K Dang Van
email: dangvan@lms.polytechnique.fr

M H Maitournam

On some recent trends in modelling of contact fatigue and wear in rail

Keywords: Multi-axial fatigue, Moving loads, Shakedown, Finite element analysis

Specific numerical methods for the computational analysis of damage induced in rail by repeated rolling are presented. The calculations of mechanical stabilized states (shakedown, ratchetting) of rail-like structures subjected to moving contact loads are performed using the "stationary methods". An association of 2-D finite element method, Fourier expansion in the longitudinal direction of the rail and steady-state assumption reduces the computational cost of such procedures. These methods constitute the key for the quantitative prediction of fatigue. Three types of damage (low-, high-cycle fatigue and damage) are encountered. Special attention to high-cycle fatigue is paid, through the use of Dang Van multi-axial fatigue criterion. The 3-D simulations of rolling contact and investigation of rail high-cycle fatigue illustrate the applicability of the methodology.

Introduction

Rolling contact fatigue (RCF) is a severe problem for railways companies. Since many decades, the french national railways company SNCF had undertaken researches to understand the conditions of initiation of fatigue cracks which may appear in the rail head. These defects are of different nature, depending on the considered period of time. From the 1960s to the early 1970s, the kidney-shape crackings are among the most frequently encountered defects on the track. They initiate in the rail head, at a depth quite far beneath the running surface (around 7 mm) and propagate downwards in the rolling direction, in a plane making an angle of about 30 • relatively to the vertical. In some circumstances, especially on heavy loaded lines or on some curves, shelling can also be observed. It is associated with severe superficial plastic deformation. Nowadays, squats and head checks are the major defects which affect the rail. Squats are encountered on straight tracks or shallow curves. They initiate in deformed zones of the running surface. When these cracks reach a certain size (around 5 mm) below the surface, they can propage downwards to form progressive transverse cracks which propagate quite fast. That is the reason why their prevention is necessary. Head checks are angle cracks observed on the high rail in curves and on crossing rails. They iniatiate near the rail gauge corner and can join up to induce a gauge corner breaking. The reasons of this difference in the type of observed damages at these different periods are, on one hand, the nature of the rail steel (in the 1960s, the rail heads contained many inclusions due to the use of Thomas process steels) and, on the other hand, the increasing of load and power per axle.

To understand the circumstance of the occurence of these defects, researches have been undertaken in order to model their initiation. It is a complex problem which necessitates to overcome two main difficulties.

(i) The first one is the evaluation of the thermomechanical quantities in the rail induced by repeated rolling contact; because of the small size of the contact area, the stresses are beyond the elastic limit, so, it is necessary to take account of the plastic deformation and residual stresses resulting from the sequence of repeated rolling. Moreover, rolling contact is a 3-D problem with friction and complex partial slip. It is well-known that different limit states may be obtained depending on the intensity of the wheel load, the friction coefficient and the material properties. These limit states are: elastic state, elastic shakedown state (the limit cycle is elastic, but early plastic deformation induces a fixed residual stress pattern), plastic shakedown and ratchetting. (ii) The second difficulty is the prediction of damage (this latter is closely linked to the mechanical limit state). This necessitates the use of multi-axial fatigue criteria able to deal with complex multi-axial loading situations. In fact the local stress and strain fields are completely tri-dimensional with no fixed directions during their evolution.

To solve the problem (i), we have developped powerful computational tools capable of determining the stabilized state of any kind of inelastic structures subjected to repeated moving loads, directly or by following the evolution pass-by-pass. The presentation of these numerical procedures can be found in [START_REF] Van | Steady-state flow in classical elastoplasticity: application to repeated rolling and sliding contact[END_REF][START_REF] Van | Elastoplastic analysis of repeated moving contact: application to railways damage phenomena[END_REF][START_REF] Van | Thermomechanical state near rolling contact area[END_REF], where they are validated by comparisons with analytical results obtained by Johnson [START_REF] Johnson | The application of shakedown principles in rolling and sliding contact[END_REF][START_REF] Johnson | Plastic flow, residual stress and shakedown in rolling contact[END_REF][START_REF] Johnson | Contact Mechanics[END_REF], Bower and Johnson [START_REF] Bower | The influence of strain hardening on cumulative plastic deformation in rolling and sliding line contact[END_REF], Hearle and Johnson [START_REF] Hearle | Cumulative plastic flow in rolling and sliding line contact[END_REF], Johnson and Jefferis [START_REF] Johnson | Plastic flow and residual stresses in rolling and sliding contact[END_REF]. Different applications to rail were presentated in [START_REF] Van | Elastoplastic analysis of repeated moving contact: application to railways damage phenomena[END_REF].

For the problem (ii) we use, in the case of high-cycle fatigue, the multi-axial fatigue criterion proposed by Dang Van, of which fundations are presented, for instance, in [START_REF] Van | Introduction to fatigue analysis in mechanical design by the multi-scale approach[END_REF][START_REF] Van | Macro-micro approach in high cycle multi-axial fatigue[END_REF][START_REF] Van | On structural integrity assessment for multi-axial loading paths[END_REF][START_REF] Ballard | High cycle fatigue and a finite element analysis[END_REF]. Its main attribute is that it can be easily identified by classical fatigue uniaxial laboratory tests like repeated tension or torsion. This criterion is essentially based on elastic shakedown hypothesis at all scales of material description (macroscopic and mesoscopic scales). The Manson-Coffin law is considered in the case of low-cycle fatigue, especially for the prediction of shelling initiation.

The purpose of this paper is to present some main methods and computation procedures, and to show some new results we have obtained in the modelling of the different defects mentioned above. Finally, we insist more on the modelling of the defect named squat.

2. Numerical methods for the determination of the limit thermomechanical state and fatigue

Presentation of the problem

The determination of the thermomechanical fields induced by repeated rolling/sliding contact is the first step for the understanding and the prediction of fatigue damage phenomena which may occur in rail head. However, this task presents many difficulties due to load level which induces plastic deformation. The simulation of such processes by using classical existing finite element softwares necessitates incremental translations of the loading. This way of calculating (used in the late 1980s) is time-consuming and cumbersome. A simplified approach based on the proposal of Zarka et al. [START_REF] Zarka | A New Approach to Inelastic Analysis of Structures[END_REF] was used by Sakae and Keer [START_REF] Sakae | Application of a direct method for a non-linear-kinematic hardening material under rolling-sliding line contact: constant ratchetting rate[END_REF] and Keer and coworkers [START_REF] Yu | A simplified direct method for cyclic strain calculation: repeated rolling-sliding contact on a case hardened half space[END_REF][START_REF] Yu | A direct analysis of two-dimensional elastic-plastic rolling contact[END_REF], nevertheless, this approach used an approximate estimation of the plastic deformation. It is the reason why we have proposed another computing procedure for the evaluation of the thermomechanical state induced by repeated rolling contact. This method, valid for 2-D and 3-D problems, relies on the steady-state assumption in the moving contact reference. It leads to the two following numerical procedures for the calculations of stresses and strains either for an arbitrary number of loading passes or directly for the stabilized state (elastic shakedown, plastic shakedown or ratchetting): (i) the pass-by-pass stationary method (PPSM) for a single load pass; this method is exact and does not suffer from the limitations of previous rolling contact finite element analyse; the calculation time for one pass is approximatively the same as the one obtained in classical plasticity for one step of translation of the contact loading; (ii) the direct stationary method (DSM) for a repeated moving load, this method goes straight to the steady-state. These procedures were extended in the case of thermomechanical coupling realized by the presence of thermal expansion in the mechanical problem and of heat sources (due to inelastic deformation, thermoelastic coupling and interfacial dissipation mechanism, like friction and micro-slip) in the thermal problem [START_REF] Van | Elastoplastic analysis of repeated moving contact: application to railways damage phenomena[END_REF]. They led to the evaluation of the temperature field in the vicinity of the contact area.

Principle of the stationary methods

The 'stationary methods' used in this paper to determine the mechanical state due to repeated rolling, and their finite elements implementation have already been published in [START_REF] Van | Steady-state flow in classical elastoplasticity: application to repeated rolling and sliding contact[END_REF][START_REF] Van | Elastoplastic analysis of repeated moving contact: application to railways damage phenomena[END_REF]. Nevertheless, for the convenience, a summary of the main features of these methods is given below.

Consider a rail subjected to contacts loads (let us say normal and tangential pressures) moving with a velocity V = V e z (Fig. 1). Assuming steady-state in a reference frame moving with the loads, the time derivative of any material quantity B becomes:

Ḃ = -V B, z (1)
The first idea consists in writing the equations governing the mechanical problem in the moving loads reference. Thus, one obtains: equations of motion

div σ = ρV 2 u, zz (2)
constitutive laws

σ = L : e + σ 0 (3)
A k = Z : α k (4) = e + p (5)
-,

p z = Λ ∂f ∂σ f (σ , A k ) ≤ 0, Λ ≥ 0, Λf(σ , A k ) = 0 (6) α k , z = Λ ∂f ∂A k (7)
Λ is the plastic multiplier, f the yield function and the plastic potential (f = 0 defines the elastic domain), σ 0 the initial stress, α k are internal parameters and A k their associated forces linked here by the tensor Z, L is the fourth-order tensor of elastic coefficients. Thanks to (Eq. ( 1)), time derivatives have been replaced by space derivatives in Eqs. ( 2), ( 6) and [START_REF] Johnson | Plastic flow, residual stress and shakedown in rolling contact[END_REF]. The integrations of Eqs. ( 6) and ( 7) are performed in space, along the direction of the motion of the loads; so that, numerically neither the load, nor the structure are translated. Two kinds of material behaviour have been used for the rail: a von Mises elastic-plastic material with linear-kinematic and isotropic hardening of which yield function f is given by:

f (σ , C p , ¯ p ) = 1 2 (σ -C p ) : (σ -C p ) -k e (¯ p ) ( 8
)
where C is the hardening modulus, k e is the shear yield stress, and ¯ p is the cumulated plastic deformation, and the non-linear-kinematic hardening law proposed by Bower [START_REF] Bower | Cyclic hardening properties of hard-drawn copper and rail steel[END_REF] to evaluate the ratchetting rate per cycle. The second crucial feature of our methods is the direct numerical search of the possible stabilized mechanical state due to repeated rolling (the stress field is periodic, and the plastic strain also in the absence of ratchetting).

The two numerical procedures for the determination of the stabilized state are derived [START_REF] Van | Steady-state flow in classical elastoplasticity: application to repeated rolling and sliding contact[END_REF]. The first one is the PPSM for the calculation of a single pass. Computing the successive passes by this method, allows to find the stabilized state and its characteristics (number of cycles before reaching the stabilized state, residual stresses, plastic deformations, etc.). This stabilized state can, of course, be ratchetting. In such a case, the ratchetting rate is immediately deduced as the increment of surface displacement or plastic strains caused by a pass. The second one is the DSM for the direct determination of the stabilized state, in the case of shakedown (elastic or plastic). The ratchetting is indicated by a non-convergence of the algorithm.

Solution methods

These methods were programed using classical finite elements method in 2-D and 3-D. For the 3-D rail problem, in order to reduce the computational cost, it has been necessary to use a combination of a 2-D finite element analysis (in the section of the rail) and a Fourier expansion along the longitudinal direction of the rail (z-direction). The obtained solution is fully tri-dimensional. The displacement is expanded (so is the loading) in the following way:

u(x, y, z) = N n=0 u c n (x, y) cos nπ z L + u s n (x, y) sin nπ z L (9)
where the number N is chosen sufficiently large to allow a good reconstruction of the input prescribed loading (contact stress distributions). Comparisons of shakedown limits given by numerical solutions and analytical ones in the case of an elastic-plastic half-space subjected to moving hertzian pressure, have also allowed us to validate the choice of N . L is the length of the considered portion of the rail. It can be a distance between two contacts (two wheels). The 3-D problem is solved iteratively, each iteration consisting in an elastic solution and a determination of plastic deformation (and internal variables). The Fourier expansion allows to reduce the elastic 3-D problem to a finite sum of 2-D finite element problems over a transverse section of the rail. Each 2-D problem is the calculation of a term u c n (x, y) or u s n (x, y) of the expansion. A re-combination of the displacements per harmonic is performed to obtain the total displacement. This latter is used to calculate the plastic deformation as in the preceding sub-section. The resulting plastic forces are then expanded for the elastic solution.

Illustrative examples

These methods allow the computation of strains and stresses in the rail due to repeated moving contacts and the quick determination of the nature of the stabilized state (elastic shakedown, plastic shakedown or ratchetting). Such analyses of repeated contacts using shakedown principles were carried out in 2-D [1,2] and compared with semi-analytical methods, by Johnson [4,[START_REF] Johnson | Plastic flow, residual stress and shakedown in rolling contact[END_REF][START_REF] Johnson | Contact Mechanics[END_REF], Bower and Johnson [START_REF] Bower | The influence of strain hardening on cumulative plastic deformation in rolling and sliding line contact[END_REF], Hearle and Johnson [START_REF] Hearle | Cumulative plastic flow in rolling and sliding line contact[END_REF], Johnson and Jefferis Fig. 1. An example of a rail mesh and a moving centered spherical pressure distribution. Fig. 2. Contours of equivalent plastic strains in the rail after one pass, for a spherical pressure distribution; a = 10 mm; P 0 /k e = 6.7; µ = 0. Fig. 3. Contours of equivalent plastic strains in the rail head at the stabilized state, for a spherical pressure distribution; a = 10 mm; P 0 /k e = 6.7; µ = 0. [START_REF] Johnson | Plastic flow and residual stresses in rolling and sliding contact[END_REF]. We present below some 3-D calculations, to show the applicability of the method. The considered rail (UIC 60 profile) is as shown in Fig. 1. In full 3-D calculations the used mesh is the one on the figure. When using the method associating finite elements and Fourier expansion, only the mesh of a transverse section of the rail is used. The load is a repeatedly moving hertzian pressure (circular contact area with radius a, maximum hertzian pressure P 0 ). Dynamic effects are neglected. The characteristics of the considered material are: E = 210 GPa; ν = 0.3, k e = 237 MPa; C = 20 GPa. Fig. 2 represents the equivalent plastic strain obtained during the first pass of the contact (P 0 /k e = 6.7; a= 10 mm and 2a is the length of the refined zone). Fig. 3 represents the same quantity in the stabilized state which is obtained directly, without simulating the successive passes. One can see that this stabilized state is a plastic shakedown: the plastic deformation is the same up-and down-stream of the contact but varies under the contact.

Numerical methods for fatigue treatment

Two kinds of fatigue criteria are used to predict crack initiation in the rail. When the rail suffers cyclic plasticity (plastic shakedown) or ratchetting low-cycle fatigue is likely to occur. For this kind of fatigue, we use the Manson-Coffin law to predict the number of cycle to crack initiation. In the case of elastic shakedown or elastic behaviour of the rail, high-cycle fatigue is likely to occur and Dang Van criterion [START_REF] Van | Introduction to fatigue analysis in mechanical design by the multi-scale approach[END_REF][START_REF] Van | Macro-micro approach in high cycle multi-axial fatigue[END_REF][START_REF] Van | On structural integrity assessment for multi-axial loading paths[END_REF] is used. The Dang Van multi-axial high-cycle fatigue criterion is based on a multi-scale approach which assumes that elastic shakedown happens before crack initiation. The considered two scales are: (i) a macroscopic scale characterized by an arbitrary elementary volume surrounding the point where fatigue analysis is made and representing for instance an element of finite element mesh (it is the usual scale considered by engineers); (ii) a mesoscopic scale corresponding to subdivision of the previous volume; the stress tensor at this scale results from the macroscopic one and the local residual stresses due to local inelastic deformation. Thanks to the shakedown assumption at the local scale, it is possible to estimate the local stress cycle from the macroscopic stress cycle. The criteria is then expressed as an inequality related to the mesoscopic stresses at all instants t of the cycle, so that damaging load can be precisely characterized. The criterion used is expressed as: tests. For instance, they are related to classical experimental fatigue strengths f -1 (in alternate bending) and t -1 (in alternate twisting) by:

a = t -1 -(f -1 /2) (f -1 /3) , b = t -1 (10)
Practically, the fatigue resistance of a structure is checked point-by-point, using two ways: These two representations are used in the section devoted to numerical analysis of rolling contact to predict the high-cycle fatigue behaviour of rails. The predictions of kidney-shape crack, squat and head check are based on this criterion.

•

Application to analysis of some rail damages

Kidney-shape cracking, shelling and wear

The 2-D calculations were performed to evaluated the stabilized state and the residual stress pattern at different depths in the rail. By that way we obtained the local stress cycle for different conditions of traffic. Application of the adequate fatigue criterion allows to predict the locus of the crack initiation in relation with the load parameters. The methodology was presented in [START_REF] Van | Elastoplastic analysis of repeated moving contact: application to railways damage phenomena[END_REF]. The results obtained are summarized in the fatigue map presented as shown in Fig. 4. This type of fatigue corresponds to the region (1) of the map: initiation in depth, high cycle fatigue in the elastic and elastic shakedown zones. The region (2) corresponds to the surface damage. When the plastic shakedown occurs (plastic deformation cycles), this regime leads shelling which is relevant to low cycle fatigue. In the case of high friction (braking zone) surface plastic flow may be severe, so that wear phenomena are predominant.

This methodology is now applied in the 3-D case. For illustration, we considerer the same structure and material as in Section 2.4. The moving load is characterized by a=10 mm and P 0 /k e = 4.25. The fatigue limits of the rail steel are: f -1 = 460 MPa and t -1 = 270 MPa. The stabilized mechanical state is an elastic shakedown. Fig. 5 shows the result of the application of the Dang Van multi-axial fatigue criterion: the most critical point is in depth, which is typical of kidney-shape crack initiation.

Analysis of squat and head checking 4.2.1. The 3-D rolling contact problem

The 2-D analysis fails in explaining the occurrence of squat cracking. In this case, the microscopic observations shows that superficial contact effects have to be taken into account. In Fig. 6 one can observe different fibers orientations (corresponding to severe local superficial plastic flow) depending on the position on the rail head surface and squat damage appears in a precise place. Dynamic computations of rail-wheel system performed by INRETS and based on Kalker method as in [START_REF] The | Normal und Tangential Spanningsberechnung beim Rollen den Kontakt für Rotationskörper mit Michtelliptischen Kontakt Flächen[END_REF], shows complex evolutions of the rail-wheel contact with different shapes of the contact area, pressure and spin distributions. The contact area changes from one to two separate areas (Fig. 7). To simulate this phenomena, we choose a simplified scenario consisting in sequences of eight different passes of different contact loading. Each one corresponds to a particular position of the wheel on the rail. The contact stress distributions consist in normal pressure (maximum value is 1.6 GPa) and important tangential stresses (essentially resulting from spin effect, the value of which can reach 500 MPa).

Evaluation of the mechanical state in the rail

The 3-D numerical calculations using the PPSM presented in Section 2 are conducted. The method which combines Fourier expansion and finite elements is used. The mesh of the tranverse section of the rail can be seen in Fig. 8. The rail steel is considered as a von Mises elastic-plastic material with linear-kinematic hardening. The material constants are: E = 210 GPa; ν = 0.3; k e = 550 MPa; C = 20 GPa. A distribution of the stresses in the central rail section, after the pass of the second type of loading (elliptical distribution of the normal and tangential stresses due to spin effect and friction: Fig. 7, second from the right) is as shown in Fig. 8.

Fatigue prediction and comparison with observations

Fatigue analysis is based on the use of the Dang Van multi-axial criterion. It was performed on and thanks to a specific fatigue analysis software. The fatigue limits of the used rail steel are: f -1 = 450 MPa and t -1 = 270 MPa. The stresses are calculated at the stabilized state reached after passes of the contacts sequence described previously. The loading path at three different points (gauge side, central zone, field side) of the running surface are as shown in Fig. 9. It can be seen that fatigue cracks are likely to occur in the central zone.

Conclusion

We have developed specific numerical procedures allowing the determination of stresses and strains in the rail subjected repeated moving contacts, directly in the stabilized state or by following their evolution pass-by-pass. To show its flexibility, the 3-D simulations have been performed in the case of elastic-plastic material with a linear-kinematic hardening. By coupling this method with appropriate fatigue criteria, a good tool for the prediction of (high or low cycle) fatigue crack initiation can be obtained. The 2-D calculations have been sufficient to predict the kidney-shape cracks occurrence, but only the 3-D calculations give the main features of the apparition of squat.

Fig. 5 .

 5 Fig. 5. Contours of Dang Van criterion (α = max t (τ (t)ap(t) -(b/b))) and loading path are the most critical point.

Fig. 4 .

 4 Fig. 4. Numerical shakedown and fatigue map for line contact and full sliding.

  The first one is the representation of the loading path (p(t), τ (t)) at each point in the (p, τ ) diagram. In this diagram, two constants, a and b, define a safety domain (no fatigue cracks) which is the region below the line (τ + ap = b). If the loading path at each point is entirely in the safety domain, there is no fatigue crack, otherwise fatigue damage occurs. • The second one is the evaluation at each point of the quantity α = max t {(τ (t) + ap(t) -b)/b}. Positive value of α means occurrence of fatigue crack.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Observations of the orientations of the fibers in the running zone (from IRSID): squat appears in the central zone.

Fig. 8 .

 8 Fig. 8. Contours of von Mises equivalent stresses near the contact area, for the second pass of the loading.

Fig. 9 .

 9 Fig. 9. Fatigue prediction: loading paths at different points of the rail surface.

Acknowledgements

The work has been partly supported by the SNCF. The authors are grateful to all the IDR2 partners, especially IRSID and INRETS.

K. Dang

 Van


Marc Fischer

On the usefulness of imprecise Bayesianism in chemical kinetics

Keywords: Imprecise probability, parameter estimation, Bayes' factors, Ockham's razor, Robust Bayesian analysis, chemical kinetics

Bayesian methods are growing ever more popular in chemical kinetics. The reasons for this and general challenges related to kinetic parameter estimation are shortly reviewed. Most authors content themselves with using one single (mostly uniform) prior distribution. The goal of this paper is to go into some serious issues this raises. The problems of confusing knowledge and ignorance and of reparametrisation are examined. The legitimacy of a probabilistic Ockham's razor is called into question. A synthetic example involving two reaction models was used to illustrate how merging the parameter space volume with the model accuracy into a single number might be unwise. Robust Bayesian analysis appears to be a simple and straightforward way to avoid the problems mentioned throughout this article.

Introduction

Chemical kinetics is the systematic study of the speed of chemical reactions [START_REF] James H Espenson | Chemical kinetics and reaction mechanisms[END_REF]. It encompasses a wide range of fields such as combustion [START_REF] Fischer | Numerical optimisation for model evaluation in combustion kinetics[END_REF], homogeneous catalysis [START_REF] Behr | Applied homogeneous catalysis[END_REF], heterogeneous catalysis Thomas and Thomas (2014), gassolid reactions [START_REF] Favergeon | A mechanism of nucleation during thermal decomposition of solids[END_REF], and biochemistry Cornish-Bowden and Cornish-Bowden (2012), to name but a few. One of its main goals is to predict the concentrations of chemical species as a function of space and/or time. A reaction rate can usually be expressed as r = k(T

) n ∏ i=1 C m i i
, where C i is the concentration of the i-th species, m i is the reaction order with respect to species i and k(T ) is the reaction rate coefficient. The latter can often be expressed as a function of the temperature T through the Arrhenius law according to which k(T ) = AT n exp(-Ea RT ) with A being the pre-exponential factor, n the temperature coefficient, Ea the activation energy, and R = 8.314 kg m 2 s -2 mol -1 K -1 the ideal gas constant. It might be necessary to consider more complex kinetic laws in specific fields such as gassolid reactions [START_REF] Favergeon | Surface nucleation and anisotropic growth models for solid-state reactions[END_REF]. In the most general case, there is a complex interaction between physics and chemistry which results in complicated systems of partial differential equations which depend on both physical and chemical unknown parameters. Chemical kinetic studies are generally designed in such a way as to simplify the mathematical description as much as possible, so that in the ideal case it becomes a system of Ordinary Differential Equations (ODE) with respect to time (e.g. in a homogeneous batch reactor [START_REF] Levenspiel | Chemical Reactor Omnibook-soft cover[END_REF] or a shock tube [START_REF] Davidson | Interpreting shock tube ignition data[END_REF] or with respect to one spatial dimension (as in the case of plug flow reactors [START_REF] David | Mixing and chemical reaction in continuous combustion[END_REF]).

At the beginning, chemical kinetics was a mostly empirical science. Modelling was limited to the estimation of kinetic parameters under conditions where only one or two reactions take place, thereby allowing for an analytical solution of the ODE system Hughes et al. (1936). With the increase in computational power, software have been developed to numerically solve the differential equation systems so that complex situations where hundreds or thousands of reactions compete with one another can be simulated [START_REF] Lutz | Senkin: A fortran program for predicting homogeneous gas phase chemical kinetics with sensitivity analysis[END_REF]. The systematic use of optimisation methods to estimate and adjust kinetic parameters is a rather recent development [START_REF] Fischer | Combustion chemistry and parameter estimation[END_REF].

There has also been a growing realisation over the last decade that having a model which nicely fits measurements cannot be the end of the story. Authors such as Frenklach Frenklach (2007b) forcefully showed that there exists a great number of competing reaction mechanisms with strongly different parameter values which yet can reproduce the experimental data at hand equally well. Reliable predictions of important quantities (such as the turnover of an enzymatic reaction or the concentration of a toxic pollutant) can only be achieved if the uncertainty of the model parameters is correctly taken into account Sheen and [START_REF] David | Combustion kinetic modeling using multispecies time histories in shock-tube oxidation of heptane[END_REF]. Like in other scientific fields, the approaches to the estimation of the parameters and their correlated uncertainties can be broadly divided into two camps, namely the frequentist and the Bayesian one, which are often (somewhat misleadingly) referred to as "determinist" and "stochastic" in the chemical kinetic community Braman et al. (2012b) The aim of the present paper is to explain why the use of precise Bayesian methods can be very problematic in situations where only few data are available as they can blur the distinction between knowledge and ignorance and how robust Bayesian analysis can help practionners avoid these difficulties. It is organised as follows. In Section 2, the different approaches are presented in the context of chemical kinetics. In Section 3, it is argued that one of the main differences between frequentism and precise Bayesianism is the presence of an Ockham factor within the Bayes factor. In Section 4, a synthetic concentration profile was used to discriminate between two reaction models and to illustrate how a naive reliance on a single (uniform) prior can lead to misleading results. The conclusion and an outlook are given in Section 5.

Frequentist, precise and imprecise Bayesian approaches

Frequentism is the view that probabilities can only be meaningfully assigned to potentially repeatable events [START_REF] Mayo | Frequentist statistics as a theory of inductive inference[END_REF]. In parameter estimation problems in chemical kinetics, the (most widely used) frequentist approach consists of estimating the feasible set of parameter values k (from a model M) which are compatible with the n available experimental data e given their uncertainty and random fluctuations [START_REF] Russi | Sensitivity analysis of uncertainty in model prediction[END_REF]; Frenklach (2007a); You et al. (2011). If we suppose that e is the realisation of the random variable vector X, for a given confidence coefficient α, the feasible set is given by

F={k | p(|X i (m i (k))-m i (k)|≥|e i -m i (k)|)≥α, i∈[|1;n|]}
To put it in modern English, this means that the observed discrepancies between model and experiment must be small enough to be accounted for by the natural fluctuations of X i around its expected value E(X i ) = m i (k). Most of the time, we suppose that the residuals follow a normal distribution (i.e. X i -E(X i ) ∼ N(0, σ i )) and are independent on one another. This naturally leads to the following definition

F ε = {k | |m i (k) -e i | ≤ εσ i }, i ∈ [|1; n|]
with σ i being the standard deviation of the i-th measurement. While such a feasible set may not have good frequentist properties, it is popular among chemical kineticists because it ensures that the model is compatible with each experimental target You et al. (2011). An example of a feasible set involving three reaction coefficients of a mechanism describing the combustion of methane CH 4 Smith et al. (Website, last access on 11.11.2014) can be seen in Figure 1. The possible values of the parameters are constrained by two experimental data sets the model has to match. As the number of experimental data increases, the feasible set becomes smaller and smaller. This, in turn, also narrows down the range of predictions of important quantities, such as heat production or the concentration of a pollutant [START_REF] Russi | Uncertainty quantification: Making predictions of complex reaction systems reliable[END_REF]. The likelihood of the measurements given one set of parameter values is usually computed under the (often implicit) assumption that the errors are normally distributed [START_REF] Turányi | Determination of rate parameters based on both direct and indirect measurements[END_REF]; Sheen and [START_REF] David | Combustion kinetic modeling using multispecies time histories in shock-tube oxidation of heptane[END_REF]. It is worth noting, however, that this remains a hypothesis that is very seldom put to the test, as the experiment is generally only repeated one or two times (if at all), which is clearly not enough to validate it [START_REF] Rj Cvetanovic | Evaluations of the mean values and standard errors of rate constants and their temperature coefficients[END_REF]. This is very unfortunate, as the errors stemming from not representing correctly the experimental fluctuations can have a deep impact on the parameter uncertainty Braman et al. (2012a). What is more, they can accumulate and strongly distort the results of the parameter estimation when hundreds of reactions and thousands of experiments are considered.

According to Bayesianism , both the measurements and the parameter values can be characterised through probability distributions [START_REF] Talbott | Bayesian epistemology[END_REF]. Based on initial probabilities (the so-called priors), posterior probabilities are computed by using Bayes theorem upon seeing new data. In the case of continuous variables, the relationship is given by

f (k | e) = L(e | k) f 0 (k) k∈K L(e | k ) f 0 (k ) dk
where f 0 (k) is the prior probability density, f (k|e) is the posterior probability density given the measurements and L(e | k) is the likelihood. For example, the posterior probability distribution of the chemical kinetic parameters of the reaction C 12 H 26 + 25 2 O 2 → 12CO + 13H 2 O are shown in Figure 2. If two models M 1 and M 2 must be compared, the Bayes factor Goodman (1999) is usually employed for finding the best one. It is defined as the ratio of the total

B = P(e|M 2 ) P(e|M 1 ) = L(e|k 2 , M 2 ) f 2,0 (k 2 |M 2 ) dk 2 L(e|k 1 , M 1 ) f 1,0 (k 1 |M 1 ) dk 1
In the field of chemical kinetics, the Bayesian approach has been applied to problems as diverse as the synthesis of pharmaceutical products [START_REF] Overstall | Bayesian prediction for physical models with application to the optimization of the synthesis of pharmaceutical products using chemical kinetics[END_REF], atmospheric reactions [START_REF] Miki | Probabilistic models and uncertainty quantification for the ionization reaction rate of atomic nitrogen[END_REF], combustion in a shock tube [START_REF] Long | Optimal design and model validation for combustion experiments in a shock tube[END_REF], and soot formation [START_REF] Alexander J Josephson | Modeling soot oxidation and gasification with bayesian statistics[END_REF], to name but a few. Somewhat surprisingly, most authors [START_REF] Plessis | Calibration of the co + oh -> co2 + h rate constant with direct and indirect information[END_REF]; [START_REF] Urzay | Uncertainty-quantifying models for chemicalkinetic rates[END_REF]; [START_REF] Niemi | Efficient bayesian inference in stochastic chemical kinetic models using graphical processing units[END_REF][START_REF] Bh Henderson | Combining bayesian methods and aircraft observations to constrain the ho.+ no 2 reaction rate[END_REF][START_REF] Holtzer | Temperature dependence of the folding and unfolding kinetics of the gcn4 leucine zipper via 13cα-nmr[END_REF][START_REF] Alexander J Josephson | Modeling soot oxidation and gasification with bayesian statistics[END_REF]; [START_REF] Miki | On the use of bayesian approach, experimental data, and artificially designed data, for the identification of missing reactions[END_REF][START_REF] Miki | Probabilistic models and uncertainty quantification for the ionization reaction rate of atomic nitrogen[END_REF][START_REF] Miki | Using bayesian analysis to quantify uncertainties in the h+ o2= oh+ o reaction[END_REF][START_REF] Miki | Systematic validation of non-equilibrium thermochemical models using bayesian inference[END_REF]; [START_REF] Mosbach | Iterative improvement of bayesian parameter estimates for an engine model by means of experimental design[END_REF][START_REF] Mosbach | Bayesian error propagation for a kinetic model of n-propylbenzene oxidation in a shock tube[END_REF] 2017); [START_REF] Wang | Combustion kinetic model uncertainty quantification, propagation and minimization[END_REF] do not explain why one should use Bayesian methods instead of frequentist ones in chemical kinetics. Several authors mention the higher simplicity, intuitiveness and appealing nature of the Bayesian approach [START_REF] Overstall | Bayesian prediction for physical models with application to the optimization of the synthesis of pharmaceutical products using chemical kinetics[END_REF][START_REF] Peter | Model selection: an overview of practices in chemical engineering[END_REF]. According to [START_REF] Sargsyan | Embedded model error representation for bayesian model calibration[END_REF], frequentism is only concerned with parameter values and ignores possible mistakes in the model assumptions. Rodionova et al. pointed out that Bayesianism readily allows researchers to divide a complex problem into a series of simpler ones [START_REF] Oe Rodionova | On one method of parameter estimation in chemical kinetics using spectra with unknown spectral components[END_REF]. According to [START_REF] Bell | A bayesian approach to calibrating hydrogen flame kinetics using many experiments and parameters[END_REF], unlike frequentist methods which do not go beyond delivering a best fit with error bars, Bayesian methods determine the set of all parameter values compatible with the data. This argument is questionable, though, as there are frequentist methods (such as the feasible set approach) that have been precisely developed in order to deal with these problems in chemical kinetics Frenklach (2007a) In most cases, however, the only information we have about a kinetic parameter is that it belongs to some interval Frenklach (2007a); [START_REF] Urzay | Uncertainty-quantifying models for chemicalkinetic rates[END_REF]; [START_REF] Braman | Bayesian analysis of syngas chemistry models[END_REF]; [START_REF] Hsu | Bayesian framework for building kinetic models of catalytic systems[END_REF]. The appropriate choice of prior probabilities in such a situation of complete ignorance appears to be one of the most problematic aspects of classical Bayesianism when understood as a universal framework [START_REF] Talbott | Bayesian epistemology[END_REF], [START_REF] Staley | An Introduction to the Philosophy of Science[END_REF]. The principle of indifference (POI) stipulates, in its most general form, that one should assign an equal plausibility to each outcome of a situation if one has no grounds for preferring one of them [START_REF] Norton | Ignorance and indifference[END_REF]. Applied to point probabilities, it naturally leads to a uniform distribution [START_REF] Norton | Ignorance and indifference[END_REF]. This, however, blurs the fundamental distinction between warranted knowledge and ignorance. If we know absolutely nothing about a coin, we cannot assume that the probability of its landing odds is 0.5 as if we knew through careful measurements that it is perfectly poised [START_REF] Walley | Towards a unified theory of imprecise probability[END_REF]; [START_REF] Salmon | The Foundations of Scientific Inference[END_REF]. What is more, let us suppose that the only thing we know about a pre-exponential factor A in chemical kinetics is that it belongs to the interval [10 10 ;10 16 ]. A straightforward application of the principle of indifference leads to p 1 = p(A ∈ [10 12 ;10 14 ]) = (10 14 -10 12 )/(10 16 -10 10 ) = 0.0099. Now, if we are completely ignorant about A, we are also ignorant about log 10 (A). However applying the principle of indifference to it leads to p 2 = p(log 10 (A) ∈ [12; 14]) = p(A ∈ [10 12 ;10 14 ])= (14 -12)/(16 -10) = 0.33333 = 33.67p 1 . The consideration of other reparametrisations can lead to an endless number of such paradoxes [START_REF] Norton | Ignorance and indifference[END_REF]. According to [START_REF] Norton | Ignorance and indifference[END_REF], this demonstrates the fact that a state of complete ignorance cannot be represented through a single probability distribution. This issue has led Jeffreys to develop a prior which is insensitive to the parametrisation of the problem [START_REF] Jeffreys | An invariant form for the prior probability in estimation problems[END_REF]. While Jeffreys' prior avoids the charge of arbitrariness, it cannot be considered a genuine representation of ignorance as it (often strongly) favours some parameter values over others [START_REF] Norton | Ignorance and indifference[END_REF]; Kass and [START_REF] Wasserman | The selection of prior distributions by formal rules[END_REF]. The impossibility of representing a state of complete ignorance through a single probability distribution is thus one of the main shortcomings of precise Bayesianism. This is where imprecise probability (forming the basis of robust Bayesianism) can be fruitfully employed [START_REF] James | Robust bayesian analysis: sensitivity to the prior[END_REF]. The probability of an event (such as that of a coin landing odds) is no longer always represented by a single value but by a probability interval (which would be [0;1] in the case of complete ignorance) [START_REF] Walley | Towards a unified theory of imprecise probability[END_REF]. The prior uncertainty about the values of a parameter is no longer described through a single distribution but through a family of distributions which must be large enough to represent ignorance [START_REF] Hall | Uncertainty-based sensitivity indices for imprecise probability distributions[END_REF]. The difference between the posteriors corresponds to the imprecision which can be reduced by increasing the size of the experimental data set.

Whilst robust Bayesian methods are now considered a respectable approach in statistics [START_REF] James O Berger | An overview of robust bayesian analysis[END_REF]; [START_REF] Radanovic | A robust bayesian truth serum for non-binary signals[END_REF], they are very rarely used in chemical kinetic studies, although a few authors are aware of the fact that the posterior can be very sensitive to the prior [START_REF] Braman | Bayesian analysis of syngas chemistry models[END_REF]; [START_REF] Galagali | Bayesian inference of chemical reaction networks[END_REF]. Instead, almost all chemical kineticists rely on a single prior which is most often uniform Plessis ( 2013 [START_REF] Braman | Bayesian analysis of syngas chemistry models[END_REF]. In that respect, they have implicitly embraced part of the philosophy of imprecise probability as the total information about the problem is no longer contained in a single probability distribution. Nevertheless, a real robust Bayesian analysis would have necessitated a much greater number of priors [START_REF] James O Berger | An overview of robust bayesian analysis[END_REF]. Likewise, in his work devoted to the Bayesian inference of chemical reaction networks [START_REF] Galagali | Bayesian inference of chemical reaction networks[END_REF], Galagali recognised that different priors (uniform, normal, exponential) might be equally justifiable and reasonable, but he limited himself to three point-mass mixture priors, which is clearly not enough to determine the whole influence that the choice of the prior can have on the posterior. To conclude, we can fairly say that the large majority of chemical kineticists applying Bayesian methods to their parameter estimation problems only use a single uniform prior distribution and that in the few cases where researchers wonder about the impact of the prior on the posterior, they only consider one or two other priors.

Ockham factors: a major difference between frequentism and precise Bayesianism

Let us suppose that we are comparing two models M 1 and M 2 which aim at accounting for the same experimental data E. k 1 and k 2 are their parameter vectors. The volumes of their parameter spaces are V M1 and V M2 , respectively and we further assume that their likelihoods are characterised by two peaks occupying the volumes V p M1 and V p M2 , respectively. The Bayes' factor B can be expressed as

B = L(E|M 2 ) L(E|M 1 ) = k2 L(E, k 2 , M 2 ) f 2,0 (k 2 |M 2 )dk 2 k1 L(E, k 1 , M 1 ) f 1,0 (k 1 |M 2 )dk 1 B = L max (E|k 2 ) L max (E|k 1 ) V p M2 V p M1 V M1 V M2 = FQ p O
if we apply the principle of indifference to f 1,0 (k 1 |M 1 ) and f 2,0 (k 2 |M 2 ). F is the ratio of the maximum likelihoods of the two models. It is the quantity that often plays the most important role in frequentist comparisons of models. This comparison of the maximum likelihoods is the approach that has been traditionally used by chemical kinetics trying to discriminate between two competing reaction mechanisms on the basis of experimental data [START_REF] William | Designs for discriminating between two rival models[END_REF][START_REF] Reilly | Statistical methods in model discrimination[END_REF], often indirectly through a comparison of the optimal values of the chi-squared distance [START_REF] Andrew L Wagner | Combustion of synthetic jet fuel: Chemical kinetic modeling and uncertainty analysis[END_REF] and sometimes more incorrectly of the leastsquared distance [START_REF] Hs Tan | The kinetics of the oxidation of propylene over a bismuth molybdate catalyst[END_REF]. The main difference between the precise Bayesian approach and the frequentist approach consists of the Ockham factor O = ( V M2 V M1 ) -1 . Following the philosophical principle of Ockham's razor, it penalises the model that is the most complex, whereby complexity is defined as the total volume of the parameter space. This reasoning can be found in an article written by Braman et al. about the quantification of uncertainty in syngas chemistry models Braman et al. (2012a). Strongly relying on the work of objective Bayesian physicist E.T. Jaynes [START_REF] Jaynes | Probability Theory: The Logic of Science[END_REF], the authors considered that a uniform prior is an appropriate representation of ignorance (even though they recognise that the posterior can be very sensitive to the choice of the prior) and that this entails Ockham's razor.

However, if the principle of indifference is not correct, that demonstration is invalid.

It is trivial to see how O is highly dependent upon the parameter representation. Let us consider, for example, that p 1 ∈ [1; 1E + 10] and p 2 ∈ [1; 1E + 100] while F = Q p = 1. If we directly apply the principle of indifference, we find that O p ≈ 1E -90, which would mean that we know with almost absolute certainty that M 1 is false. If, however, we apply the POI to log 10 (p 1 ) and log 10 (p 2 ), we find that O ln(p) ≈ 0.091. And if we apply the POI to p -1 1 and p -1 2 , we find that O p -1 ≈ 1. It is important to keep in mind that if we are totally ignorant about the distribution of the values of p i , we should logically be equally ignorant about the values of log 10 (p i ) and p i -1 . According to Wolpert Wolpert (1995), this Bayesian demonstration of Ockham's razor fails because O depends on the choice of the variable. The only way to make the ratio of the posteriors independent on the parametrisation would be to always select the following prior probabilities for the models: p(M 1 ) = V M1 and p(M 2 ) = V M2 , which would effectively shave away the razor.

How not to mix ignorance and knowledge

To see how this problem can negatively affect the work of chemical kineticists, a synthetic experimental data set was used to discriminate between two models. Let us consider an experimental profile that we want to simulate with two models M 1 and M 2 . The standard deviation of any measurement e i is given by σ i = 0.065e i . Let us further assume that there are only two reaction models which could account for the data • M 1 : An isomerisation towards product P 1 :

R → P 1 d[R] dt = -k 1 [R]
• M 2 : A recombination towards product P 2 :

R + R → P 2 d[R] dt = -2k 2 [R] 2
and that products P 1 and P 2 cannot be detected. In each case, the ordinary differential equation given above must be solved, given the initial condition [R] 0 = 1200 mol/ m 3 . Since it involves the rate coefficient k i , the computed temporal evolution of R (and thus the likelihood as well) are also bound to depend on it.

For theoretical and empirical reasons, we know the lower and upper bound of each kinetic coefficient while lacking any information about the distribution of the possible values. While an analogy with other reactions allows us to set narrow bounds for k 1 (k 1 ∈ [2; 2E + 3])s -1 , the uncertainty concerning k 2 is far larger in that k 2 ∈ [5E + 04; 5E + 09] mol.cm -3 .s -1 . We want to determine whether or not the data allow us to identify which of the two models is the most plausible.

Frequentist approach

The optimal curves obtained via the maximum likelihood method are contrasted with the measurements in Figure 3. The likelihood function of M 1 and M 2 can be seen in Figure 4 and 5, respectively. For ε = 2 and

F ε = {k | |m i (k) -e i | ≤ εσ i }, i ∈ [|1; n|]
, the feasible set approach leads one to discard M 1 as its feasible set is empty. The feasible set of M 2 is given by k 2 ∈ [8.51E +05; 1.22E +06] mol.cm -3 .s -1 . The ratio between the maximum likelihood of M 2 and M 1 is F = 322.91.

Precise Bayesian approach

Applying the principle of indifference directly to k 1 and k 2 leads to the following uniform priors: P(k 1 |M 1 ) = 1/(2 3 -2) = 0.00050 and P(k 2 |M 2 ) = 1/(5 9 -5 4 ) = 2 -10 . After having computed the likelihood function, we can compute the Bayes factor B (see Section 3). We find that B k = 0.312, which means that we ought to privilege M 1 over M 2 , even though the predictions of M 1 are farther from the measurements, and crucially, even though M 1 (unlike M 2 ) fails to reproduce the shape of the experimental profile. This is an example of Lindley's paradox where frequentist and Bayesian methods lead to conflicting results [START_REF] Robert | On the jeffreys-lindley paradox[END_REF]. This is caused by the fact that the knowledge consisting of the experimental data (which boils down to aleatory uncertainty) was illegitimately mixed up with our ignorance over the parameter values (epistemic uncertainty). Several authors have pointed out that while aleatory uncertainty can often be represented through a single probability distribution (provided there is enough data), epistemic ignorance cannot be treated in this manner [START_REF] He | Mixed aleatory and epistemic uncertainty quantification using fuzzy set theory[END_REF]; Abdallah et al. ( 2013

Robust Bayesian approach

It is precisely in such ambiguous situations that imprecise probabilities can prove extremely fruitful [START_REF] Walley | Towards a unified theory of imprecise probability[END_REF]. Instead of only considering a single prior probability density, we will now consider six of them, namely f k , f x , f l , f y , f z , and f w . They were obtained by reparametrising the problem and attributing a flat prior to the new parameter, following

Variable O B k 4.00E-07 3.12E-01 x = 1/k 2.50E+04 1.67E+04 l = log10(k) 6.00E-01 3.29E+02 y = 1/log10(k) 2.75E+01 3.74E+03 z = k 0.25 2.19E-02 7.08E+01 w = k 2 1.60E-13 1.98E-04
Table 1: Dependency of O and B on the prior the same logics as in Sections 2 and 3. These new variables are such that x = 1/k, l = log10(k), y = 1/log10(k), z = k 0.25 , and w = k 2 . Of course, more sophisticated priors can be used or built up in the framework of robust Bayesian analysis [START_REF] Scott Ferson | Constructing probability boxes and dempster-shafer structures[END_REF]; [START_REF] James O Berger | An overview of robust bayesian analysis[END_REF]. The main advantage of the priors defined in this section is that they are all uniform with respect to one variable and are equally justifiable, if we adopt a precise Bayesian framework 1 , in that being totally ignorant about the values of k logically entails being also totally ignorant about the values of x, l, y, z, and w Norton (2008). The priors for M 1 and M 2 can be seen in Figure 6 and 7, respectively. The Bayes factor was recomputed along with the Ockham factor. The results can be seen in Table 1. It is striking that the values of the Bayes' factor are strongly dependent on the prior. This is due to the Ockham factor O that heavily penalises M 2 in the case of k and w = k 2 whereas it favours it in the case of x = 1/k and y = 1/log10(k). Interestingly, l = log10(k) leads to the value of B l = 3.29E + 02 that is the closest to the maximum likelihood ratio F = 322.91, which is a purely experimental quantity (if we know the physical distribution of the measurement uncertainty). Nevertheless, 1. From an imprecise Bayesian standpoint, none of the priors alone can represent ignorance. Only all priors taken together are an adequate representation of our lack of knowledge. there is no reason to suppose that priors based on log10(k) will always have that desirable quality.

Discussion

The simplicity of this example allows one to rapidly visualise, analyse and understand the nature of the problem. The consequences of mixing knowledge and ignorance are, however, often not so easy to recognise (see [START_REF] Norton | Cosmic confusions: Not supporting versus supporting not[END_REF] [START_REF] Miki | On the use of bayesian approach, experimental data, and artificially designed data, for the identification of missing reactions[END_REF][START_REF] Miki | Probabilistic models and uncertainty quantification for the ionization reaction rate of atomic nitrogen[END_REF][START_REF] Miki | Using bayesian analysis to quantify uncertainties in the h+ o2= oh+ o reaction[END_REF][START_REF] Miki | Systematic validation of non-equilibrium thermochemical models using bayesian inference[END_REF]; [START_REF] Panesi | Calibration of rates parameters for multi-temperature models using bayesian formulation[END_REF]Najm et al. (2009); [START_REF] Prager | Uncertainty quantification in the ab initio rate-coefficient calculation for the ch3ch (oh) ch3+ oh= ch3c (oh) ch3+ h2o reaction[END_REF]; [START_REF] Sargsyan | Bayesian inference of spectral expansions for predictability assessment in stochastic reaction networks[END_REF], and some of them even state that this is the only logical choice if everything we know about a parameter is its lower and upper bound [START_REF] Hsu | Bayesian framework for building kinetic models of catalytic systems[END_REF]. Prior sensitivity analyses are very rare and limited to the use of one or two additional priors [START_REF] Galagali | Bayesian inference of chemical reaction networks[END_REF]; Braman et al. (2012b). As such, it is unclear whether the results in all these works only reflect the experimental data or both the data and the prior.

In their work about the reaction pathways underlying the electrochemical reduction of nitrate on tin, Katsounaros et al. described the Bayes factor as the pure impact of the data on the model relative plausibility [START_REF] Katsounaros | Reaction pathways in the electrochemical reduction of nitrate on tin[END_REF], like many other authors in the general field of parameter estimation Alatri (1999); [START_REF] Rouder | Bayesian inference for psychology, part IV: parameter estimation and bayes factors[END_REF]; [START_REF] Morey | Bayes factor approaches for testing interval null hypotheses[END_REF]. In the presence of a prior distribution based on empirical information, Bayes factors can indeed be a very useful tool to discriminate between models which is far superior to the traditional p-value tests Goodman (1999).

However, in the case of the two reaction models we just considered here (see Fig. 3), it is clearly wrong to say that it is the data themselves which disfavour M 2 (B k = 0.312). It would, on the contrary, be more appropriate to say that our decision to try to represent our ignorance through a flat prior on k smothered the experimental information at our disposal. In their work on the Bayesian analysis of syngas chemistry models, Braman et al. mentioned and emphasised the danger of overfitting highly complex models and argued that the closeness of an optimised model to all measurements cannot be the sole arbiter of its worth and truth [START_REF] Braman | Bayesian analysis of syngas chemistry models[END_REF]. They went on to praise the ability of precise Bayesianism to deliver us a single number combining the lack of complexity of a model and its ability to reproduce experimental data. However, the inability of precise Bayesianism to represent true ignorance might appear to be too high a price to pay for that [START_REF] Salmon | The Foundations of Scientific Inference[END_REF].

Even if we do not equate ignorance with uniform probability, there are other reasons to strongly favour simple models over complex ones. Kelly argued that systematically choosing the simpler of two equally accurate models maximises our convergence rate to the true model, regardless of whether it is simple or complex [START_REF] Kelly | Ockham's razor, empirical complexity, and truth-finding efficiency[END_REF]. While overfitting complicated chemical models should always be avoided, the simplicity of a model is not necessarily related to its plausibility Sober (2015).

Conclusion and outlook

Chemical kinetics determines many processes and phenomena that play a crucial role in the modern world, such as the combustion of fossil and renewable fuels [START_REF] Fischer | An assessment of chemical kinetics for bio-syngas combustion[END_REF], catalytic reforming [START_REF] Dou | Steam reforming of crude glycerol with in situ co2 sorption[END_REF], the reduction of pollutant emissions from car and truck engines [START_REF] Ip Kandylas | Mathematical modelling of precious metals catalytic converters for diesel no x reduction[END_REF], the growth of multicellular cancer [START_REF] Bajzer | Mathematical modeling of tumor growth kinetics[END_REF], the depletion of ozone in the stratosphere [START_REF] Solomon | On the depletion of antarctic ozone[END_REF], and other atmospheric reactions influencing global warming Sulbaek [START_REF] Mads P Sulbaek Andersen | Atmospheric chemistry of isoflurane, desflurane, and sevoflurane: kinetics and mechanisms of reactions with chlorine atoms and oh radicals and global warming potentials[END_REF]. As a consequence, a good knowledge of chemical kinetic parameter uncertainty might prove crucial to face the challenges of the modern world.

Over the last decades, an increasing number of authors have applied both frequentist and Bayesian methods to chemical kinetic parameter estimation (including optimal design) [START_REF] Frenklach | Collaborative data processing in developing predictive models of complex reaction systems[END_REF]; [START_REF] Wang | Combustion kinetic model uncertainty quantification, propagation and minimization[END_REF]. The main advantage of the Bayesian approach is its ability to take into account genuine prior information (such as a frequency distribution of the rate coefficients of reactions of a certain type) and to provide decision makers with quantities that are intuitively much more appealing than their frequentist counterparts (compare, for instance, the frequentist and Bayesian notion of confidence interval). However, in situations where prior experimental knowledge is limited or non-existent, the problem cannot be described through a single prior probability distribution. Bayesian robust analysis, the systematic study of the influence of the choice of the prior on the posterior variables of interest, has been applied to a large number of topics such as climatology [START_REF] Tomassini | Robust bayesian uncertainty analysis of climate system properties using markov chain monte carlo methods[END_REF], the monitoring of clinical trials Greenhouse and Waserman (1995), pipeline failure [START_REF] Cagno | Using ahp in determining the prior distributions on gas pipeline failures in a robust bayesian approach[END_REF], and accident proneness [START_REF] Sixto Rios Insua | Bayesian forecasting for accident proneness evaluation[END_REF], to name but a few. However, the large majority of chemical kineticists resorting to Bayesian methods only use one single (mostly uniform) prior distribution. In Section 2 and 3, it was shown that in the absence of prior information, relying on a single uniform prior blurs the distinction between knowledge and ignorance. A simple example of chemical kinetics involving the comparison of two reactions to account for the same experimental profile was drawn upon to illustrate how the complexity of a model (i.e. the volume of its parameter space) and its accuracy (i.e. the closeness of its predictions to the measurements) are two irreducible quantities that were artificially merged into a single number because of the use of a uniform prior. There are many other ways in which the posterior might be strongly influenced by the choice of the prior. When considering the probability that an explosion might occur or that a toxic chemical might be formed, it is vital to make sure it is the experimental data that are speaking and not the prior. The purpose of this article was to draw attention to this fact and to encourage chemical kineticists employing Bayesian methods to systematically determine the sensitivity of their results to the prior.

An important question needs now to be addressed. It is well known that the influence of the prior is bound to decrease as the number of experimental data rises. This is commonly referred to as washing out the priors [START_REF] Nola | Bayesian methodology in science[END_REF]. Is it not reasonable to expect that, from a given number of measurements, the choice of the prior will no longer matter anyway? If every new measured variable were equally sensitive to every reaction, this is something we could feel relatively confident about. Unfortunately, we know that this is rarely the case in chemical kinetics. For instance, if we consider the combustion of methane CH 4 , carbon monoxide CO and hydrogen H 2 , the reaction H + O2 → OH + O will almost always play a very significant role whereas the reaction HO 2 + CO → OH + CO 2 can be neglected for most macroscopic experimental targets such as ignition delay times and laminar flame speeds involving either CH 4 or CO as fuel [START_REF] Peters | Reduced Kinetic Mechanisms for Applications in Combustion Systems[END_REF] so that only few experimental data will constrain its values. And yet, it can play an important role in the combustion of bio-syngas (CO -CO 2 -CH 4 -H 2 mixtures) at high pressures and middle temperatures [START_REF] Fischer | An assessment of chemical kinetics for bio-syngas combustion[END_REF] so that accurate values of its kinetic parameters would be required for, say, minimising the formation of pollutants in a combustor under these conditions. As a consequence, even in cases where we have a lot of measurements at our disposal, the choice of the prior will still have a strong influence on the parameters most measured variables are not sensitive to so that it cannot be "washed away" easily. A systematic application of robust Bayesian analysis to chemical kinetic problems appears thus to be always advisable. There is also a need for chemical kinetic studies where both frequentist and Bayesian methods are applied to the same problem in order to see the practical differences caused by the choice of the fundamental framework. To the best of the author's knowledge, no such work can be found in the literature about chemical kinetics.

The results of such studies would give us important insights into how to optimally represent the uncertainty of chemical kinetic parameters in a way that would make them safer for decision makers to rely on.

Figure 1 :

 1 Figure 1: Example of feasible set

Figure 2 :

 2 Figure 2: Posterior predictive marginal and joint distributions of Arrhenius rate parameters

  ; Najm et al. (2009); Panesi et al. (2011); Prager et al. (2013); Sargsyan et al. (2009); Sondak et al. (

  ;[START_REF] Russi | Sensitivity analysis of uncertainty in model prediction[END_REF];You et al. (2011). Several authors[START_REF] Braman | Bayesian analysis of syngas chemistry models[END_REF];[START_REF] Duque-Bernal | Kinetic study on the homogeneous esterification of acetic acid with isoamyl alcohol[END_REF];[START_REF] Hakim | Probabilistic parameter estimation in a 2-step chemical kinetics model for n-dodecane jet autoignition[END_REF] have emphasised that frequentist methods are prone to overfitting, i.e. to favouring very complex models with many parameters which perfectly match the measurements. By merging relative accuracy and simplicity into one single number (the Bayes factor), precise Bayesian methods avoid this problem. However, as we shall see in Section 3 and 4, this is not necessarily always a strength as this can lead one to mistake ignorance for knowledge. Other authors mention the strongest advantage of the Bayesian framework which is its ability to integrate genuine prior information into the problem at hand Hsu et al. (2009); McMullen and Jensen (2011); Verheijen (2003); Janković (2015). For instance, while studying the homogeneous esterification of acetic acid with isoamyl alcohol, Duque-Bernal et al. used frequency data about similar esterification reactions found in the literature to define the prior distribution of the activation energy of the specific reaction they were studying Duque-Bernal et al. (2013). Similarly, in their study of the polymorphic transformation of L-glutamic acid crystals, Hermanto et al. used preliminary maximum likelihood estimates to inform their prior Hermanto et al. (2008). In such situations, traditional frequentist methods would have only considered the lower and upper bound, thereby failing to exploit important and useful prior information.

  );[START_REF] Urzay | Uncertainty-quantifying models for chemicalkinetic rates[END_REF];[START_REF] Bell | A bayesian approach to calibrating hydrogen flame kinetics using many experiments and parameters[END_REF];Braman et al. (2012b);[START_REF] Hakim | Probabilistic parameter estimation in a 2-step chemical kinetics model for n-dodecane jet autoignition[END_REF];[START_REF] Hsu | Bayesian framework for building kinetic models of catalytic systems[END_REF][START_REF] Alexander J Josephson | Modeling soot oxidation and gasification with bayesian statistics[END_REF];[START_REF] Miki | On the use of bayesian approach, experimental data, and artificially designed data, for the identification of missing reactions[END_REF][START_REF] Miki | Probabilistic models and uncertainty quantification for the ionization reaction rate of atomic nitrogen[END_REF][START_REF] Miki | Using bayesian analysis to quantify uncertainties in the h+ o2= oh+ o reaction[END_REF][START_REF] Miki | Systematic validation of non-equilibrium thermochemical models using bayesian inference[END_REF]; Panesi et al. (2011); Najm et al. (2009); Prager et al. (2013); Sargsyan et al. (2009). For their study, Emerson Holtzer et al. used bounded Gaussian priors they set up arbitrarily Holtzer et al. (2001). Braman et al. systematically used both a uniform and a normally distributed prior for their estimation of the chemical kinetic parameters of syngas combustion

  Figure 3: Optimised profiles of R.

  Figure 5: Likelihood (M 2 )

  );[START_REF] Bae | An approximation approach for uncertainty quantification using evidence theory[END_REF][START_REF] Bree R Mathon | Dempster-shafer theory applied to uncertainty surrounding permeability[END_REF].

Figure 6 :

 6 Figure 6: Prior distributions for p 1 (M 1 )

Figure 7 :

 7 Figure 7: Prior distributions for p 2 (M 2 )

  David H. Wolpert. On the bayesian 'occam's factors' argument for occam's razor', in computational learning theory and natural learning systems iii, t. petsche et al. Technical report, Computational Learning Theory and Natural Learning Systems, Volume III: Selecting good models, 1995. X. You, T. Russi, A. Packard, and M. Frenklach. Optimization of combustion kinetic models on a feasible set. Proceedings of the Combustion Institute, 33:509-516, 2011.

Acknowledgments

I am most thankful to Dr. Matthias Troffaes, professor Gert de Cooman and professor Frank Coolen for our useful and pleasant exchanges.