMs due to the advective nature of the system and the source location uncertainty. In this context, we have designed and thoroughly evaluated in a previous study, a data-driven ROM combining Proper Orthogonal Decomposition (POD) with Gaussian Process Regression (GPR) to forecast tracer concentration field first-and second-order statistics for a simplified turbulent atmospheric boundary-layer flow over a surface-mounted obstacle [START_REF] Nony | Reduced-order modeling for parameterized large-eddy simulations of atmospheric pollutant dispersion[END_REF]. Bayesian optimisation aided by POD optimised the GPR hyperparameters through the hierarchy of POD modes. Our analysis showed that the POD-GPR model was able to capture the large range of spatial scales seen in the POD modes, even with a limited training budget of approximately a hundred LES snapshots, while maintaining acceptable accuracy. However, when further reducing the LES training budget, the emulated tracer concentration fields could lose their physical consistency, in particular in regions where advection processes are dominant, for instance near the emission sources upstream of the obstacle [START_REF] Nony | Reduced-order models under uncertainties for microscale atmospheric pollutant dispersion in urban areas: exploring learning algorithms for high-fidelity model emulation[END_REF]. Based on this finding, the present study aims at improving the physics consistency and accuracy of POD-GPR capabilities under a small training LES data budget. To this end, more advanced deep-learning approaches based on convolutional autoencoder are utilised to overcome POD data compression limitations, and all of the LES statistics are exploited to explore the potential of multi-fidelity ROM combining high-fidelity, expensive LES solutions with lower-fidelity, less expensive RANS solutions.

The outline of this study is as follows. Section 2 presents the selected test case and describes the numerical configuration to generate the LES database. Section 3 introduces the ROM approaches using LES training data only. First, we recall the POD-GPR standard approach (Sect. 3.1) and improve it by substituting POD with a convolutional autoencoder (Sect. 3.2). Such an approach still demonstrates a lack of physics consistency when trained on a sparse LES database. In response, Sect. 3.3 introduces an alternative approach, which aims to combine emulation of relevant flow quantities from LES that are injected in a lower fidelity Reynolds-average representation of tracer dispersion. Similar ideas of exploiting the rich LES information with a simplified transport equation (TE) for tracer dispersion are found in the literature: for instance, [START_REF] Du | A novel approach to simulate pollutant dispersion in the built environment: Transport-based recurrence CFD[END_REF] proposed a simplified TE-ROM model based on flow statistics precomputed from LES data to give access to fast tracer dispersion inference, while preserving the rich information from the LES dataset. Finally, Sect. 4 explores a multi-fidelity ROM approach mixing sparse training 50 LES and 450 TE-ROM solutions using convolutional autoencoders alongside co-kriging and autoregressive processes to reduce the computational cost, while maintaining an acceptable level of accuracy.

DATABASE OF PARAMETERISED LARGE-EDDY SIMULATIONS AND PERFOR-MANCE EVALUATION CRITERIA

In this section, we present the two-dimensional test case for which we have generated a LES parameterised simulation database. This database corresponds to an ensemble of LES snapshots with multiple parameter entries, from which we aim at learning a ROM to accurately predict the dispersion quantities of interest. In this perspective, we also present the criteria to evaluate the quality of ROM predictions using the LES simulations as references.

Description of the test case

We simulate pollutant dispersion induced by the interactions between a fully-developed neutral turbulent boundary-layer flow and an isolated square-shaped obstacle (Fig. 1) in a twodimensional domain, as in [START_REF] Nony | Reduced-order modeling for parameterized large-eddy simulations of atmospheric pollutant dispersion[END_REF]. The pollutant is a passive tracer released at a constant rate from a point-source emission (the emission source axial position is x src and height is z src ). The turbulent boundary-layer flow is obtained by adding synthetically-generated wind fluctu-ations to a time-averaged vertical wind profile corresponding to neutral atmospheric conditions:

     u inlet (z) = u τ κ log 1 + z z 0 , u inlet (z = z c ) = u zc , (1)
where u τ (m s -1 ) is the friction velocity, κ = 0.41 is the dimensionless von Kármán constant, u zc (m s -1 ) is the velocity magnitude at reference height (z = z c = 10 m) and z 0 (m) is the aerodynamic roughness length. Assuming equilibrium with the inlet profile, the ground boundary condition (z = 0 m) is defined using a rough law-of-the-wall treatment based on the same aerodynamic roughness length z 0 .

Choice of parametric uncertainty

As in [START_REF] Nony | Reduced-order modeling for parameterized large-eddy simulations of atmospheric pollutant dispersion[END_REF], we consider four uncertain parameters gathered into the vector µ = (u zc , z 0 , x src , z src ) T ∈ R 4 , which acts as input to the ROMs (Sects. 3-4). Based on typical ranges of parameters found in atmospheric dispersion studies, we consider the following loguniform distribution for z 0 and uniform distribution for u zc :

log(z 0 ) ∼ U log(10 -3 ), log(10 -1 ) m, u zc ∼ U([3, 9]) m s -1 . (2)
The emission source can be located upstream, above or downstream of the obstacle. Its location uncertainty is characterised by the following uniform distributions from which the obstacle area is removed:

x src ∼ U([-3.5, 3.5]) m, z src ∼ U([0.2, 2.0]) m.
(3)

Ensemble of large-eddy simulations snapshots

In this work, we rely on the AVBP 1 code developed by CERFACS [START_REF] Gicquel | High performance parallel computing of flows in complex geometries[END_REF]) to generate the LES parameterised simulations (flow and tracer concentration field statistics) denoted by K les = {K 1 , . . . , K N h } T ∈ R N h , where N h represents the number of grid elements in the domain of interest (N h ≈ 240, 000 to properly resolve the flow/obstacle interactions).

The AVBP code solves the filtered compressible Navier-Stokes equations for flow dynamics and additional advection-diffusion equations for passive scalar dispersion on unstructured mesh. Subgrid-scale turbulence is represented using the standard Smagorinsky model [START_REF] Smagorinsky | Computational urban flow predictions with bayesian inference: Validation with field data[END_REF]. Each solution snapshot is obtained by averaging the LES instantaneous fields over a time window corresponding to 40 vortex shedding periods in the wake of the obstacle to guarantee statistical convergence for any inflow boundary condition [START_REF] Nony | Reduced-order models under uncertainties for microscale atmospheric pollutant dispersion in urban areas: exploring learning algorithms for high-fidelity model emulation[END_REF].

A large database of 600 LES-based snapshots (corresponding to a total cost of 0.6 million CPU hours) is built, in order to train and test the ROM approaches. Each snapshot corresponds to a different realisation of the input parameter vector µ following Halton's low-discrepancy sequence. For each snapshot, different quantities of interest are stored: the time-averaged tracer concentration field but also airflow field statistics such as the time-averaged airflow field and the turbulent kinetic energy. The analysis of this LES database shows that varying the tracer source location leads to very different regions of tracer accumulation, implying that there is a strong nonlinearity of the concentration field response to changes in the emission source location and to a lesser extent to the other parametric changes, which is a challenge for ROM [START_REF] Nony | Reduced-order modeling for parameterized large-eddy simulations of atmospheric pollutant dispersion[END_REF].

Reduced-order model performance evaluation

Dataset splitting The full LES database made of N snapshots is split into two subsets following Halton's sequence ordering: i) a training dataset (80%, i.e. N train = 450) to calibrate the ROM parameters; and ii) a test dataset to evaluate the ROM capacity to predict LES quantities of interest for new scenarios of input parameters (20%, i.e. N test = 150). We analyse in this study how the ROM performance is modified when reducing the size of the training dataset to only 50 LES snapshots (N train = 50) to move closer to a feasible budget for a multi-query LES framework on a realistic application. In this restricted framework, the test dataset remains the same to avoid introducing bias.

Performance evaluation To quantify the ROM performance in the physical space, we evaluate the Q 2 criterion on each feature. The reconstruction/prediction error is weighted by the variance over the LES samples for each grid element i:

Q 2 i = 1 - ∥K les,i -K rb,i ∥ 2 2 ∥K les,i -Ê[K les,i ]∥ 2 2 , ∀i = 1, . . . , N h , (4)
where K rb,i corresponds to the ROM prediction at the ith grid element, whose objective is to best represent the LES counterpart K les,i . The Q 2 i criterion is estimated over the training dataset for verification and over the test dataset for evaluating the ROM prediction capacity. To help with the analysis, we also derive a global score from the variance-weighted local Q 2 criterion as

Q 2 = N h i=1 ω i Q 2 i , ω i = V(K les,i ) N h j=1 V(K les,j ) , (5)
where V(K les,i ) corresponds to the variance unbiased estimation over the snapshots. This weighted average matches the usual explained variance criterion of the POD [START_REF] Nony | Reduced-order models under uncertainties for microscale atmospheric pollutant dispersion in urban areas: exploring learning algorithms for high-fidelity model emulation[END_REF].

SINGLE-FIDELITY REDUCED-ORDER MODELLING APPROACHES

The very high computational cost associated with LES predictions motivates the offline construction of accurate ROM, which can produce in the online phase new ensemble predictions of the field statistics (i.e. for unexplored values of the uncertain parameters) at a very low computational cost. However, this is a challenging task since the LES quantities of interest are of very large dimension N h , the number of available LES snapshots is limited (N ≪ N h ), and the LES response to the input parameters µ may be subject to strong nonlinearity due to the complexity of flow topology and tracer concentration patterns induced by the obstacle.

In this section, we introduce two ROM approaches for learning the parametric model response. We first recall a purely data-driven ROM approach directly learning the mapping between the uncertain parameters and the time-averaged tracer concentration fields based on LES data as in [START_REF] Nony | Reduced-order modeling for parameterized large-eddy simulations of atmospheric pollutant dispersion[END_REF], and we propose an improvement in the form of a convolutional neural network. We then propose and investigate an alternative physics-constrained ROM approach, which injects emulated flow quantities learned from LES data into a lower-fidelity Reynolds-averaged tracer transport equation; this equation corresponds to RANS formalism and predicts ensemble-averaged tracer concentration fields.

3.1 Data-driven POD-GPR reduced-order modelling approach based on LES data

Methodology

Due to the high dimension of the time-averaged tracer concentration fields, the data-driven ROM is built in a two-step approach following the choices made in [START_REF] Nony | Reduced-order modeling for parameterized large-eddy simulations of atmospheric pollutant dispersion[END_REF]. The first step consists in projecting the LES fields onto a reduced-basis space of dimension L (with L < N ≪ N h ) using a standard proper orthogonal decomposition (POD) approach [START_REF] Sirovich | Turbulence and the dynamics of coherent structures. I -Coherent structures. II -Symmetries and transformations. III -Dynamics and scaling[END_REF][START_REF] Berkooz | The proper orthogonal decomposition in the analysis of turbulent flows[END_REF]. Within this POD framework, the approximated LES fields K rb are represented as linear combinations of the POD modes {ψ l } l=1,••• ,L :

K rb = L l=1 k l (µ) ψ l (x, z) , (6)
where k l (µ) ∈ R is the l-th reduced coefficient to emulate with respect to the uncertain parameters µ and to whiten [START_REF] Kessy | Optimal whitening and decorrelation[END_REF].

Based on the comparative study in [START_REF] Nony | Reduced-order models under uncertainties for microscale atmospheric pollutant dispersion in urban areas: exploring learning algorithms for high-fidelity model emulation[END_REF] showing the better performance of Gaussian processes compared to other metamodeling techniques such as polynomial chaos and decision trees, emulation of each reduced coefficient k l is carried out using a Gaussian process regression (GPR) model [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF]. We consider L independent and noisy GPR models based on anisotropic Matérn covariance kernels (here characterised by the smoothness hyperparameter ν = 5/2); anisotropy means that we consider a different length-scale for each input parameter using the automatic relevance determination (ARD) formulation. If we note

(U, K l ) = {(µ (n) , k (n) l ), n = 1, . . . , N } the
training dataset dedicated to the l-th reduced coefficient and (U * , K * l ) the test dataset counterpart, the GPR inference formula for the test reduced coefficients results from the posterior distribution

K * l | U, K l , U * ∼ N (m * l , cov(K * l )) , (7)
with m * l = r l (U * , U) [r l (U, U) + s 2 l I] -1 K l cov(K * l ) = r l (U * , U * ) -r l (U * , U) [r l (U, U) + s 2 l I] -1 r l (U, U * ), (8)
where s 2 l is the noise variance and r l is the Matérn kernel. The hyperparameters of each GPR model may be optimised separately to best fit the characteristic length-scale of each POD mode [START_REF] Nony | Reduced-order modeling for parameterized large-eddy simulations of atmospheric pollutant dispersion[END_REF]; this is done using a ten-restart gradient descent on the marginal loglikelihood [START_REF] Rasmussen | Gaussian Processes for Machine Learning[END_REF].

In this work, POD is implemented using the randomised truncated singular value decomposition [START_REF] Halko | Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions[END_REF] from the scikit-learn library [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF]. GPR is also implemented using scikit-learn. It was shown in [START_REF] Nony | Reduced-order modeling for parameterized large-eddy simulations of atmospheric pollutant dispersion[END_REF] that a large number of reduced basis modes are required to accurately represent the LES ensemble variance at all points of the domain of interest (in particular, in the area upstream of the obstacle where the emission sources is sometimes located -Fig. 1).

POD-GPR prediction results

To quantify the performance of the POD-GPR approach, we employ the Q 2 ensemble performance metrics to accurately assess the ROM performance (Sect. 2.4). For all ROM implementations, Q 2 results can be found in Table 1.

To further validate the POD-GPR ROM, we present in Fig. 2 the emulation results for a test snapshot with nominal uncertain parameter values (this nominal snapshot has been selected to represent the ensemble mean of uncertainty distributions and to capture the various physical phenomena occurring near the obstacle such as recirculation areas, turbulent diffusiondominated regions, and advection-dominated regions). Figure 2a shows the LES reference field; Fig. 2b shows the emulated field obtained with the POD-GPR ROM trained on the full dataset of 450 LES. This example demonstrates the capability of the ROM to accurately reproduce the main tracer concentration structures, particularly in regions where turbulent diffusion effects are dominant (for instance, in the wake of the obstacle far away from the emission source). These regions carry the majority of the ensemble variance and are easily predicted, thus leading to a high Q 2 global score of 96.7% on the test ensemble. Still, this global score masks some prediction errors that occur locally: i) in the vicinity of the emission source, where the estimated tracer concentration is substantially underestimated; ii) in the recirculation area upstream of the obstacle, where the tracer concentration levels are overestimated; and iii) in tracer-free regions upstream of the obstacle due to the noisy high-order POD modes [START_REF] Nony | Reduced-order modeling for parameterized large-eddy simulations of atmospheric pollutant dispersion[END_REF].

We now examine how the POD-GPR prediction changes when the ROM is trained on the limited dataset of 50 LES. The associated global Q 2 -score of 83.1% may first appear satisfactory but when examining the nominal snapshot prediction in Fig. 2c, it becomes clear that the ROM fails to accurately emulate the tracer concentration magnitude near the emission source, with noisy spurious structures appearing in tracer-free regions upstream of the emission source (white lines). Furthermore, the tracer concentration is significantly underestimated at the emission source and upstream of the obstacle, thus indicating a low quality of the ROM prediction. This result illustrates that physical consistency is lost upstream of the obstacle when reducing the training dataset. This non-physical behaviour is worsened by further reducing the training dataset [START_REF] Nony | Reduced-order models under uncertainties for microscale atmospheric pollutant dispersion in urban areas: exploring learning algorithms for high-fidelity model emulation[END_REF] and is mainly due to POD, which is unsuitable for strongly nonlinear patterns that require a very large number of POD modes in the reduced basis (L = 100 for the full training dataset) while being incompatible with sparse training data. To mitigate physicallyinconsistent emulations, we propose to improve the ROM structure by replacing POD with a deep learning convolutional autoencoder (AE).

Improving the data-driven reduced-order modelling approach using deep learning

In this section, we explore the ability of convolutional autoencoders [START_REF] Murata | Nonlinear mode decomposition with convolutional neural networks for fluid dynamics[END_REF][START_REF] Fukami | Convolutional neural network based hierarchical autoencoder for nonlinear mode decomposition of fluid field data[END_REF] to enhance the data-driven ROM performance by improving average concentration field compression, while meeting the constraint of reduced training dataset for LES.

Convolutional autoencoder implementation

Convolutional autoencoders cannot be directly applied to unstructured mesh since their convolutional kernels rely upon a Cartesian grid. For this reason, before proceeding to the learning stage, we linearly interpolate the LES field data onto a uniform Cartesian grid, whose spatial discretisation is consistent with that of the unstructured mesh, resulting in a total of 25,960 grid points and corresponding to tensors of dimension R 295×88 .

Figure 3 shows the convolutional autoencoder architecture we use in this work and that is inspired by the work by [START_REF] Murata | Nonlinear mode decomposition with convolutional neural networks for fluid dynamics[END_REF]. This architecture includes convolutional layers at the periphery and dense layers in the core layers close to the latent vector. Without lack of generality, the size of the latent space is empirically set to L = 10.

The convolutional autoencoder was implemented using the Keras/Tensorflow libraries. Train- for feature extraction and dimension reduction, and ii) a dense multilayer perceptron (middle column) for highlevel feature abstraction. γ corresponds to the latent vector of dimension L. Each rounded box describes a layer operation and the corresponding input/output tensor dimensions are specified underneath in brackets.

(3×| 𝛾 |) (2×| 𝛾 |) (3×| 𝛾 |) (2×| 𝛾 |) (912
ing was performed by minimising the mean-square error via the ADAM descent scheme, with an initial learning rate of 10 -3 , which was subsequently manually decayed to 10 -4 and 10 -6 . The convolutional autoencoder is combined with a GPR model for each latent variable, forming the AE-GPR ROM.

AE-GPR prediction results

Figure 2d shows the emulated nominal snapshot obtained with the AE-GPR ROM when considering sparse training data (i.e. using only 50 LES snapshots for the ROM training). This new ROM version improves the prediction results obtained with the POD-GPR approach at equivalent computational budget (Fig. 2c). The plume shape better matches that of the reference LES solution (Fig. 2a), both downstream of the obstacle and upstream near the emission source. Furthermore, the low-concentration contour line is found to be more faithful to the LES solution.

Still, it should be noted that near the emission source and near the upstream face of the obstacle, the tracer concentration peak is slightly shifted downstream from the actual source location, which is in violation of basic consideration of the physics of this problem.

Globally, the AE-GPR ROM achieves a global Q 2 score (84.8%) that is slightly higher than the POD-GPR ROM (83.1% -Table 1). The increase in performance may seem modest in view of the prediction improvements in the tracer concentration patterns (Fig. 2d), but these improvements are localised and do not represent a large fraction of the total variance. This motivates the implementation of scores that characterise position and shape errors, which is beyond the scope of the present work. It also points to the importance of obtaining physically sound predictions, in particular in regions with strong flow gradients, recirculations, etc. In the following, we focus on designing a ROM approach that better satisfies physical constraints and gives access to more plausible tracer concentration predictions.

Physics-constrained reduced-order model based on LES-informed Reynolds-averaged tracer transport equations

In this section, we introduce an innovative hybrid approach referred to as RANS-TE, combining a ROM approach with physical equations. It consists of two steps: i) emulating relevant airflow statistics (e.g. mean flow field, turbulent kinetic energy) based on a POD-GPR ROM trained on LES data (as in Sect. 3.1), and ii) integrating these emulated statistics into a simplified scalar transport model via a Reynolds-average transport equation for the tracer, whose numerical resolution is much cheaper than simulating a full LES. In addition to saving computational time, this hybrid approach allows the flow dynamics to be decoupled from the tracer transport, thus separating the atmospheric parametric uncertainties from the source location parametric uncertainties. As tracer uncertainties are no longer sampled in the LES training database, the data-driven ROM approach can solely focus on atmospheric uncertainties (i.e. the reference velocity magnitude u zc and the aerodynamic roughness length z 0 ) in order to map the relevant airflow statistics. These emulated airflow statistics are then injected into the scalar transport equations to predict the average tracer concentration fields for different scenarios of source location.

LES-informed Reynolds-averaged tracer transport formulation

The key idea of the proposed RANS-TE hybrid approach is to solve the tracer transport equations by feeding some relevant information from the data-driven ROM trained onto the LES database for various flow scenarios. It is worth noting that supplementing DNS-or LESbased data into lower-order equations such as RANS is a growing strategy found in the literature [START_REF] Steiner | Classifying regions of high model error within a datadriven rans closure: Application to wind turbine wakes[END_REF][START_REF] Amarloo | Frozen propagation of reynolds force vector from high-fidelity data into reynolds-averaged simulations of secondary flows[END_REF]. Additionally, inverse modelling can be used to infer corrective fields to improve RANS closures accuracy [START_REF] Parish | A paradigm for data-driven predictive modeling using field inversion and machine learning[END_REF].

In the present work, the main flow statistics (velocity components, kinetic energy, and turbulent dissipation), denoted with a ⋆ superscript, are estimated from 50 LES snapshots and predicted using a POD-GPR framework based on L = 10 POD modes and an anisotropic Matérn kernel for GPR as detailed in Sect. 3.1. We note that much fewer POD modes are required here compared to the data-driven ROM approach presented in Sect. 3.1 as the learning only deals with atmospheric uncertainties and does not need to directly handle source uncertainties. For the tracer concentration, a more conventional Reynolds-averaged transport equation is solved as

u j ⋆ ∂K ∂x j = - ∂ ∂x j K ′ u ′ j . (9)
In this equation, the carrier velocity field u j ⋆ is extracted from LES (as denoted by the ⋆ superscript), instead of being obtained via the resolution of the full RANS equations. Unclosed terms related to second-order velocity-tracer cross-correlations (turbulent diffusion term K ′ u ′ j ) remain to be modelled. A standard closure based on the Boussinesq assumption involving the mean tracer gradient is adopted here:

K ′ u ′ j = - ν RANS T Sc RANS T ∂K ∂x j , ( 10
)
where ν RANS T is the turbulent eddy-viscosity, and Sc RANS T is the turbulent Schmidt number. As proposed by [START_REF] Yoshizawa | A Reynolds-averaged turbulence modeling approach using three transport equations for the turbulent viscosity, kinetic energy, and dissipation rate[END_REF], ν RANS T is solved through an additional transport equation for the turbulent eddy-viscosity that takes the following form:

u j ⋆ ∂ν RANS T ∂x j = C µP k ⋆ tke -C µϵ 1 τ ⋆ T ν RANS t + ∇ • ν RANS T σ ν ∇ν T , (11)
where C µP , C µϵ , σ ν are modelling constants associated with turbulent production, dissipation, and diffusion; u j ⋆ is the time-averaged velocity jth component; and k ⋆ tke is the turbulent kinetic energy. τ * T is a relevant turbulence time scale [START_REF] Yoshizawa | A Reynolds-averaged turbulence modeling approach using three transport equations for the turbulent viscosity, kinetic energy, and dissipation rate[END_REF] extracted from LES based on the ratio of turbulent kinetic energy k * tke to turbulent dissipation ϵ * . More details about this RANS modelling and implementation approach can be found in [START_REF] Nony | Reduced-order models under uncertainties for microscale atmospheric pollutant dispersion in urban areas: exploring learning algorithms for high-fidelity model emulation[END_REF].

RANS-TE prediction results

In this section, the RANS-TE ROM is implemented, for which airflow closure terms are approximated from a limited training dataset of 50 LES snapshots sampling the atmospheric uncertainties and the averaged concentration fields are obtained through the numerical resolution of the scalar transport equations for each source location sample. To evaluate the RANS-TE prediction performance, we rely on the quantitative Q 2 metrics and the nominal snapshot example as before.

Figure 4 shows the tracer concentration prediction obtained with the RANS-TE approach (Fig. 4a), and compares it to the reference LES solution (Fig. 4b). There is a good agreement between the two solutions upstream of the obstacle, although the RANS-TE model slightly overestimates tracer concentration. The shape of the wake is also accurately represented by the ROM. However, downstream of the obstacle, the ROM prediction diverges significantly from the LES solution, especially in the recirculation area near the obstacle. This induces a lower global Q 2 score for the RANS-TE approach (69.6%) than for the POD-GPR ROM approach (83.1% -Table 1).

The loss of performance in the downstream recirculation area is mostly due to the tracer turbulent flux closure, which is not intended to represent accurately turbulence transport for a two-dimensional setup, but it is not due to the airflow ROM lack of accuracy [START_REF] Nony | Reduced-order models under uncertainties for microscale atmospheric pollutant dispersion in urban areas: exploring learning algorithms for high-fidelity model emulation[END_REF]. Indeed, the emulation of the closure fields using POD-GPR reduces the Q 2 performance by only 2%.

Discussion about single-fidelity approaches

So far we have designed single-fidelity ROM approaches using LES data, cf. Table 1 for quantitative results. We first demonstrated the capability of a standard POD-GPR approach to accurately predict tracer concentration intensity and wake structure in regions dominated by turbulent transport. However, this approach had difficulty in capturing tracer concentration magnitude and patterns in advection-dominated regions upstream of the obstacle, where the concentration plumes are narrow, as nonlinearities are poorly handled by POD. The AE-GPR approach (based on a convolutional autoencoder for dimension reduction instead of POD) achieved better performance under small training dataset constraints. In a second step, to both reduce the training computational cost and better handle the tracer emission source uncertainties, the hybrid RANS-TE approach combining a data-driven ROM approach for airflow statistics and RANS transport equations was proposed. Although Q 2 performances were lower than for the AE-GPR ROM approach, the RANS-TE hybrid approach demonstrated its ability to accurately predict tracer concentration patterns in advection-dominated regions upstream of the obstacle. The complementarity of the RANS-TE and AE-GPR approaches in turbulent diffusion-and advection-dominated regions motivates the exploration of multi-fidelity ROM approaches, which could benefit from the advantages of each single-fidelity approach to reduce ROM prediction errors.

MULTI-FIDELITY REDUCED-ORDER MODELLING APPROACH

This section examines the potential for a more robust and efficient data-driven ROM by combining LES (Sect. 2.3) and RANS with transport equation (RANS-TE) modelling (Sect. 3.3). This type of approach mixing solutions of different fidelity levels is known as multi-fidelity [START_REF] Goodfellow | Deep Learning[END_REF]. The primary motivation is to harness the cheap computational cost of the low-fidelity RANS-TE model in order to more accurately predict average tracer concentration fields. In particular, multi-fidelity takes advantage of generating a larger training database at a reduced computational cost (compared to single-fidelity approaches presented in Sect. 3). The RANS-TE model is here considered as the "low-fidelity" solver with faster evaluation, while the "high-fidelity" LES solver is very accurate but computationally expensive. The proposed multi-fidelity ROM is referred to as MF-ROM in the following.

Methodology

The present framework builds its multi-fidelity approach on a mixed training database made of 50 LES simulations along with 450 RANS-TE solutions. These RANS-TE solutions may be viewed as somewhat biased data as discussed in Sect. 3.3.2 as they themselves rely on LES data to be generated in the first place. In practice, the same 50 LES snapshots are also used to emulate the airflow field statistics for the RANS-TE approach .

We focus on several extensions to multi-level response ROMs, which feature a similar nonintrusive structure to the POD-GPR approach. Multi-fidelity is introduced to reduce the dimensionality of the high-dimensional mean tracer concentration fields K les (µ) and K te (µ) obtained from LES and RANS-TE, respectively. Additionally, a multi-fidelity regression model is used to map the uncertainty parameters µ onto the resulting compressed coefficients k les (µ) and k te (µ) in the latent space. Statistical approaches such as POD and GPR are initially tailored for single-level data. As an alternative, more sophisticated GPR procedures, such as those combining co-kriging and autoregressive models [START_REF] Gratiet | Multi-fidelity Gaussian process regression for computer experiments[END_REF][START_REF] Brevault | Overview of Gaussian process based multi-fidelity techniques with variable relationship between fidelities, application to aerospace systems[END_REF], as well as convolutional autoencoders (AEs), which employ pre-training techniques [START_REF] Goodfellow | Deep Learning[END_REF], may be utilised for this multi-fidelity purpose.

Pre-training for convolutional autoencoder

We implement the convolutional AE introduced in Sect. 3.2 using a pre-training transfer learning procedure [START_REF] Goodfellow | Deep Learning[END_REF]. The transfer learning here consists of taking a model that has already been trained for a given task and completing the training on new data to solve a different, but similar task. In our case, this procedure involves splitting the network training process into two steps. Firstly, the network weights are trained on the large dataset of 450 low-fidelity RANS-TE solutions K te until loss function convergence is reached. The best weights obtained in this step are then saved and used to initialise the second step of the training process, which is conducted based on the high-fidelity solutions until loss function convergence is reached again. In the end, the final network weights are the ones that best approximate the original 50 high-dimensional K les statistical fields. ADAM gradient descent is employed in both steps with an initial learning rate set to 10 -3 , and manually stepwise-decreased to a final value of 10 -6 . The resulting decoder is then used in the final multi-fidelity ROM (MF-ROM) provided some latent space variables.

Multi-fidelity Gaussian processes using co-kriging and autoregressive models

The GPR framework is enhanced to model the mapping from the uncertain input parameters to the latent variables, i.e. µ → k MF-ROM ≈ k les . Here, co-kriging with autoregressive models [START_REF] Kennedy | Predicting the output from a complex computer code when fast approximations are available[END_REF][START_REF] Gratiet | Multi-fidelity Gaussian process regression for computer experiments[END_REF] is used for multi-fidelity by training L = 10 independent GPR models on a multi-level formulation. Co-kriging combines data from different levels of fidelity using a weighted combination of the lower-fidelity data and a nonlinear mapping of the higher-fidelity data. Autoregressive models use the lower-fidelity data to learn correlations between input variables, and model the nonlinear mapping of the higherfidelity data.

In the first step, a standard GPR model (as in Sect. 3.1) is trained to emulate the low-fidelity latent variables k l,te (µ). Then, using co-kriging, assuming the joint process (k l,les (µ), k l,te (µ)) is Gaussian given some parameters, and the Markov property, the high-fidelity latent variables k l,les (µ) are derived using autoregression from the low-fidelity latent variables k l,te (µ) (Le [START_REF] Gratiet | Multi-fidelity Gaussian process regression for computer experiments[END_REF]:

k l,les (µ) ≈ k l,MF-ROM (µ) = ρ l (µ) k l,te (µ) + δ l (µ), for l = 1, . . . , L, (12)
where δ l (µ) and k l,te (µ) are assumed to be independent standard Gaussian processes with Matérn 3/2 kernels. The hyperparameters for δ l (µ) and k l,te (µ) are denoted by θ l,δ and θ l,te , respectively. The scaling factor ρ l (µ) may be assumed to be constant, and is calibrated during the training process. From such modelling assumptions, a Bayesian prediction model for the high-fidelity latent variables k l,MF-ROM (µ * ) can be derived from the posterior Gaussian distribution of k MF-ROM (µ) given the dataset collection of RANS-TE and LES solutions D, the scale factor ρ, and the hyperparameters θ l,te and θ l,δ :

k l,MF-ROM (µ * ) | D, ρ l , θ l,te , θ l,δ ∼ N (m l (µ * ), r l (µ, µ * )), (13)
with m l and r l the mean and covariance function of the Gaussian process. A closed form for the expression of m l (µ * ) is described in the work of Le [START_REF] Gratiet | Multi-fidelity Gaussian process regression for computer experiments[END_REF], allowing for a fast numerical implementation of the emulation strategy.

Co-kriging and autoregressive models can then be used to model the mapping between uncertain parameters µ and the latent mode representation of the time-averaged tracer response in a multi-fidelity context. This can be performed by optimising the Gaussian process hyperparameters using the Maximum Likelihood Likelihood (MLL) procedure, with multiple restarts (10 restarts) for improved accuracy.

Multi-fidelity results

To assess the performance of the multi-fidelity ROM, we use a Q 2 score evaluation on the test dataset and analyse the emulated nominal snapshot as in previous ROM prediction evaluations.

Figure 5 shows the prediction results for the nominal snapshot. This illustrates the superior performance of the multi-fidelity approach in comparison to the single-fidelity procedures trained under the sparse data constraint (Figs. 2cd). Namely, the emulated plume shape is in better agreement with the LES solution (Fig. 2a) and does not replicate the discrepancies of the RANS-TE, POD-GPR, and AE-GPR solutions. Downstream, the MF-ROM does not replicate the strong errors obtained from the RANS-TE procedure. The upstream region is also better reconstructed than the AE-GPR solution. In particular, the concentration peak is accurately located at the emission source position, which was not the case for the single-fidelity AE-GPR approach (for which the peak concentration was slightly shifted downstream -(Fig. 2d). Furthermore, fewer artificial noise structures were generated in the upstream area for the multifidelity ROM prediction compared to the POD-GPR solution (Fig. 2c). Still, the presence of a noisy low-concentration K = 10 -4 concentration isoline in Fig. 5 implies that the convolutional autoencoder has not yet achieved convergence. The MF-ROM model achieves a global score Q 2 global = 92.6%, which is a substantial improvement over the results obtained with single-fidelity ROM approaches based on 50 training LES snapshots (Table 1). This suggests that the MF-ROM approach benefits from the complementary information coming from both LES and RANS-TE snapshots, thereby increasing the accuracy of the convolutional AE and the parametric mapping of the latent space variables. Interestingly, the performance of the MF-ROM approach is even relatively close to that of the single-fidelity AE-GPR approach trained on 100 LES snapshots (Table 1). However, the latter is almost twice as expensive as the MF-ROM in terms of training cost, which makes the multifidelity approach promising for field-scale pollutant dispersion applications based on LES data.

CONCLUSION

In this study, we proposed a series of improvements to the finely tuned POD-GPR ROM approach introduced in [START_REF] Nony | Reduced-order modeling for parameterized large-eddy simulations of atmospheric pollutant dispersion[END_REF], using cutting-edge data-driven techniques based on LES data, for a simplified problem of pollutant dispersion in a turbulent boundary-layer flow. We observed that POD was not optimal to map the highly nonlinear relationship between the uncertain emission source location and the tracer concentration field values, leading to poor compression and a high required number of POD modes in the reduced basis. Convolutional autoencoders allow to significantly reduce the latent space dimension (by a factor of about ten) while improving ROM prediction accuracy. Still, ROM predictions can suffer in this case from a lack of physical consistency for sparse LES training data. To overcome this limitation, which is an issue in field-scale pollutant dispersion applications, we have introduced a hybrid TE-ROM approach combining machine learning and a transport equation in order to constrain the emulated tracer concentration fields. Although this hybrid approach improves the physical consistency of the emulated fields in regions dominated by advection close to the emission source, the emulated fields are degraded in regions dominated by turbulent diffusion processes in the downstream recirculation area due to the limitations of the TE closure model formulation.

Interestingly, we have demonstrated that this novel TE-ROM approach still presents an interesting opportunity for multi-fidelity modelling. Using an augmented dataset of only 50 LES and 450 TE-ROM solutions, we have designed a multi-fidelity ROM combining a convolutional autoencoder with transfer learning and multi-fidelity GPR models. This novel TE-ROM achieves the best results of all tested ROM approaches at equivalent computational budget.

In future work, we plan to optimise the ratio between high-fidelity and low-fidelity solutions by leveraging modern techniques, taking advantage of active learning in order to maximise performance with minimal computational cost to move towards realistic field-scale pollutant dispersion applications.

Figure 1 :

 1 Figure 1: Schematic of an atmospheric turbulent boundary-layer flow around a surface-mounted square obstacle. Text boxes point to the uncertain parameters, i.e. pollutant source location, ground roughness and incoming flow velocity.

Figure 2 :Filters

 2 Figure 2: Normalised average tracer concentration field obtained for the nominal snapshot corresponding to µ = (5.8 m s -1 , 2.8 × 10 -2 m, -1.0 m, 0.8 m). (a) LES reference. (b) POD-GPR ROM prediction when considering the full training dataset (450 LES snapshots) to compare to (c) POD-GPR ROM prediction obtained for the limited training dataset (50 LES snapshots). (d) AE-GPR ROM prediction also obtained for the limited training dataset (50 LES snapshots). White lines correspond to the 5 × 10 -4 -contour line to highlight the presence of low-magnitude noisy structures.

Figure 3 :

 3 Figure 3: Convolutional autoencoder architecture consisting of i) convolutional layers (first and last columns)for feature extraction and dimension reduction, and ii) a dense multilayer perceptron (middle column) for highlevel feature abstraction. γ corresponds to the latent vector of dimension L. Each rounded box describes a layer operation and the corresponding input/output tensor dimensions are specified underneath in brackets.

Figure 4 :

 4 Figure 4: Nominal snapshot normalised average tracer concentration field. (a) RANS-TE ROM prediction. White lines indicate the presence of low-magnitude noisy structures (5 × 10 -4 -contour lines). (b) Prediction absolute error measuring the discrepancies between the RANS-TE prediction and the LES reference solution.

Figure 5 :

 5 Figure 5: Nominal snapshot normalised average tracer concentration field. (a) MF-ROM prediction. The white lines indicate the presence of low-magnitude noisy structures, with the 5 × 10 -4 -contour line indicating the magnitude of these structures. (b) Prediction absolute error relative to the reference LES solution.

Table 1 :

 1 Single-and multi-fidelity ROM Q 2 global scores (in %) obtained by comparison to the LES test database. (size and nature of training simulations database are mentioned in parentheses).

	ROM architectures	Training data	Q 2 (%)
	POD-GPR	450 LES	96.7
	POD-GPR	50 LES	83.1
	AE-GPR	100 LES	94.0
	AE-GPR	50 LES	84.8
	RANS-TE	50 LES	69.6
	MF-ROM	50 LES + 450 RANS-TE 92.6

AVBP documentation, see http://www.cerfacs.fr/avbp7x/


Hénoïk Willot
email: henoik.willot@hds.utc.fr

Sébastien Destercke
email: sebastien.destercke@hds.utc.fr

Khaled Belahcène
email: khaled.belahcene@centralesupelec.fr

ISIPTA

Keywords: robust classifier, explainability, prime implicants, Imprecise Probabilities, Naive credal classifier

In this paper, we investigate how robust classification results can be explained by the notion of prime implicants, focusing on explaining pairwise dominance relations. By robust, we mean that we consider imprecise models that may abstain to classify or to compare two classes when information is insufficient. This will be reflected by considering (convex) sets of probabilities. By prime implicants, we understand a subset of attributes, minimal w.r.t. inclusion, that we need to know or specify before reaching a specified conclusion (either of dominance or non-dominance between two classes). After presenting the general concepts, we derive them in the case of the well-known naive credal classifier.

Introduction

Two key aspects of trustworthy AI are the ability to provide robust and safe inferences or predictions, and to be able to provide explanations as of why those have been made.

Regarding explainability, the notion of prime implicants corresponds to providing minimal sufficient condition to make a given statement, e.g., the attributes that need to be instantiated to make a classification. They have been successfully proposed as components of explanations for large classes of models such as graphical ones [START_REF] Shih | A symbolic approach to explaining bayesian network classifiers[END_REF], with very efficient procedure existing for specific structures such as the Naive one [START_REF] Marques-Silva | Explaining Naive Bayes and Other Linear Classifiers with Polynomial Time and Delay[END_REF]. In contrast with other methods such as SHAP [START_REF] Van Den Broeck | On the tractability of shap explanations[END_REF] that tries to compute the average influence of attributes, prime implicants have the advantage to be wellgrounded in logic, and to provide certifiable explanation (in the sense that the identified attributes are logical, sufficient reasons).

However, explainable AI tools have been mostly applied to precise models, at least in the machine learning domain (this is less true, e.g., in preference modelling [START_REF] Belahcene | Explaining robust additive utility models by sequences of preference swaps[END_REF]). Yet, in applications involving sensitive issues or in which the decision maker wants to identify ambiguous cases, it may be preferable to use models that will return sets of classes when information is insufficient rather than always returning a point-valued prediction. Several frameworks such as conformal prediction [START_REF] Vineeth Balasubramanian | Conformal prediction for reliable machine learning: theory, adaptations and applications[END_REF], indeterminate classifiers [START_REF] José | Learning nondeterministic classifiers[END_REF] or imprecise probabilistic models [START_REF] Corani | Bayesian networks with imprecise probabilities: Theory and application to classification[END_REF] have been proposed to handle such issue. While some explanation methods for such models have been recently proposed [START_REF] Lev V Utkin | An imprecise shap as a tool for explaining the class probability distributions under limited training data[END_REF][START_REF] Zhang | Explaining cautious random forests via counterfactuals[END_REF], none of them explicitly adopts a logical standpoint regarding explanations, meaning that the present work is complementary to those.

Imprecise probabilistic models in particular have the interest that they are direct extensions and generalisations of probabilistic classifiers, hence one can directly try to transport well-grounded explanation principles existing for precise probabilistic classifier to this setting. This is what we intend to do in this paper for prime implicant explanations.

We will start by introducing how the idea of prime implicants can be adapted to classifiers considering sets of probabilities as their uncertainty models. Section 2 will be a short reminder of the robust classification setting, and will introduce our notations. In Section 3, we will present the idea of prime implicant, as well as how it can answer various explanatory needs. As the formulated problems are likely to be computationally challenging for generic models, we focus in Section 4 on the naive credal classifier, that generalise the naive Bayes classifier. We show that for such a model, computing and enumerating prime implicants can be done in polynomial time, thanks to its independence assumption and decompositional properties. We also provide an experiment in Section 5 illustrating our approach. Note that this work builds upon some first results published in [START_REF] Willot | Explaining robust classification through prime implicants[END_REF], that focused exclusively on a unique use of prime implicants and did not contain any experiment. In contrast, the study of this paper provides much more use of prime implicants, study some basic properties of their behaviour and provide some first experiments on a real-world data set.

Preliminaries on Robust Classification

In this section, we lay down our basic notations and provide necessary reminders about imprecise probabilities.

We consider a usual discrete multi-class problem, where we must predict a variable 𝑌 taking values in Y = {𝑦 1 , . . . , 𝑦 𝑚 } using 𝑛 input variables 𝑋 1 , . . . , 𝑋 𝑛 that respectively takes values in X 𝑖 = {𝑥 1 𝑖 , . . . , 𝑥 𝑘 𝑖 𝑖 }. We note X = × 𝑛 𝑖=1 X 𝑖 and x ∈ X a vector in this space. When considering a subset 𝐸 ⊆ {1, . . . , 𝑛} of dimensions, we will denote by X 𝐸 = × 𝑖 ∈𝐸 X 𝑖 the corresponding domain, and by x 𝐸 the values of a vector on this sub-domain. We will also denote by -𝐸 := {1, . . . , 𝑛} \ 𝐸 all dimensions not in 𝐸, with X -𝐸 , x -𝐸 following the same conventions as X 𝐸 , x 𝐸 . We will also denote by (x 𝐸 , y -𝐸 ) the concatenation of two vectors whose values are given for different elements. Notation (x 𝐸 , •) means that all features in -𝐸 can take any value. If 𝐸 = {1, . . . , 𝑛}, then we will simply ignore the subscript.

In the rest of the paper, we will often refer to partially ordered sets, their corresponding relations and sets of sufficient elements that allows to asset them. Those sufficient elements will here be composed of a vector of specified feature values and of a set of probabilities. We will denote by 𝑦 ⪰ 𝑝, (x) 𝑦 ′ the fact that considering the model 𝑝 and the vector (x) is sufficient to state (or implies) 𝑦 ⪰ 𝑦 ′ .

In the case of precise classifiers, we have 𝑦 ⪰ 𝑝, (x) 𝑦 ′ when the condition1

𝑝 (𝑦|x) 𝑝 (𝑦 ′ |x) ≥ 1 (1)
is met, or in other words when 𝑝 (𝑦|x) ≥ 𝑝 (𝑦 ′ |x). However, probabilistic classifiers can be deceptively precise, for instance when only a small number of data are available to estimate them, or when data become imprecise. This is why, in this paper, we consider generalised probabilistic settings, and more specifically imprecise probability theory, where one considers that the probability 𝑝 belongs to some subset P, often assumed to be convex (this will be the case here). One then needs to extend the relation ⪰ 𝑝 to such a case, and a common and robust way to do so is to require ⪰ 𝑝 to be true for all elements 𝑝 ∈ P. In this case, 𝑦 is said to robustly dominate 𝑦 ′ upon observing a vector x, written 𝑦 ≻ P, (x) 𝑦 ′ , when the condition inf

𝑝∈ P 𝑝 (𝑦|x) 𝑝 (𝑦 ′ |x) ≥ 1 (2)
is met, or in other words when 𝑝(𝑦|x) ≥ 𝑝(𝑦 ′ |x) for all 𝑝 ∈ P. Going from the precise to imprecise probabilities can introduce incomparabilities between classes, written 𝑦 ≻≺ P, (x) 𝑦 ′ when both inf

𝑝∈ P 𝑝 (𝑦|x) 𝑝 (𝑦 ′ |x) < 1 and inf 𝑝 ′ ∈ P 𝑝 ′ (𝑦 ′ |x) 𝑝 ′ (𝑦|x) < 1. ( 3
)
1Using dominance expressed this way will be useful in the sequel. We will also restrict ourselves to 0/1 loss functions here.

Explaining Robust Classification through Prime Implicants

Explaining the conclusion or deduction of an algorithm, and in particular of a learning algorithm, has become (again) an important issue [START_REF] Burkart | A survey on the explainability of supervised machine learning[END_REF]. A notion that can play a key role in explanation mechanisms is the one of prime implicants, i.e., which elements are sufficient before drawing a given conclusion. In this section, we detail how prime implicants can be used to answer the needs of different explanatory mechanisms, within the setting of robust, imprecise probabilistic classifiers.

Prime Implicants as Validatory Explanations

When observing a vector x 𝑜 and making a prediction about whether 𝑦 dominates 𝑦 ′ , finding a prime implicant confirming that 𝑦 dominates 𝑦 ′ corresponds to finding the values of x 𝑜 that are sufficient to state that 𝑦 dominates 𝑦 ′ , and that are minimal with this property. With this idea in mind, we will say that a subset 𝐸 ⊆ {1, . . . , 𝑛} := ⟦1, 𝑛⟧ of attributes (where 𝐸 are the indices of the considered attributes) is a validatory implicant of

𝑦 ≻ P, (x 𝑜 𝐸 ,•) 𝑦 ′ iff inf x 𝑣 -𝐸 ∈ X -𝐸 inf 𝑝∈ P 𝑝 (𝑦|(x 𝑜 𝐸 , x 𝑣 -𝐸 )) 𝑝 (𝑦 ′ |(x 𝑜 𝐸 , x 𝑣 -𝐸 )) > 1, (4)
that is if dominance holds for any values of attributes whose indices are outside 𝐸, and any probability 𝑝 ∈ P. This means that knowing x 𝑜 𝐸 alone is sufficient to deduce 𝑦 ≻ 𝑦 ′ . A set 𝐸 is a prime implicant iff we satisfy (4) and for any 𝑖 ∈ 𝐸, we have

inf x 𝑣 -𝐸∪{𝑖} ∈ X -𝐸∪{𝑖} inf 𝑝∈ P 𝑝 (𝑦|(x 𝑜 𝐸\{𝑖 } , x 𝑣 -𝐸∪{𝑖 } )) 𝑝 (𝑦 ′ |(x 𝑜 𝐸\{𝑖 } , x 𝑣 -𝐸∪{𝑖 } )) ≤ 1, (5)
that is if removing any attribute from 𝐸 makes our deduction invalid, so that 𝐸 is a minimal sufficient condition for 𝑦 ≻ P, (x 𝑜 𝐸 ,•) 𝑦 ′ to hold for any completion of -𝐸. In the sequel, it will prove useful to consider the function that associates to each possible subset the value of the ratio between the obtained posterior probabilities. This function 𝜙 𝑣 is defined by :

𝜙 𝑣 (𝑦, 𝑦 ′ , x 𝑜 , 𝐸) := inf x 𝑣 -𝐸 ∈ X -𝐸 inf 𝑝∈ P 𝑝 (𝑦|(x 𝑜 𝐸 , x 𝑣 -𝐸 )) 𝑝 (𝑦 ′ |(x 𝑜 𝐸 , x 𝑣 -𝐸 ))
.

To ease the use of this function, we will omit the observed vector x 𝑜 when context is clear and we will write "𝑦, 𝑦 ′ " as a subscript meaning that class 𝑦 is the numerator and 𝑦 ′ the denominator, i.e., 𝜙 𝑣 𝑦,𝑦 ′ (𝐸) := 𝜙 𝑣 (𝑦, 𝑦 ′ , x 𝑜 , 𝐸) Note that when the set P reduces to a singleton, that is when we consider precise classifiers instead of robust ones, then our notion of prime implicant reduces to previously proposed ones [START_REF] Marques-Silva | Explaining Naive Bayes and Other Linear Classifiers with Polynomial Time and Delay[END_REF], and our approach is therefore a formal generalisation of those.

Monotony with respect to imprecision. one can note that the notion of validatory prime implicant is monotonic with respect to imprecision, in the following sense Proposition 1 Consider two credal sets P ′ ⊆ P, then 4) is true for 𝐸 and P, it must be true for 𝐸 and P ′ , as the infimum is taken over a smaller domain.

𝑦 ≻ P, (x 𝑜 𝐸 ,•) 𝑦 ′ =⇒ 𝑦 ≻ P ′ , (x 𝑜 𝐸 ,•) 𝑦 ′ Proof Immediate, since if Equation (
This means that a validatory implicant will remain so if we consider a more precise model (obtained, e.g., by observing additional data). However, if a subset 𝐸 was prime for P, it does not need to be so for P ′ , meaning that the size of validatory prime implicants should decrease as imprecision decreases. This is somehow natural, as a more informative model should need less measurements to provide a conclusion.

Prime Implicants as Contrastive Explanations

Another quite common way to audit or explain a statement "Why X is P?" is by answering the implicit question "Why X is P and not Q?" [START_REF] Miller | Explanation in artificial intelligence: Insights from the social sciences[END_REF][START_REF] Guidotti | Counterfactual explanations and how to find them: literature review and benchmarking[END_REF]. This can classically be answered by finding a counter-factual, i.e., a modification of the example with sufficient changes so as to change our conclusion.

Replying to this question in a minimal way can be seen as the task of finding a minimal set of attributes or features for which a modification could change our decision. We will call 𝐸 ⊆ ⟦1, 𝑛⟧ a contrastive prime implicant if modifying the attributes within 𝐸 is a minimal sufficient condition to change our decision, that is, if inf

x 𝑐 𝐸 ∈ X 𝐸 inf 𝑝∈ P 𝑝 (𝑦|(x 𝑐 𝐸 , x 𝑜 -𝐸 )) 𝑝 (𝑦 ′ |(x 𝑐 𝐸 , x 𝑜 -𝐸 )) < 1, (7)
and if for any 𝑖 ∉ 𝐸, we do have inf

x 𝑐 𝐸\{𝑖} ∈ X 𝐸\{𝑖} inf 𝑝∈ P 𝑝 (𝑦|(x 𝑐 𝐸\{𝑖 } , x 𝑜 -𝐸∪{𝑖 } )) 𝑝 (𝑦 ′ |(x 𝑐 𝐸\{𝑖 } , x 𝑜 -𝐸∪{𝑖 } )) ≥ 1, (8)
that is there is at least one modification of feature values in 𝐸 that lead to a different decision, and any change done within a subset of it would not change the decision. Denoting x 𝑐 𝐸 the argument of Equation [START_REF] Chin | Contrastive constraints guide explanation-based category learning[END_REF], 𝐸 is a contrastive implicant if 𝑦 ⊁ P, (x 𝑐 𝐸 ,x 𝑜 -𝐸 ) 𝑦 ′ . We also consider the function 𝜙 𝑐 𝑦,𝑦 ′ that associate to each possible subset the value

𝜙 𝑐 𝑦,𝑦 ′ (𝐸) := inf x 𝑐 𝐸 ∈ X 𝐸 inf 𝑝∈ P 𝑝 (𝑦|(x 𝑐 𝐸 , x 𝑜 -𝐸 )) 𝑝 (𝑦 ′ |(x 𝑐 𝐸 , x 𝑜 -𝐸 )) (9)
One of the interesting aspects of considering imprecise models is that contrastive explanations do not necessary lead to reversing the initial preference (which is the case for precise models). Indeed, modifying the conclusion 𝑦 ≻ P,x 𝑜 𝑦 ′ by considering the modified vector (x 𝑐 𝐸 , x 𝑜 -𝐸 ) can lead to two quite different situations, resulting either in 𝑦 ′ ≻ P, (x 𝑐 𝐸 ,x 𝑜 -𝐸 ) 𝑦 (reversing of preference) or 𝑦 ≻≺ P, (x 𝑐 𝐸 ,x 𝑜 -𝐸 ) 𝑦 ′ (weakening of preference) and we will define two notions of contrastive explanations.

Given that 𝐸 is a contrastive prime implicant, we say that it is also a reversing prime implicant if in addition we have2 inf

𝑝∈ P 𝑝 (𝑦 ′ |(x 𝑐 𝐸 , x 𝑜 -𝐸 )) 𝑝 (𝑦|(x 𝑐 𝐸 , x 𝑜 -𝐸 )) ≥ 1, (10)
as this contrastive prime implicant change the initial statement or conclusion into its reverse. Otherwise, if it does satisfy Equations ( 7) and ( 8), but not [START_REF] José | Learning nondeterministic classifiers[END_REF], we say that 𝐸 is a weakening prime implicant, as it changes a preference between two classes into incomparability. The vector (x 𝑐 𝐸 , x 𝑜 -𝐸 ) also provides us with a contrastive example for which the decision would change.

Monotony with respect to imprecision. as with validatory implicants, the notion of contrastive implicants is monotonic with respect to imprecision, but in the other direction.

Proposition 2 Consider two credal sets P ⊆ P ′ , then 7) is true for 𝐸 and P, it must be true for 𝐸 and P ′ , as the infinimum is taken over a larger domain, and is of lower value. This means that a contrastive implicant and the associated example (x 𝑐 𝐸 , x 𝑜 -𝐸 ) will remain so if we consider a more imprecise model. However, if a subset 𝐸 was prime for P, it does not need to be so for P ′ , meaning that the size of contrastive prime implicants should decrease as imprecision increases. Again, this is somehow intuitive, as a dominance obtained for a more imprecise model should be easier to modify than the same dominance obtained from a more precise model. It should also be noted that the argument x 𝑐 𝐸 obtained for P in Equation ( 7) may actually change when considering P ′ .

𝑦 ⊁ P, (x 𝑐 𝐸 ,x 𝑜 -𝐸 ) 𝑦 ′ =⇒ 𝑦 ⊁ P ′ , (x 𝑐 𝐸 ,x 𝑜 -𝐸 ) 𝑦 ′ Proof Immediate, since if Equation (

Remark 3 Equation (7) corresponds to finding some minimal changes that could modify our conclusion, and can therefore be viewed as a tool to analyse the robustness of this decision. As such, it can be useful to analyse the model and its robustness. However, answering the question

"what should I change to be sure to reverse the dominance" would necessitate another notion where the satisfaction of Equation [START_REF] José | Learning nondeterministic classifiers[END_REF] is enforced, that we will not consider here, leaving it for future work.

2Recall that x 𝑐 𝐸 is the argument of Equation ( 7).

Prime Implicants as Explanations of Doubt

For precise models, the statement "Why X is P?" that we have to explain is typically a precise assignment or a dominance relation between two classes. In the case of robust classification, the question "Why X is neither P nor Q?", and for what reasons cannot I classify X precisely, also makes sense.

In this case, we say that 𝐸 ⊆ ⟦1, 𝑛⟧ is a prime implicant of doubt if

sup x 𝑑 -𝐸 ∈ X -𝐸 inf 𝑝∈ P 𝑝 (𝑦|(x 𝑜 𝐸 , x 𝑑 -𝐸 )) 𝑝 (𝑦 ′ |(x 𝑜 𝐸 , x 𝑑 -𝐸 )) < 1, (11a)
and

sup x 𝑑 -𝐸 ∈ X -𝐸 inf 𝑝∈ P 𝑝 (𝑦 ′ |(x 𝑜 𝐸 , x 𝑑 -𝐸 )) 𝑝 (𝑦|(x 𝑜 𝐸 , x 𝑑 -𝐸 )) < 1, (11b)
that is any change performed outside of 𝐸 (in particular the changes for the most favourable values for 𝑦 in Equation (11a) and the most favourable for 𝑦 ′ in Equation (11b))

will not modify the fact that the two classes are incomparable given our model and knowledge of it. It is further more minimal if for any 𝑖 ∉ 𝐸, we do have either

sup x 𝑑 -𝐸∪{𝑖} ∈ X -𝐸∪{𝑖} inf 𝑝∈ P 𝑝 (𝑦 ′ |(x 𝑜 𝐸\{𝑖 } , x 𝑑 -𝐸∪{𝑖 } )) 𝑝 (𝑦|(x 𝑜 𝐸\{𝑖 } , x 𝑑 -𝐸∪{𝑖 } )) ≥ 1, (12a) or sup x 𝑑 -𝐸∪{𝑖} ∈ X -𝐸∪{𝑖} inf 𝑝∈ P 𝑝 (𝑦|(x 𝑜 𝐸\{𝑖 } , x 𝑑 -𝐸∪{𝑖 } )) 𝑝 (𝑦 ′ |(x 𝑜 𝐸\{𝑖 } , x 𝑑 -𝐸∪{𝑖 } )) ≥ 1, (12b)
We also consider the function 𝜙 𝑑 𝑦,𝑦 ′ that associate to each possible subset the value

𝜙 𝑑 𝑦,𝑦 ′ (𝐸) := sup x 𝑑 -𝐸 ∈ X -𝐸 inf 𝑝∈ P 𝑝 (𝑦|(x 𝑜 𝐸 , x 𝑑 -𝐸 )) 𝑝 (𝑦 ′ |(x 𝑜 𝐸 , x 𝑑 -𝐸 )) (13)
𝜙 𝑑 𝑦,𝑦 ′ corresponds to Equation (11a) and 𝜙 𝑑 𝑦 ′ ,𝑦 to Equation (11b) and both will be used in the computations as we will see in Section 4 for the naive credal classifier. In general, the vectors x 𝑑 -𝐸 for which the bounds of (11a)-(11b) are obtained will be different.

Monotony with respect to imprecision. as before, we can easily show that implicants of doubt are somehow monotonic with imprecision, in the following sense Proposition 4 Consider two credal sets P ⊆ P ′ , then

𝑦 ≻≺ P, (x 𝑜 𝐸 ,•) 𝑦 ′ =⇒ 𝑦 ≻≺ P ′ , (x 𝑜 𝐸 ,•) 𝑦 ′
Proof Immediate, since if Equations (11a)-(11b) are true for 𝐸 and P, it must be true for 𝐸 and P ′ , as the infinimum is taken over a larger domain, and is of lower value.

It should be remarked that while those implicants are also of validatory nature, in the sense that they confirm our conclusion, their monotonicity is not in the same direction as the validatory implicants of dominance relations. This is however not surprising: as imprecision increases, it becomes easier to obtain that two classes are incomparable, hence prime implicants should decrease in size as credal sets become more imprecise.

Remark 5

We could also have considered contrastive implicants of doubt that would transform the incomparability into a dominance relation, as done for example in [START_REF] Zhang | Explaining cautious random forests via counterfactuals[END_REF]. Such implicants would be of the same kind as the ones mentioned in Remark 3, as they would answer the question "what should I change to be sure to have a dominance relation".

Remark 6

In contrast with the previous implicants trying to either verify or contradict a dominance relation between two classes, it may be that 𝐸 = ∅ is the only prime implicant of 𝑦 ≻≺ 𝑦 ′ , in which case doubt is simply due to inherent imprecision of our information (think, for instance, about the case of total ignorance).

Short Discussion about the 3 Types of Prime Implicants

We defined three functions 𝜙 𝑐 𝑦,𝑦 ′ , 𝜙 . This means that they can be seen as value functions associated to 𝐸, and that finding a prime implicant amounts to the task of finding a minimal "bundle of items"3 𝐸 such that 𝜙 𝑣 𝑦,𝑦 ′ (𝐸) ≥ 1, 𝜙 𝑐 𝑦,𝑦 ′ (𝐸) < 1 or 𝜙 𝑑 𝑦,𝑦 ′ (𝐸) < 1, therefore allowing us to map the finding of robust prime implicants to an item selection problem or to knapsack problems where we have to fill the sack until it reaches a certain value. Unfortunately, in general, the log-functions4 of each problem will not be additive, as we will not have log

• 𝜙 𝑦,𝑦 ′ (𝐸 ∪ {𝑖}) = log • 𝜙 𝑦,𝑦 ′ (𝐸) + log • 𝜙 𝑦,𝑦 ′ ({𝑖}).
We will nevertheless show in Section 4 that it is the case for the Naive credal classifier.

While providing a minimal subset of features such that a preference/dominance is preserved or changed can be considered to some extent as satisfactory for the user [START_REF] Jay | The role of explanation in discovery and generalization: evidence from category learning[END_REF][START_REF] Chin | Contrastive constraints guide explanation-based category learning[END_REF] (as long as those features have a meaning for the user), the same cannot really be said about non-dominance or incomparability. In such a case, the user will probably not be satisfied by the mere fact that features values in 𝐸 are sufficient to claim incomparability, and will request to know why this incomparability happens.

In a machine learning setting, it makes sense to differentiate between incomparability due to ambiguity, where a small change in our knowledge representation P would lead to a decision, from incomparability due to lack of knowledge, where it would require significantly more knowledge to obtain a decision. These two types of uncertainty sources are often referred to as epistemic and aleatoric uncertainties, and those can be quantified [START_REF] Hüllermeier | Quantification of credal uncertainty in machine learning: A critical analysis and empirical comparison[END_REF].

It seems reasonable that the complementary explanation to incomparability should differ according to the dominating source of uncertainty or indecision. In particular:

• if the indecision is mainly due to aleatoric uncertainties, it is clear that collecting more data is unlikely to solve the issue, and that it would be important to identify those features that generate the ambiguity. In this case, it would seem preferable to provide a contrastive explanation (in the sense of Section 3.2) rather than recommending the collection of further data, so as to answer the question: "which features generate my ambiguity?".

• if the indecision is mainly due to epistemic uncertainties, a possible way to answer this question is to know how many further data points would we need to collect (and which ones) in order to reach a conclusion rather than producing none. The question we would answer would then be "what data should I collect to gain knowledge?" It is clear to us that providing formal reasons as to why an incomparability is observed, and proposing tools in this direction is a worthwile undertaking, and that our proposal could be useful to the analyst as a way to audit the model (why is my model doubting, and what could I do about it?). It is less clear that the notion proposed in this paper is instrumental to the end-user. Indeed, once x 𝑜 is known, letting the end-user know that we could have known earlier (i.e., with less measurement) that we could not reach a decision is not very helpful. However, our approach can also be considered to detect from partial observations, and before measuring all features, that incomparability will ensue whatever happens, therefore sending an early signal that with this model and this degree of cautiousness, taking more measurements is fruitless.

A definite goal we have in mind for future work is to go beyond the definition of prime implicants of doubts, and investigate problems such as active learning or feature acquisition in which they could offer an operational advantage.

The Case of the Naive Credal Classifier

We now study the specific case of the Naive credal classifier [START_REF] Zaffalon | The naive credal classifier[END_REF], and show that in this case, computing prime implicants becomes easy, as such a computation can be brought back to selecting items with an additive value functions, or equivalently to simple knapsack problems.

Generic Case

The basic idea of the Naive credal classifier (the same as its precise counterpart) is to assume that attributes are independent of each other given the class. This modelling assumption means that

𝑝 (𝑦|x) = 𝑛 𝑖=1 𝑝 𝑖 (x 𝑖 |𝑦) × 𝑝 𝑌 (𝑦) 𝑝 (x)
once we apply the Naive assumption and Bayes rule. This means in particular that

𝑝 (𝑦|x) 𝑝 (𝑦 ′ |x) = 𝑝 𝑌 (𝑦) 𝑝 𝑌 (𝑦 ′ ) 𝑛 𝑖=1 𝑝 𝑖 (x 𝑖 |𝑦) 𝑝 𝑖 (x 𝑖 |𝑦 ′ )
with every 𝑝 𝑖 (•|𝑦) being independent of 𝑝 𝑖 (•|𝑦 ′ ), and every 𝑝 𝑖 (•|𝑦), 𝑝 𝑗 (•|𝑦) independent for 𝑖 ≠ 𝑗. When switching to credal models, one considers sets of conditional distributions P X 𝑖 (•|𝑦) and a set P Y of priors rather than precise probabilities. We will abuse the notation P X 𝑖 by P 𝑖 and 𝑝 X 𝑖 by 𝑝 𝑖 for the sake of conciseness.

In our Equations ( 4), ( 7) and (11a, 11b) we have two optimisation problems, one in X 𝐸 (or X -𝐸 ) and one in P. Thanks to the independence assumptions, the two problems can be solved independently. Let us now see how the common part of the three Equations, the problem in P, transform in this case. We do have inf

𝑝∈ P 𝑝 (𝑦|x) 𝑝 (𝑦 ′ |x) = inf 𝑝 𝑌 ∈ P Y 𝑝 𝑌 (𝑦) 𝑝 𝑌 (𝑦 ′ ) 𝑛 𝑖=1 inf 𝑝 𝑖 ∈ P 𝑖 𝑝 𝑖 (x 𝑖 |𝑦) 𝑝 𝑖 (x 𝑖 |𝑦 ′ ) (14)
Once again, thanks to our Independence assumption, each term of the equation can be taken independently of the others (different variables) and inside each feature the numerator is independent of the denominator (different conditioning element). Moreover, as our probability sets P 𝑖 are convex, finding the minimum and maximum value is usually easy. Finally, we get that Equation ( 14) becomes inf

𝑝 𝑌 ∈ P Y 𝑝 𝑌 (𝑦) 𝑝 𝑌 (𝑦 ′ ) 𝑛 𝑖=1 𝑝 𝑖 (x 𝑖 |𝑦) 𝑝 𝑖 (x 𝑖 |𝑦 ′ ) (15)
where 𝑝 (x) = inf 𝑝∈ P 𝑝 (x) and 𝑝 (x) = sup 𝑝∈ P 𝑝 (x). Let's note 𝑝 𝑦,𝑦 ′ = inf 𝑝 𝑌 ∈ P Y 𝑝 𝑌 ( 𝑦) 𝑝 𝑌 ( 𝑦 ′ ) . We can now rewrite our functions as :

𝜙 𝑣 𝑦,𝑦 ′ (𝐸) = 𝑝 𝑦,𝑦 ′ 𝑖 ∈𝐸 𝑝 𝑖 (x 𝑜 𝑖 |𝑦) 𝑝 𝑖 (x 𝑜 𝑖 |𝑦 ′ ) 𝑖 ∈ -𝐸 inf x 𝑣 𝑖 ∈ X 𝑖 𝑝 𝑖 (x 𝑣 𝑖 |𝑦) 𝑝 𝑖 (x 𝑣 𝑖 |𝑦 ′ ) (16a) 𝜙 𝑐 𝑦,𝑦 ′ (𝐸) = 𝑝 𝑦,𝑦 ′ 𝑖 ∈𝐸 inf x 𝑐 𝑖 ∈ X 𝑖 𝑝 𝑖 (x 𝑐 𝑖 |𝑦) 𝑝 𝑖 (x 𝑐 𝑖 |𝑦 ′ ) 𝑖 ∈ -𝐸 𝑝 𝑖 (x 𝑜 𝑖 |𝑦) 𝑝 𝑖 (x 𝑜 𝑖 |𝑦 ′ ) (16b) 𝜙 𝑑 𝑦,𝑦 ′ (𝐸) = 𝑝 𝑦,𝑦 ′ 𝑖 ∈𝐸 𝑝 𝑖 (x 𝑜 𝑖 |𝑦) 𝑝 𝑖 (x 𝑜 𝑖 |𝑦 ′ ) 𝑖 ∈ -𝐸 sup x 𝑑 𝑖 ∈ X 𝑖 𝑝 𝑖 (x 𝑑 𝑖 |𝑦) 𝑝 𝑖 (x 𝑑 𝑖 |𝑦 ′ ) (16c
) As we see in Equations (16a), (16b) and (16c), in the case of the NCC the optimisation on X 𝐸 (or X -𝐸 ) is independent of the computation of 𝜙 𝑦,𝑦 ′ (𝐸). It follows that the results are unique and can be computed before choosing the items in 𝐸. We can represent them by unique "worst opponent" vectors, depending only on classes 𝑦 and 𝑦 ′ (the former in the numerator and the later at the denominator) :

x 𝑣:𝑦,𝑦 ′ = × 𝑛 𝑖=1 arg inf x 𝑣 𝑖 ∈ X 𝑖 𝑝 𝑖 (x 𝑣 𝑖 |𝑦) 𝑝 𝑖 (x 𝑣 𝑖 |𝑦 ′ ) x 𝑐:𝑦,𝑦 ′ = × 𝑛 𝑖=1 arg inf x 𝑐 𝑖 ∈ X 𝑖 𝑝 𝑖 (x 𝑐 𝑖 |𝑦) 𝑝 𝑖 (x 𝑐 𝑖 |𝑦 ′ )
x 𝑑:𝑦,𝑦 ′ = × 𝑛 𝑖=1 arg sup

x 𝑑 𝑖 ∈ X 𝑖 𝑝 𝑖 (x 𝑑 𝑖 |𝑦) 𝑝 𝑖 (x 𝑑 𝑖 |𝑦 ′ )
When we solve the problem of selecting 𝐸, in the case of validatory and contrastive prime implicants, we will only use the "worst opponent" vectors with "𝑦, 𝑦 ′ ", whereas we also need the converse "𝑦 ′ , 𝑦" for prime implicants of doubt. We will then refer to x 𝑣 and x 𝑐 instead of x 𝑣:𝑦,𝑦 ′ and x 𝑐:𝑦,𝑦 ′ . We can also note that x 𝑣 and x 𝑐 are equal in the case of NCC, the two problems of finding validatory and contrastive prime implicants only differing by the fact that in the validatory case, elements of 𝐸 are fixed, while they are modified in the contrastive case. It should be noted that this uniqueness of "worst opponent" is not true for more generic models, in the sense that the arguments of Equations ( 4), [START_REF] Chin | Contrastive constraints guide explanation-based category learning[END_REF] and (11a, 11b) in X 𝐸 will typically depend on 𝐸.

Coming back to the NCC, we can also see that each of our function is additive on their log form. Indeed, for instance for 𝜙 𝑣 𝑦,𝑦 ′ we have :

log 𝜙 𝑣 𝑦,𝑦 ′ (𝐸 ∪ {𝑖}) -log 𝜙 𝑣 𝑦,𝑦 ′ (𝐸) = (log 𝑝 𝑖 (x 𝑜 𝑖 |𝑦) -log 𝑝 𝑖 (x 𝑜 𝑖 |𝑦 ′ )) -(log 𝑝 𝑖 (x 𝑣 𝑖 |𝑦) -log 𝑝 𝑖 (x 𝑣 𝑖 |𝑦 ′ )) (17)
As this value is independent of any feature (inside or outside 𝐸) different from 𝑖. We can therefore define contribution functions 𝐺 𝑣 , 𝐺 𝑐 and 𝐺 𝑑 𝑦,𝑦 ′ mapping each feature 𝑖 to the contribution of adding 𝑖 to 𝐸 :

𝐺 𝑣 (𝑖) = (log 𝑝 𝑖 (x 𝑜 𝑖 |𝑦) -log 𝑝 𝑖 (x 𝑜 𝑖 |𝑦 ′ )) -(log 𝑝 𝑖 (x 𝑣 𝑖 |𝑦) -log 𝑝 𝑖 (x 𝑣 𝑖 |𝑦 ′ )) (18a) 𝐺 𝑐 (𝑖) = (log 𝑝 𝑖 (x 𝑐 𝑖 |𝑦) -log 𝑝 𝑖 (x 𝑐 𝑖 |𝑦 ′ )) -(log 𝑝 𝑖 (x 𝑜 𝑖 |𝑦) -log 𝑝 𝑖 (x 𝑜 𝑖 |𝑦 ′ )) (18b) 𝐺 𝑑 𝑦,𝑦 ′ (𝑖) = (log 𝑝 𝑖 (x 𝑜 𝑖 |𝑦) -log 𝑝 𝑖 (x 𝑜 𝑖 |𝑦 ′ )) -(log 𝑝 𝑖 (x 𝑑:𝑦,𝑦 ′ 𝑖 |𝑦) -log 𝑝 𝑖 (x 𝑑:𝑦,𝑦 ′ 𝑖 |𝑦 ′ )) (18c)
We see that, by definition, values of functions 𝐺 𝑣 's are at least zero because we replace the worst opponent value with a better value (the observed one)5 and is at most 0 for 𝐺 𝑐 and 𝐺 𝑑 𝑦,𝑦 ′ . It follows that

log 𝜙 𝑣 𝑦,𝑦 ′ (𝐸) = log 𝜙 𝑣 𝑦,𝑦 ′ (∅) + ∑ 𝑖 ∈𝐸 𝐺 𝑣 (𝑖) (19a) log 𝜙 𝑐 𝑦,𝑦 ′ (𝐸) = log 𝜙 𝑐 𝑦,𝑦 ′ (∅) + ∑ 𝑖 ∈𝐸 𝐺 𝑐 (𝑖) (19b) log 𝜙 𝑑 𝑦,𝑦 ′ (𝐸) = log 𝜙 𝑑 𝑦,𝑦 ′ (∅) + ∑ 𝑖 ∈𝐸 𝐺 𝑑 𝑦,𝑦 ′ (𝑖) (19c)
We will note the log-contributions of the empty set by 𝐶 𝑣 , 𝐶 𝑐 and 𝐶 𝑑 𝑦,𝑦 ′ . Using these additive rewriting, we will now investigate how to compute our three types of prime implicants (validatory, contrastive and doubt), and the associated complexity.

Validatory Prime Implicants

From Equation (19a), we have that log 𝜙 𝑣 𝑦,𝑦 ′ (𝐸) = 𝐶 𝑣 + 𝑖 ∈𝐸 𝐺 𝑣 (𝑖) and our goal (4) is to find subsets 𝐸 ⊆ ⟦1, 𝑛⟧ such that log 𝜙 𝑣 𝑦,𝑦 ′ (𝐸) ≥ 0. It follows that we want to optimise 𝐸 so that the sum of positive contributions is greater than 𝐶 𝑣 . Finding a smallest prime implicant is then computationally easy, as it amounts to order the 𝐺 𝑣 (𝑖) ′ 𝑠 in decreasing order, and add them until 𝑖 ∈𝐸 𝐺 𝑣 (𝑖) ≥ -𝐶 𝑣 . The whole procedure is summarised in Algorithm 1.

The complexity of Algorithm 1 is linear over the ordered contributions, in number of attributes. Computing the contributions remains easy as the only complexity is to compute the "worst case" vector x 𝑣 , whose components x 𝑣 𝑖 requires |𝑋 𝑖 | = 𝑘 𝑖 evaluations on each dimensions. As sets P are typically polytopes defined by linear constraints, finding the values 𝑝 and 𝑝 amounts to solve linear programs, something that can be done in polynomial time. For some specific cases such as probability intervals [START_REF] Campos | Probability intervals: a tool for uncertain reasoning[END_REF] (induced, e.g., by the classical Imprecise Dirichlet Model [START_REF] Bernard | An introduction to the imprecise dirichlet model for multinomial data[END_REF]), this can even

5Indeed, log 𝑝 𝑖 (x 𝑣 𝑖 | 𝑦) -log 𝑝 𝑖 (x 𝑣 𝑖 | 𝑦 ′ ) < log 𝑝 𝑖 (x 𝑜 𝑖 | 𝑦) - log 𝑝 𝑖 (x 𝑜 𝑖 | 𝑦 ′ ) by definition.
Input: 𝐶 𝑣 ; 𝐺 𝑣 Output: 𝑋 𝑝𝑙 = (𝐸, x 𝑜 𝐸 ): explanation in terms of attribute Order 𝐺 𝑣 in decreasing order, with 𝜎 the associated permutation

𝑖 ← 1 while 𝜙 𝑣 𝑦,𝑦 ′ (𝐸) + 𝐶 𝑣 < 0 do 𝑖 ← 𝑖 + 1 𝐸 ← 𝐸 ∪ {𝜎 -1 (𝑖)} 𝜙 𝑣 𝑦,𝑦 ′ (𝐸) ← 𝜙 𝑣 𝑦,𝑦 ′ (𝐸) + 𝐺 𝑣 (𝜎(𝑖)) end 𝑋 𝑝𝑙 ← (𝐸, x 𝑜 𝐸 ) return (𝑋 𝑝𝑙)
Algorithm 1: Compute first available prime implicants explanation be done in linear time. Therefore, the overall method is polynomial, with a linear pre-treatment over the sum of 𝑘 𝑖 's, followed by a sorting algorithm, after which Algorithm 1 is linear over the number of attributes.

Contrastive Prime Implicants

The case of the contrastive prime implicants is straightforward once we solved the validatory prime implicants. Indeed, as suggested by the similarity between the definitions of 𝜙 𝑣 𝑦,𝑦 ′ and 𝜙 𝑐 𝑦,𝑦 ′ in Equations ( 16a) and (16b) and the definitions of x 𝑣 and x 𝑐 , we almost compute the same thing, the difference being that the role of 𝐸 for 𝜙 𝑣 𝑦,𝑦 ′ is fulfilled by -𝐸 for 𝜙 𝑐 𝑦,𝑦 ′ . We obtain that 𝐶 𝑐 > 0 whereas we had 𝐶 𝑣 < 0, as 𝐶 𝑐 is obtained when we observe the full vector x 𝑜 , and that 𝐺 𝑐 (𝑖) ≤ 0. To use Algorithm 1, we only need to change the while condition to 𝜙 𝑐 𝑦,𝑦 ′ (𝐸) + 𝐶 𝑐 > 0, and the vector 𝐺 𝑣 to be ordered in ascending order.

That such strong duality relations hold in general is unlikely, even if validatory and contrastive explanations are known to be linked in general [START_REF] Ignatiev | On relating'why?'and'why not?'explanations[END_REF].

Prime Implicants of Doubt

From the definition of prime implicant of doubt in Equations (11a) and (11b) we have to investigate simultaneously two problems, one in favour of 𝑦 against 𝑦 ′ and one in favour of 𝑦 ′ against 𝑦. To do so we have two functions 𝜙 𝑑 𝑦,𝑦 ′ and 𝜙 𝑑 𝑦 ′ ,𝑦 , which in the case of the NCC are additive:

log 𝜙 𝑑 𝑦,𝑦 ′ (𝐸) = 𝐶 𝑑 𝑦,𝑦 ′ + ∑ 𝑖 ∈𝐸 𝐺 𝑑 𝑦,𝑦 ′ (𝑖), log 𝜙 𝑑 𝑦 ′ ,𝑦 (𝐸) = 𝐶 𝑑 𝑦 ′ ,𝑦 + ∑ 𝑖 ∈𝐸 𝐺 𝑑 𝑦 ′ ,𝑦 (𝑖).
𝐶 𝑑 𝑦,𝑦 ′ and 𝐶 𝑑 𝑦 ′ ,𝑦 are obtained when we assume observing the "worst case opponents" x 𝑑:𝑦,𝑦 ′ and x 𝑑:𝑦 ′ ,𝑦 , the two vectors the most in favour of 𝑦 against 𝑦 ′ and of 𝑦 ′ against 𝑦. In practice, we then want to find which features of x 𝑜 are sufficient to observe so that both dominance relationships (if they hold for some vectors x 𝑑:𝑦,𝑦 ′ , x 𝑑:𝑦 ′ ,𝑦 , which may not be the case as hinted by Remark 6) are broken, i.e., both 𝑦 ⊁ P, (x 𝑜 𝐸 ,x 𝑑:𝑦,𝑦 ′ -𝐸 ) 𝑦 ′ and 𝑦 ′ ⊁ P, (x 𝑜 𝐸 ,x

𝑑:𝑦 ′ ,𝑦 -𝐸
) 𝑦. This problem can be represented as a 2-dimensional Knapsack where the objects are the features and the two Knapsacks corresponds to the dominance of 𝑦 over 𝑦 ′ and the converse. We obtain the following formulation min

𝑛 ∑ 𝑖=1 𝑥 𝑖 subject to 𝑛 ∑ 𝑖=1 𝑥 𝑖 * 𝐺 𝑑 𝑦,𝑦 ′ (𝑖) ≤ -𝐶 𝑑 𝑦,𝑦 ′ , (20)
𝑛 ∑ 𝑖=1 𝑥 𝑖 * 𝐺 𝑑 𝑦 ′ ,𝑦 (𝑖) ≤ -𝐶 𝑑 𝑦 ′ ,𝑦 , ∀𝑖 ∈ {1, . . . , 𝑛} 𝑥 𝑖 ∈ {0, 1}.
This problem can be solved by using efficient MILP solver, which may provide fast solutions for the average case, even if the problem worst complexity remains NP-hard. The indexes with a non zero associated 𝑥 𝑖 are the components of 𝐸, i.e. are our prime implicants.

NCC with the Imprecise Dirichlet Model

In this section we will present the Imprecise Dirichlet Model [START_REF] Bernard | An introduction to the imprecise dirichlet model for multinomial data[END_REF], which is a classical model of representation of domain of probabilities, and study how the prime implicants will behave in this case.

The main idea of the IDM is to build a cautious interval around a precise probability distribution. Let's note the number of observation of an event 𝑋 by 𝑛 𝑋 , same notations for a conditional event 𝑋|𝑌 by 𝑛 𝑋 |𝑌 and 𝑁 the total number of observation. We obtain that the probability of witnessing 𝑋 is 𝑛 𝑋 𝑁 . We introduce the meta-parameter 𝑠 of the IDM which can be interpreted as a number of "unwitnessed" observations. As these could be or not witnessed for 𝑋 the probability of 𝑋 belongs to the interval

𝑛 𝑋 𝑁 +𝑠 , 𝑛 𝑋 +𝑠 𝑁 +𝑠 .
We can easily see that, in the case of the IDM, the 𝑠 hyper-parameter allows us to go from fully precise (𝑠 = 0) to fully imprecise (𝑠 = ∞), meaning that the monotonicity properties we mentioned so far can easily be checked by modifying its value.

First Experiments

This Section will present an illustrative case based on the data from the Zoo dataset from UCI repository [START_REF] Dua | UCI machine learning repository[END_REF] using the NCC alongside the IDM. To avoid probabilities of 0 we will regularize them by mixing them with a uniform distribution (using a coefficient 𝜖 = 0.05 to weight this uniform). The experiment will be separated in two parts. The first one will focus on trying to answer with a quantitative study to the questions "How do the different implicants behave, in size, absence, number, based on the kind of implicants we search, and on whether we justify a relation consistent with the ground truth ?" and "Is the size of pairwise explanations dependent of the imprecision and the number of predicted classes ?". Second one will illustrate how a discussion with a user could occur based on this data for the different types of explanations.

The Zoo dataset is a classification dataset containing 101 samples of animals with 16 input features and the class. The classes are numbers from 1 to 7 corresponding to Mammal, Bird, Reptile, Fish, Amphibian, Bug and Invertebrate. We used 14 features for classification ( 'feathers': { 𝑓 𝑒, ¬ 𝑓 𝑒}, 'eggs': {𝑒, ¬𝑒}, 'airborne': {𝑎𝑖, ¬𝑎𝑖}, 'aquatic': {𝑎𝑞, ¬𝑎𝑞}, 'predator': {𝑝, ¬𝑝}, 'toothed': {𝑡𝑜, ¬𝑡𝑜}, 'backbone': {𝑏, ¬𝑏}, 'breathes with lungs': {𝑙, ¬𝑙}, 'venomous': {𝑣, ¬𝑣}, 'fins': { 𝑓 𝑖, ¬ 𝑓 𝑖}, 'legs': {0, 2, 4, 5, 6, 8}, 'tail': {𝑡𝑎, ¬𝑡𝑎}, 'domestic': {𝑑, ¬𝑑}, 'at least catsize': {𝑐, ¬𝑐}), all binary except for the number of legs. To have sufficient classification errors, we removed 2 features ('hair' and 'milk') from the original features.

Quantitative Study

We performed this study using a 4-Fold cross-validation with a stratified data separation, due to the samples by class being very unbalanced, e.g. 41 for Mammals and 4 for Amphibians.

Are Validatory and Contrastive explanations size dependent of miss-classification ?

The idea behind this question is to verify the shape of explanations when the observation is well classified against when it is miss-classified. If 𝑦 is the true class, we could expect explanation for an observed dominance 𝑦 ≻ P,x 𝑜 𝑦 ′ that are "true" to differ from observed dominance 𝑦 ′ ≻ P,x 𝑜 𝑦 that are false. In Figure 1, we plot the size of such explanations.

First note that the monotonicity in terms of imprecision are well observed: as 𝑠 increases, the size of validatory and contrastive explanations respectively increases and decreases.

Then, we can see that there is no significant difference between the distributions when the prediction is correct or not, except maybe for a bigger variability in the case of wrong prediction. While further experiments would be needed to confirm this, it seems that the length of explanation is not related to whether they explain a correct or incorrect prediction, suggesting that one would have to check their plausibility. Are Doubt explanation dependent of the number of undominated items? We will now focus on prime implicants of doubt explanations. As said in Section 3.4, incomparabilities may arise from lack of knowledge or from ambiguity about the observed element. A question is then to know whether this affects the length of our explanation. As a proxy, we plotted in Figure 2 the length of implicants explaining incomparabilities against the number of undominated classes, with the idea that this is a reasonable proxy of ambiguity versus lack of knowledge (the more the number of undominated, the more incomparabilities are likely to be due to lack of knowledge). Again, while the Figure 2 does show the expected monotonicity, it seems that the size of pairwise explanation is not especially affected by the final number of classes in the prediction. As we used a proxy, this independence would however have to be confirmed by more precise assessment of whether our incomparability is mainly due to epistemic or aleatoric uncertainty.

Illustrative Explanations

Let us now present some results we can get from the experiments. We will focus on values 𝑠 ∈ {0.5, 1, 2} and on 3 animals: Giraffe, Seal and Tortoise. We will denote NCC𝑠 the corresponding classifier.

Giraffe as a non-ambiguous problem. Described by (¬ 𝑓 𝑒, ¬𝑒, ¬𝑎𝑖, ¬𝑎𝑞, ¬𝑝, 𝑡𝑜, 𝑏, 𝑙, ¬𝑣, ¬ 𝑓 𝑖, 4, 𝑡𝑎, ¬𝑑, 𝑐), the Giraffe is a prototypical example of Mammals, as all NCC0.5, NCC1 and NCC2 classifies it as a Mammal only. To illustrate the validatory prime implicants explanations, we will take a look into the preference "Mammal ≻ P,𝐺𝑖𝑟 𝑎 𝑓 𝑓 𝑒 Bird". For NCC0.5, a sufficient reason to classify the Giraffe as Mammal and not Bird is that it has no feathers (¬ 𝑓 𝑒), does not produce eggs (¬𝑒) and is toothed (𝑡𝑜). With the increasing cautiousness of NCC1 we need to add the fact that the Giraffe has 4 legs to the explanation and for NCC2 we then add that it is not airborne. All advanced reasons correspond to attributes of Mammals and not of Birds. A contrastive explanation showing how "robust" our classification is for NCC0.5 and NCC1 that we change the values of the features feathers, eggs and toothed. So, an animal like the giraffe, but with feathers, laying eggs and no teeth could be either a mammal or a bird.

Seal as an ambiguous animal. Described by (¬ 𝑓 𝑒, ¬𝑒, ¬𝑎𝑖, 𝑎𝑞, 𝑝, 𝑡𝑜, 𝑏, 𝑙, ¬𝑣, 𝑓 𝑖, 0, ¬𝑡𝑎, ¬𝑑, 𝑐), it is classified as a Mammal for NCC0.5, but NCC1 and NCC2 are more cautious by predicting the set {𝑀𝑎𝑚𝑚𝑎𝑙, 𝑅𝑒 𝑝𝑡𝑖𝑙𝑒, 𝐹𝑖𝑠ℎ, 𝐴𝑚 𝑝ℎ𝑖𝑏𝑖𝑎𝑛}. Let us now investigate the comparison between Mammal (the true class) with Fishes.

For NCC0.5 a sufficient reason to classify as a Mammal is that the Seal has lungs, does not produce eggs, is (at least) catsized, is not venomous, has no feathers and has a backbone. Note that this time explanations contain element that support Mammal but can nevertheless be met in fishes as well (e.g., has no feathers). The decision is also less robust, as contrastive explanation shows that flipping one of the features ['lungs', 'eggs', 'catsized', 'venomous'] is enough to make Mammal and Fish incomparable.

When going to NCC1, Mammal ≻≺ P,𝑆𝑒𝑎𝑙 Fish can be explained by the fact that the Seal does not produce eggs, is aquatic, breathes, has fins, has no legs. Interestingly, we can see that the explanation shows that the seal is somehow ambiguous, having some typical features of fishes (𝑎𝑞, 𝑓 𝑖) as well as of Mammals (¬𝑒, 𝑙).

Tortoise as a mistaken animal. Described by (¬ 𝑓 𝑒, 𝑒, ¬𝑎𝑖, ¬𝑎𝑞, ¬𝑝, ¬𝑡𝑜, 𝑏, 𝑙, ¬𝑣, ¬ 𝑓 𝑖, 4, 𝑡𝑎, ¬𝑑, 𝑐), it is wrongly labelled by NCC0.5 and NCC1 as a Mammal rather than a Reptile. NCC2 is much lesser precise and predicts that a Tortoise can be every class except for Fish.

If we investigate the reasons why NCC0.5 believes the Tortoise is a Mammal we obtain the validatory prime implicant not venomous, has 4 legs, is catsized, breathes, is not a predator, has a backbone, is not airborne (for NCC1 we add that it has no feathers, is not aquatic and has no fins.)

The explanation is reasonable but quite long, and does not use the fact that a Tortoise lay eggs (the Platypus being one of the mammal, it is possible for mammals to lay eggs). Also, the Reptile class is poorly represented (4 examples) and most by "serpent like" animals with no legs, pretty venomous, small (so not catsized) and predators.

Finally, when increasing to NCC2, we obtain that the doubt between Reptile and Amphibian is not caused by any feature (empty prime implicant of doubt). This clearly shows that Reptile and Amphibian are indistinguishable "by default" and are underrepresented, as the Amphibian and reptile classes have respectively 3 and 4 learning observations which is too little compared to s=2.

Discussion and perspectives

Considering explanations for imprecise classifier opens up many questions, for instance in relation with the possibility of observing incomparability, or of increasing/decreasing the imprecision of a model. In this paper, we focused on prime implicants, extending notions proposed so far in the precise setting. We introduced three notions of prime implicants in the case of pairwise comparison, answering the questions "Why X is P?", "Why X is P and not Q?" and "Why is X neither P nor Q?". When applying them to the Naïve credal classifier, we obtain that the computations are computationally easy (at least for validatory and contrastive explanations).

Complexity beyond the NCC case While Section 4 showed that the notions of Section 3 could be computed relatively easily for some case, it is clear that applying them in general will present many challenges, as the problem of finding prime implicants with minimal cardinality or of enumerating prime implicants is known to be NP-hard. Coming up with methods to extract such prime implicants from other credal classification methods therefore constitute an important avenue for future research. We are however relatively confident that the notion of (prime) implicants could be used in numerous cases, notably for the following reasons:

• while considering NCC results in an additive structure making the problme of finding prime implicants linear in the number of features, other structural assumptions may also make this task easy. For instance, monotone classifiers [START_REF] Marques-Silva | Explanations for monotonic classifiers[END_REF] and decision trees [START_REF] Izza | On explaining decision trees[END_REF] also exhibit a structure that allows for efficient algorithms in the precise case;

• while finding prime implicants for other classifiers such as random forest can turn out to be quite complex [START_REF] Izza | On explaining random forests with sat[END_REF], relaxing the notion of prime implicants to weaker notions such as being a prime implicant for a majority of trees may allow for efficient, polynomial search methods [START_REF] Audemard | Trading complexity for sparsity in random forest explanations[END_REF];

• many classification problems do not include that many features, meaning that even if the problem of finding or enumerating prime implicants is computationally challenging, one could still use, e.g., ILP formulations of the problem with powerful solver to obtain a solution.

Some perspectives

In the future, we would like to focus on various questions not investigated here, such as for which robust models (e.g., including some dependence statements) do computations remain tractable? What happens with interaction between attributes ? When trying to explain the complete partial order, should we use pairwise or holistic (i.e., prime implicants explaining the non-dominated classes at once) explanations? How do we select what dominance to explain in such a case ? There are also several other explanation mechanisms we could consider [START_REF] Audemard | On tractable xai queries based on compiled representations[END_REF]. We also want to investigate the case with prediction costs. Indeed, we can associate costs to making a prediction 𝑦 ′ when the truth is 𝑦. During this paper we used the usual 0/1 cost for all types of miss-classification.

Finally, we also feel that we have only skimmed the surface of the role of incomparability explanations, in the sense that the operational role and advantage of such explanations still remain to be explored.

Figure 1 :

 1 Figure 1: Explanation length according to prediction truth.Green triangles are mean values.

Figure 2 :

 2 Figure 2: Length of pairwise doubt prime implicants by size of prediction and values of IDM (ball size is normalized with respect to the number of examples having the same number of undominated classes)

© H. Willot, S. Destercke & K. Belahcène.

3Each index of an attribute being associated to an item. 4As we will deal later with joint probabilities and independence, using log transform will allow us to turn products into sums.