Guillaume Baudart

Automatic Guide Generation for Stan via NumPyro

Stan is a very popular probabilistic language with a state-of-the-art HMC sampler but it only offers a limited choice of algorithms for black-box variational inference. In this paper, we show that using our recently proposed compiler from Stan to Pyro, Stan users can easily try the set of algorithms implemented in Pyro for black-box variational inference. We evaluate our approach on PosteriorDB, a database of Stan models with corresponding data and reference posterior samples. Results show that the eight algorithms available in Pyro offer a range of possible compromises between complexity and accuracy. This paper illustrates that compiling Stan to another probabilistic language can be used to leverage new features for Stan users, and give access to a large set of examples for language developers who implement these new features.

MOTIVATION

The Stan probabilistic language [START_REF] Carpenter | Stan: A probabilistic programming language[END_REF]] enjoys broad adoption by the statistics and social sciences communities [START_REF] Bradley | Bayesian methods for data analysis[END_REF][START_REF] Gelman | Data analysis using regression and multilevel/hierarchical models[END_REF][START_REF] Gelman | Bayesian data analysis[END_REF]. A Stan program defines a function from data and parameters to the value of a special variable target that represents the log-density of the model. Given the observed data, the posterior distribution of the parameters can then be inferred using specialized inference algorithms, e.g., NUTS [START_REF] Hoffman | The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo[END_REF] (No U-Turn Sampler), an optimized Hamiltonian Monte Carlo (HMC) algorithm that is the preferred inference method for Stan.

Pyro [START_REF] Bingham | Pyro: Deep Universal Probabilistic Programming[END_REF]] and its JAX-based counterpart NumPyro [START_REF] Phan | Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro[END_REF] on the other hand are generative probabilistic languages. They describe generative models, i.e., stochastic procedures that simulate the data generation process. Generative PPLs are increasingly used in machine-learning research and are rapidly incorporating new ideas, such as Stochastic Gradient Variational Inference (VI), or Bayesian neural networks in what is now called Deep Probabilistic Programming [START_REF] Baudart | Deep Probabilistic Programming Languages: A Qualitative Study[END_REF][START_REF] Bingham | Pyro: Deep Universal Probabilistic Programming[END_REF][START_REF] Tran | Deep Probabilistic Programming[END_REF].

Variational Inference tries to find the member 𝑞 𝜃 * (𝑧) of a family Q = 𝑞 𝜃 (𝑧)

𝜃 ∈Θ of simpler distributions that is the closest to the true posterior 𝑝 (𝑧 | x) [START_REF] Blei | Variational Inference: A Review for Statisticians[END_REF]. Members of the family Q are characterized by the values of the variational parameters 𝜃 . The fitness of a candidate is measured using the Kullback-Leibler (KL) divergence from the true posterior, which VI aims to minimize:

𝑞 𝜃 * (𝑧) = argmin 𝜃 ∈Θ KL 𝑞 𝜃 (𝑧) ∥ 𝑝 (𝑧 | x) .
Pyro natively supports variational inference and lets users define the family Q (the variational guide) alongside the model. However, manually defining a correct and efficient guide is complex and error prone [START_REF] Lee | Towards verified stochastic variational inference for probabilistic programs[END_REF]. Stan, on the other hand, offers ADVI [START_REF] Kucukelbir | Automatic Differentiation Variational Inference[END_REF], an implementation of black-box VI where guides are automatically synthesized from the model using a mean-field or a full-rank approximation. But, ADVI is only efficient for a subclass of models.

As an intermediate solution, Pyro developers recently introduced a zoo of autoguides: variational guides that are automatically synthesized from the model using different heuristics [START_REF] Webb | Improving Automated Variational Inference with Normalizing Flows[END_REF]. Users can now try a range of synthesized guides on a given model before attempting to craft their own. We recently proposed new backends for the Stanc3 Compiler to Pyro and NumPyro1 and showed how to extend Stan with support for explicit variational guides [START_REF] Baudart | Compiling Stan to Generative Probabilistic Languages and Extension to Deep Probabilistic Programming[END_REF]. In this paper, we show that our compiler and runtime can be used to test NumPyro autoguides on Stan models, and evaluate our approach on PosteriorDB a database of Stan models with corresponding data, and reference posterior samples [START_REF] Vehtari | PosteriorDB: a database with data, models and posteriors[END_REF].

EXAMPLE

The multimodal example shown in Figure 1 is a mixture of two Gaussian distributions with different means but identical variances. This example is particularly challenging for NUTS. Using multiple chains, NUTS finds the two modes, but the chains do not mix and the relative densities are incorrect. This is a known limitation of HMC.2 This model is also challenging for Stan's ADVI since the synthesized guide cannot approximate multi-modal distribution as illustrated in the histograms of Figure 1 (by default, the runtime detects that ADVI does not converge and throws an exception).

Using our compiler from Stan to NumPyro we can try Pyro autoguides on the same example. The following code illustrates our runtime.

1 from stannumpyro import NumPyroModel 2 import numpyro.infer.autoguide as autoguide 3 from numpyro.infer import Trace_ELBO 4 from numpyro.optim import Adam 5 from jax.random import PRNGKey 6 7 numpyro_model = NumPyroModel("multimodal.stan") 8 guide = autoguide.AutoBNAFNormal(numpyro_model.get_model()) 9 svi = numpyro_model.svi(Adam(step_size=0.0005), Trace_ELBO(), guide) 10 svi.run(PRNGKey(0), {}, num_steps=10000, num_samples=10000)

The file multimodal.stan contains the Stan program of Figure 1. From this file, we create a NumPyroModel object which compiles the model and loads the NumPyro code (line 7). We then synthesize a guide for the model using the AutoBNAFNormal heuristic (line 8). Following NumPyro's API, to launch the inference, we first create a SVI object from an optimizer (Adam), a loss function (Trace_ELBO), and the guide (line 9). Given a random seed PRNGKEY(0), and the input data (here an empty dictionary since the model has no data block), the run method first computes num_steps optimization steps and then draws num_samples samples from the posterior distribution (line 10). NumPyro offers eight different heuristics to synthesize a guide from the model:3 • AutoNormal (similar to AutoDiagonalNormal), and • AutoDelta (MAP estimates).

Figure 2 clearly shows that these heuristics yield different results on the example of Figure 1. Only two of them successfully identify the two modes.

EVALUATION

PosteriorDB [START_REF] Vehtari | PosteriorDB: a database with data, models and posteriors[END_REF] provides reference samples for 49 pairs (model, dataset). Due to missing functions in our implementation of the standard library (e.g., ODE solvers), we cannot test six models. For each of the remaining 43 models, we run 100,000 inference steps (using Adam(step_size=0.0005)) and generate 10,000 samples from the posterior distribution. To evaluate inference accuracy, for each parameter 𝑥 in the posterior distribution we compute the relative error w.r.t. to the reference samples. For multidimensional parameters we compute the error for every component.

err = |mean(𝑥 ref ) -mean(𝑥)| stddev(𝑥 ref )
The evaluation code is available at https://github.com/deepppl/evaluation-autoguide. Table 1 summarizes the results. For each model, we report the maximal relative error across parameters. Relative errors below 0.3 -the criteria used by regression tests for Stan4 -appear in green. On this set of benchmarks, the autoguides relying on normalizing flows AutoBNAFNormal and AutoIAFNormal outperform other autoguides. AutoDelta can be used with all models without raising runtime errors, but it only computes a MAP estimate. AutoLaplaceApproximation also runs without error on all models but returns less accurate distributions based on a MAP estimations. Runtime errors correspond to NaN values. Overall, the autoguides demonstrate a range of possible compromises between complexity and accuracy.

For comparison, we also report the results of Stan ADVI with the mean-field and full-rank algorithms. Compared to NumPyro, instead of running a fixed number of iterations, the Stan runtime relies on an adaptive optimization sequence and throws an exception when the algorithm fails to converge.

CONCLUSION

In this paper, we showed that by compiling Stan models to NumPyro, Stan users now have access to a large variety of automatically synthesized guides for variational inference. As illustrated on 43 benchmarks from PosteriorDB, these new guides offer new compromises between complexity and accuracy compared to Stan ADVI full-rank and mean-field algorithms. 5This paper illustrates that compiling Stan to another probabilistic language can be used to leverage new features for Stan users, and give access to a large set of examples for language developers who implement these new features.

Fig. 1 .

 1 Fig. 1. Multimodal example in Stan, and graph of expected posterior distribution and histograms of the inferred posterior distributions using Stan NUTS and Stan ADVI with the mean-field and full-rank algorithms.

Fig. 2 .

 2 Fig.2. Inference results using NumPyro autoguides on the model of Figure1after 100,000 inference steps with 10,000 samples.

Table 1 .

 1 Relative errors w.r.t. the reference samples of PosteriorDB. We report the maximum relative error across parameters in green if it is below 0.3, a ✗ indicates a runtime error.

	Dataset -Model	B N A F N o r m a l D e lt a	D ia g o n a lN o r m a l I A F N o r m a l L a p la c e A p p r o x im a t io n L o w R a n k M u lt iv a r ia t e N o r m a l M u lt iv a r ia t e N o r m a l N o r m a l	S t a n M e a n F ie ld S t a n F u ll R a n k
	arK-arK	0.26	0.44	0.09 0.22	0.38	0.06	0.13	0.17	✗	0.33
	arma-arma11	✗	0.26	✗	✗	0.16	✗	✗	✗	✗	✗
	sblri-blr	1.96	0.50	0.70 2.46	0.38	0.64	0.76	0.64	✗	✗
	sblrc-blr	1.05	0.51	0.18 1.52	0.40	0.55	1.58	0.27	✗	✗
	dogs-dogs	0.10	0.12	0.13 0.16	0.11	0.05	0.08	0.13	0.38	0.80
	dogs-dogs_log	✗	0.12	0.07	✗	0.12	0.03	0.05	0.07	0.93	✗
	earnings-earn_height	0.30	24.65 23.88	✗	24.26 24.21 24.08 24.55	✗ 18.71
	eight_schools-eight_schools_centered	0.13	1.13	0.65 0.07	1.13	0.52	0.60	0.62	1.17	1.39
	eight_schools-eight_schools_noncentered	0.07	1.13	0.22 0.07	13.03	0.17	0.18	0.26	0.22	0.21
	garch-garch11	2.68	1.15	2.68 2.71	0.74	1.80	1.68	2.69	0.81	0.34
	bball_drive_event_0-hmm_drive_0	0.28 290.68	0.29	✗	0.42	0.43	0.27	0.10	0.48 14.31
	bball_drive_event_1-hmm_drive_1	0.53	0.59	0.52 0.77 768.80	0.52	0.63	0.48	9.68	4.59
	hmm_example-hmm_example	0.16	0.37	0.46	✗	0.25	0.05	0.30	0.36	1.73	0.55
	kidiq-kidscore_interaction	0.62	0.63	2.47 0.82	0.49	2.46	2.21	2.47	2.61	✗
	kidiq_with_mom_work-kidscore_interaction_c	0.73	48.31 46.66 0.23	46.42 48.03 49.68 46.71 44.70 56.46
	kidiq_with_mom_work-kidscore_interaction_c2 0.67	35.30 34.10 0.33	33.91 34.35 35.09 34.13	✗ 93.98
	kidiq_with_mom_work-kidscore_interaction_z	0.70	48.03 46.41 0.44	46.15 48.49 49.75 46.45	2.87 48.65
	kidiq_with_mom_work-kidscore_mom_work	0.65	16.47 15.94 0.17	15.79 16.05 16.04 15.95	1.94	7.23
	kidiq-kidscore_momhs	0.19	16.61 15.84	✗	15.66 15.96 16.02 15.83	6.76	2.18
	kidiq-kidscore_momhsiq	0.27	0.17	2.44 0.25	0.12	2.28	1.82	2.44	✗	4.07
	kidiq-kidscore_momiq	0.10	0.15	2.20	✗	0.12	2.13	1.75	2.21	3.28	3.65
	kilpisjarvi_mod-kilpisjarvi	2.12	1.99	1.97	✗	1.95	1.98	7.00	1.98	✗	✗
	earnings-log10earn_height	0.12	0.09	0.17	✗	0.08	0.15	0.56	0.19	✗	✗
	earnings-logearn_height	0.13	0.09	1.08	✗	0.08	1.11	1.21	1.08	✗	✗
	earnings-logearn_height_male	0.25	0.11	4.22 0.16	0.08	4.06	3.74	4.21	1.33 10.73
	earnings-logearn_interaction	0.56	0.12	5.52 0.81	0.11	5.55	5.99	5.51	✗	✗
	earnings-logearn_interaction_z	0.67	0.12	0.27 0.42	0.10	0.10	0.27	0.15	1.90	1.28
	earnings-logearn_logheight_male	0.45	0.12	1.22 0.54	0.09	1.21	1.17	1.22	0.66	✗
	mesquite-logmesquite	0.15	0.93	0.03 0.22	0.80	0.05	0.06	0.05	✗	0.11
	mesquite-logmesquite_logva	0.34	0.65	0.06 0.37	0.52	0.05	0.05	0.05	0.14	0.05
	mesquite-logmesquite_logvas	0.14	0.93	0.13 0.27	0.80	0.11	0.06	0.13	✗	0.09
	mesquite-logmesquite_logvash	0.22	0.84	0.08 0.26	0.72	0.06	0.08	0.05	✗	✗
	mesquite-logmesquite_logvolume	0.32	0.45	0.13	✗	0.33	0.11	0.13	0.04	0.22	0.08
	low_dim_gauss_mix-low_dim_gauss_mix	0.68	0.13	0.22 0.75	0.08	0.21	0.23	0.27 63.08 62.71
	mesquite-mesquite	0.16	8.12 11.02 6.39	7.89 11.80 11.25 10.94 23.32 17.18
	nes1988-nes	0.18	0.27	0.06 0.33	0.24	0.33	0.06	0.11	0.88	2.09
	nes1972-nes	0.19	0.21	0.05 0.33	0.18	0.38	0.05	0.15	0.27	1.30
	nes1984-nes	0.16	0.23	0.04 0.30	0.20	0.36	0.04	0.14	1.25	1.31
	nes1980-nes	0.14	0.29	0.03 0.19	0.25	0.31	0.03	0.13	1.05	0.74
	nes2000-nes	0.12	0.37	0.04 0.19	0.32	0.25	0.03	0.12	1.01	1.28
	nes1976-nes	0.19	0.24	0.05 0.32	0.21	0.35	0.06	0.13	2.61	1.89
	nes1992-nes	0.18	0.23	0.05 0.27	0.21	0.35	0.05	0.13	2.19	0.85
	nes1996-nes	0.17	0.25	0.04 0.25	0.21	0.34	0.05	0.14	1.29	2.54
	Average	0.46	11.72	5.30 0.71	22.89	5.43	5.59	5.32	6.16 11.30
	Successes	25	21	22	16	20	14	21	22	4	5
	Mismatches	16	22	20	16	23	28	21	20	25	27
	Errors	2	0	1	11	0	1	1	1	14	11

https://github.com/deepppl/stanc3

https://mc-stan.org/users/documentation/case-studies/identifying_mixture_models.html

http://num.pyro.ai/en/stable/autoguide.html

https://github.com/stan-dev/performance-tests-cmdstan

https://discourse.mc-stan.org/t/intermediate-between-mean-field-and-full-rank-advi

ACKNOWLEDGMENTS

We would like to thank Eli Bingham, Fritz Obermeyer, and Du Phan for suggesting us this application of the Stan to NumPyro compiler.


O De Mouzon

D Dubois

H Prade

Twofold Fuzzy Sets in Single and Multiple Fault Diagnosis, Using Information about Normal Values

This paper proposes a general approach to diagnosis based on fuzzy pattern matching, making use of consistency and inclusion hased indices in the setting of possibility tbeory. The approach was first developed for biuary attributes and single faults. lt was then generalized to any kind of attributes (including multidimen sional ones). The paper presents a refined representation (wbere a distinction is made between effects that are possible for sure and effects that are just not impossible, and where information about (ab)normal values is used). Moreover, an extension to multiple fault diagnosis and to "cascading faults" is outlined.

INTRODUCTION

Therc has becn a continuous interest in fuzzy set-bascd ap proaches to diagnosis for about 25 years, starting with the early . work of Sanchez [START_REF] Sanchez | Solutions in composite fuzzy relation equations: ap plication to medical diagnosis in Brouwerian logic[END_REF], and continuing with the development of applications in medicine (sce [START_REF] Buisson | Practical Applications of Fuzzy Technologies, chapter Approximate reasoning in computer-aided medical de cision systems[END_REF] for a rcview) and in industrial processes (e.g., see [START_REF] Ulieru | lntelligentManufacturing and Fault Diagnosis (Il). Soft computing approaches to fault diagnosis[END_REF]). This paper presents some extensions of a diagnosis approach based on possibility thcory and on fuzzy pattern matching. It somewhat departs from previous proposais by emphasizing the use of possibility distributions for represent ing uncerlain pieces of knowledge, rather than using fuzzy re lations for modelling the intensity of symptoms. This approach, which was first proposed for binary attributes [3], has been ex tended more recently to numcrical attributcs in [START_REF] Dubois | A general approach to diagnosis in a fuzzy setting[END_REF]. The intercst of this approach is discussed and exemplified in [START_REF] De Mouzon | Using consistency and abduction based indices in possibilistic causal diagnosis[END_REF] for single fault diagnosis. Severa] further extensions of this approach are presented be low. Section II provides the background and briefly discusses the handling of multidimensional attributes. Section III deals with some rcfincments in knowledge rcpresenlation. Il first in troduces the use of twofold fuzzy sets (instead of simple fuzzy sets) for a better representation aiming al distinguishing be twecn effects which are possible for sure, and effccts which are just not forbidden. Besides, it allows for the representation of normal behaviour information, which can be useful for the diag nosis. Section IV takes advantage of the latter improvement in order to cope with multiple-fault diagnosis. Finally, Section V briefly discusses the detection of "cascading faults".

Il POSSIBILISTIC DIAGNOSIS

The framework developed for single-fault diagnosis is first presented and then extended to multidimensional attributes.

A Background

The approach is based on fuzzy relational knowledge: The relation between the value of each considered (binary, discrete or continuons) attribute a and each possible fault fis described by a possibility distribution rc[: Ua--+ [0, 1], where Ua is the domain of attribute a. This relation can be causal, thus describ ing the effect of / on a by telling which values of a are (more or less) possible when f (alone) occurs. But it is not necessarily so, as it may also represent a necessary condition on attribute a for f to appear (e.g., testis and womb cancer can respectively appcar only on men and women). In fac!, for concrete applica tions, il is not feasible to ask for ail rc[ relations, especially when faults and attributes are numerous. So only known relations are described through rd distributions. When rr.[ is not available, it either means that the value of a is not affected by the presence of f, or that this relation is not yet known. Tuen, rc[ can be taken as equal to 1 everywhere: no information enables us to discard a value for attribute a when fault f is prescnt. The observations may also be pervaded with imprecision and uncertainty: a possibility distributionµ� : Ua --t (0, 1] repre scnts the (more or Jess) possible values of attribute a according to the observation O (e.g., reflecting sensor reliability). Two fuzzy pattern matching indices [ 4] between effects ( rc{,) and observations(µ�) are computed for the diagnosis: • cons" evaluatcs how consistent the hypothesis of the presence of a fault (f) is with the observations:

cons"(f) = min cons"[, a0! (1)
where .!'t is the sel of ail the attributes and cons"[ evaluates the consistency off with the observation of a:

cons"[== sup min (µ�(u),1t[{u)).

B Multidimensianal attributes

The approach straightforwardly extends to multidimensional attributes, at the formai level, using joint possibility distribu tions such as rcL a 2 from Va 1 x V a 2 to [O, 1] and conjunctions of observations, i.e. µ� 1 ," 2 (u1, u2) == min(µ� 1 (u1) ,µ� 2 (u2) ). How ever, it is useful in practice to identify the cases where the computation can still be done al the level of one-dimension at tributes. Indeed, it is often not nccessary to use explicit multidi mensional distributions: Two major kinds of multidimensional attributes can be distinguished and represented in terms of one dimension possibility distributions. The first kind conccms formula-linloed multidimensional at tributes: For instance, assume that when fault f is present, attributes a 1 and a2 should no longer be approximately equal. The possible values (u1 and uz) of a1 and az for this symptom offault f may be represented as 1Tii 1 ,a 2 (u1,u2) == n;(u1 -u2), where a' is the new attribute a' = a1 -az and the closer to Ou' is, the doser rr;, (u' ) is to O. The second kind concerns projection-decamposable multidi mensional distributions: Tuen, a symptom is describcd as sev eral one-dimension manifestations linked together with logi cal connectors. For instance, when fault f occurs, both at tribute a1 and a2 are high or attribute a2 is close to O. Then, n1i_a 2 = max(min(1t 1 ! 1 , itt), rr''tz), where rr!, expresses that a1 is high, n' t2 expresses that az is high and n" t2 expresses that a2 is close to O. Of course, the two kinds of attributes can be mixed.

III REFINEMENTS OF THE REPRESENTATION

This section first introduces the use of twofold fuzzy sets for the knowledge representation. This enables us to make the dis tinction between what is just not impossible (because it is not rulcd out by the experts' beliefs) and what is known as fcasi ble (because it has already been observed). Moreover, under the single-fault hypothesis, the fact that fault f bas no effect on attributc a means that a stil! takes normal values and this infor mation can be useful to discard the prcsencc off when a takes abnormal values. So Section B introduces the representation of (ab)normal values of attributcs and its impact on the diagnosis. Note also that a good diagnosis should explain all the abnormal behaviours observed on the attributes. For simplicity, the results below arc prescnted for one-dimension attributes only and could be easily extended to multidîmensional attributes.

A Twofold fuzzy sets

In the initial representation framework, n! expresses the known restrictions on the values of attribute a linked to the pres ence of fault f. This idea of possibility may correspond to val ues which are just not known as impossible, since they are not ruled out by /. ln particular, when nothing is known, n! is 1 everywhere. However, it may be useful to identify among the values restricted by n{ those which are really known as feasible, because they have been observed in usual cases.

Thus a second possibility distribution (Dt: V a --+ (0, 1)) can bt! used for describing how typical the feasible values are known to be. For instance, we may know that whcn someonc has in fluenza, some range of body tcmperaturc is guaranteed possible (according, for example, to former typical observations on peo ple suffering from the same disease ), while some other values around may not be a priori forbidden, although we arc not sure that they can really take place. li will be assumed that a feasible value v for attributc a when fault f is prcscnt (i.e. D[ ( v) > 0) should be such that nt ( v) = 1, for consistency reasons. So, the support of D[ is included in the core of n{ Il entails ô[ ::::; rc{ When nothing is known, Dt is O cverywhere. Indced, 6{, ex presses to what extent a value of attribute a is known as feasible whcn f occurs.

Two other indices, cons 6 and rel", may be defined (following cons" [START_REF] Boverie | Using fuzziness for causal diagnosis in en gine dyno test bcnches[END_REF] and rel" [START_REF] Cayrac | Handling uncertainty with possibility theory and fuzzy sets in a satellite fault diagnosis ap plication[END_REF], resp.) with cons 6! and re/ 6 ! (resp.) by replacing rc[ by &t (in cons rr!, (2) and rel rr!, (4), resp.). coni (resp. rel 0 ) expresses to what extent it is possible (resp. certain) to have observed only typical effects or conditions of a fault. This enables us to make the distinction betwecn a fault f that has an unknown cffect on an attribute a (nt == 1 and o! == 0) and a fault f' that has no effect on attribute a (nf == 1 and ô( == 1), which was not possible in the initial framcwork. Note that the pair (nt.Dt) could be summarized as a unique possibility distribution whose core would be the corc of &t and whose support would be the support of n{ This is a way of putting together the ideas of feasibility and non-impossibility. Yet, the pair ( rct, Dt) provides a more refincd represcntation. As pointed out in [START_REF] Cayrol | Fuzzy pattern matching[END_REF], cons" 2: rel it holds, so cons ° 2: rez' 6 still holds. As it! > Dt, it follows that cons" > cons 0 and rel" > re/ 0 .

So cons 11 6 and rel [START_REF] Ulieru | lntelligentManufacturing and Fault Diagnosis (Il). Soft computing approaches to fault diagnosis[END_REF] • See [START_REF] De Mouzon | Causal diagnosis in engine dyno test benches: A possibilistic treat ment[END_REF] for other refincmcnts using discri-min and lcximin, which amount to vector comparisons of cons"{, (or rel"[).

B Information on (ab)normal values

In fac!, it is not fully satisfying to represent the information that the values of attribute a and the prcsencc of fault f' arc not linked with distributions rrf = 1 and af = 1. Indeed, when f al one is present, a should take normal values in this case.

So, some information on (ab)normal behaviour of attributes is needed. Let P a : U a --t [O, 1] be the possibility distribution expressing which values of attribute a are known as (more or less) abnormal. For short, it is assumed here that the normal behaviour of attribute ais defined as Tl a = 1 -P a • Now, the former knowlcdge reprcsentation may benefit from this new information. When rtb and ôb are not givcn (i.e. a (j.

'l( J ), it is because the expert states one of the following cases: f (alone) is not linked to values of a, so a should keep normal values when f oceurs. We gct the twofold fuzzy set:

n[ = 1-Pa; ô[ (u) = 1 if Pa(u) = 0; ô[ is O else. ( 5
)
Let 'l{_ J denote the set of attributes which are f;f_ot linked to f.

/ could be linked to values of a but we do not know to which ones, which mcans that no value of a is known as impossi ble (resp. feasibte) when f atone occurs: i.e. Ttt (resp. 6b) is l (resp. 0) everywhere. Let 'U t dcnote the set of attributes which have an !l_nknown link to f.

'JCJ stilt corresponds tu any other case for the attributes (Know11 links). So, 'tff E Y: ,('lCt,'ll t ,9-lt) is a 3-partition of 5l. 'l1 f should be as small as possible (as every link between the presence of faults and the values of the attributes should be known, cven only roughty) and 'l{_ f can be very large. Then, this information on rc[ and ô[ is used with the former (cons" and re/ 6 ) indices. So ret 6 is O as soon as 'l1 f :p 0 and cons" in equation (1) is computed with min instead of min. aE 'K.rU�r aE>t The foltowing toy example shows the benefits of this new in formation for the diagnosis. Let 5I = { a1, ai} and fi and h be two faultss.t.'l(J, = {ai},9,l fi = {a2} and'l(h = {a1,az}, with ( h "'h ) _ ( fi "'fi ) ( . _ ,, [, _ , nt2 d h < p ( . 7ta 1 , ua 1 --1ta 1 , ua 1 so cons 1 -cons 1 , an 1t a 2 _ az 1.e.

IV MULTIPLE-FAULT DIAGNOSIS

This section discusses the practical handling of multiple faults, which is often left aside, sinec it has not the computa tional simplicity of the "single fault assumption". By multiple faults is meant a combination of faults that are present simul taneously. For instance: A person may have both measles and white tonsillitis. Theoretically speaking, the approach to single fault diagnosis can detect and identify multiple faults as well, by describing, for each possible combination of faults, the associ ated links to attributes values. Yet, if any set of multiple faults are possible, then one would have to define the associated links for 2" faults, if n is the number of single faults. Such a knowl edge base would be difficult to obtain in practice and would be quite redundant as in many cases the effeets linked to multiple faults are just the "sum" of the effects linked to the different single faults. For instance, measles gives fever and read spots on the body skin, and white tonsillitis gives fever (also) and white spots in the throat. Then, when someone suffers from both measles and white tonsillitis, fever, red spots on the body skin and white spots in the throat are expectcd.

ln the fottowing we propose a general approach to the han dling of multiple faults. The idea is to represent the effects of single faults onty and then to be able to compute the effeets of any multiple faults (may be in a rather imprecise way).

The simples! hypothesis, called "superposition hypothesis", is that effects of multiple faults are just the sum of the effects of the single faults involved in the multiple faults, i.e. the effects of all the involved single faults are present. ln this case, when multiple faults arc present, each of them can be selected using the indices cons" and rel" of the single fault case. The idea is to look for sets of faults with minimal cardinality (in prac tice one fault sets, first, then two faults sets and so on) whose joint effects are consistent and relevant w.r.t. the observations. This procedure is inspired from the parsimonious covering pro cedure, which was first suggested by Peng and Reggia in [START_REF] Peng | Abductîve inference models for di agnostic problern-solving[END_REF]. From a crisp point of view, the "superposition hypothesis" means that the effects of multiple fautts on an attribute are the intersection of the effects of the corresponding single faults on the same attributc (the effeets being represented by sets of pos sible values).

Note that when f E F and a E 9-l f , then fis not tinked to a and so it has no incidence on the computation of the link between F and a. If the above situation holds V f E F for attribute a, then Fis not linked to a (a E 'J.f.J.): nf and 6f fottow [START_REF] De Mouzon | Using consistency and abduction based indices in possibilistic causal diagnosis[END_REF]. Otherwisc:

VF Ç:F ,VaE.:il,VuE Va, n F (u) = min n[(u) a fE{gEF\aE'l(_g} (6)
and o;;•(u) = min ô[(u). fE(gEFlaE'l(_gU'll g } Note that the minimum in [START_REF] De Mouzon | Extensions of the fuzzy pattern matching approach to diagnosis[END_REF] should always be based on f E {g E F J a E �U 'll g }, But from a computational point ofview, the f such that a E 'l1 f can be left asidc (rc{ = 1). On the eon trary, if {gE FI a E 'll g } f. 0, then �=O.

Note that pair ( n;., 6;;°) is stitl a twofold fuzzy set. The fuzzy intersection is justified for both n; and ô1;' as rt;. represents the fuzzy set of values which are made impossible by none of the f E F and 6f the fuzzy set of values which are known feasible for all the f E F.

Yet, the "superposition hypothesis" is not always acceptable: Sorne effects cannot superpose as they are contradictory (they lead to rcf = 0, i.e. attribute a may not have a value when F is present!), For instance, let disease a give fever, disease 13 give hypothermia. What can be computed for the cffect on the body temperature when those two diseases are simultaneously present? Will the effect of one fault be stronger than the effect of the other? or wilt both effects compensate into a normal body temperature? or will the simultaneous presence of the Iwo dis eases lead to something different, as a very strong fever? Here, the "superposition hypothesis" cannot hold.

In fact, all these types of effect combination can be mod eled. For instance, "one effect is stronger" (but we do not know which one: It may depend on the person or on the states of the diseases) is captured by the union, in the crisp case, and leads to: '<IF Ç '.f ,'<la E�,'vu E U 0 ,n:t'(u) = maxJEf;:EFlaE 'Kg }n:{(u) and of(u) = maxfE{gEFlaE 1(. U '11,}ô{(u). Note that (rc;,oi) is still a twofold fuzzy set. Ali types of effect combination can be computed. The important point is that the calculation must yield a twofold fuzzy set. Another combination of interest could be the worsening of the effects. For instance, if one disease gives fever and an other too, the presence of both diseases might give a strong fever.

Il might seem that wc are back to the initial problcm: For cv ery multiple faults, and each attribute, the type of combination must be defined. This is !rue when there are no means to "guess" the type of combination. Yet, in most of the cases (in the domain where this multi-fault diagnosis is applied), the "superposition hypothesis" holds, unless it cornes to a contradiction. In the cases wherc a contradiction is reached, ail types of combination can be found. But, as a general approach is nceded, we might just say that when "superposition hypothesis" does not hold for F on a, F bas an effect on a but it cannol be computed precisely. So, � = 1 (a could possibly take any value when F is prescnt) and � = 0 (no value is more characteristic than another). Of course, when this effect combination computation is not satis fying for multiple fault '.f on attribute a, it is still possible to define � and � in the knowlcdge base.

V CASCAüING FAULTS Another interesting phenomenon in diagnosis problems is the possible cascade from one fault to another: Fault fi may have some effects such that after a while fault [2 will occur so on ... ). For instance, if you do not stop early enough your in fluenza, you may get bronchitis. Thcn, you might have both influenza and bronchitis at the same lime. Using multiple-fault diagnosis, these cascading faults can be identified as several successive single or multiple faults. Here: First influenza takes place as a single disease, then both in fluenza and bronchitis. Yet, the cascade itself is not recognized through the diagnosis although it could be a good help in the diagnosis process. lndecd, a cascade is rathcr a single fault inducing other faults than a multiple fault involving several independent faults. In order to take this into account in the parsimonious covering proccdure, a cascade (no matter how many faults it induces or whether it is cycling or not) is counted as only one fault.

For this, let C be a binary relation in :f x :J, with C (/1, fz) = 1 if and only if fault fi may induce fault [2. Tuen, as soon as 3f, 3tocons"(f,to) > 0 and cons"(!/)= 0 for t 1 < to, it means that the presence off, or of a cascade starting with f (the possi ble cascades are given by C) is consistent with the observations. Here, cons"(f,t) is the result of cons"(!) at time t. The corre sponding extension is also valid for rel 0 .

In practice the diagnosis system first only searches for single faults. After one is found (fi), it searches for this single fault or cascades starting with this single fault. If necessary, it searches for a multiple fault of cardinality 2, made of fi (or one of its cascade) and another single fault (/2). Tuen it searches for a multiple fault of cardinality 2 made of fi (or one of its cascade) and /2 (or one of ils cascade), and so on. The underlying hy-pothesis is that a fault stays until it has been corrected, once it has occurred.

Another improvement is tu have C (fi, fz) expressing the lime for h to appear after /1 has taken place (may be in a fuzzy way). This additional information would be a useful ingredient to incorporate in cons 11 and re/ 0 in order to identify cascades.

If the hypothesis -that a fault lasts until it is corrected -does not hold, cascades have to be dcscribed as chroniclcs, giving also (possibly) fuzzy time periods during which each fault lasts before it disappears (see [START_REF] De Mouzon | Extensions of the fuzzy pattern matching approach to diagnosis[END_REF]).

VI CONCLUSION

This paper has shown how the basic fuzzy pattern matching approach to diagnosis problem could be refined in order to al low for a more refined representation framcwork (distinguishing betwcen effects which are surely possible and effccts which arc not impossible), to take advantage of normal behavior informa tion, and to cope with multiple faults.

The approach has been implcmcntcd on a car engine dyno test bcnch problem, where observations arrive in real lime. Pre liminary results are promising [START_REF] Boverie | Using fuzziness for causal diagnosis in en gine dyno test bcnches[END_REF].

  aEXf

a2

  takes abnormal values, if /z is present), while rtbt = 1 -Pa 2 • h h If cons""' = cons""' = 1 and µ� 2 :S 1 -Pa 2 (1.e. a2 has normal values), thcn the diagnosis will select /1 (with re/ 0 ) and discard h (with cons"), as expected. h h f Now, suppose that cons lla 1 = cons" 0 1 = l andµ� 2 :S: Oa� (i.e. a2 takes ab normal values linkcd with the prescnce of [2). The diag nosis should lead to h Yet, with rcfl = 1 and ô[l = 1, as in the approach of Sections A or II, h and fi are both proposed. Us ing the information on (ab)normal values (rc[l = 6bl = 1-PaJ leads to selccting h with rel 0 (cons"(f1 ) = 1 and rel r, (f2) 2: 0.5 as µ� 2 :S oli :S n[D and discarding fi with co1is" (cons"(f1) :S 0.5 as µ� 2 :S n[� � p" 2 and nbl = 1 -Paz>- Moreover, information on (ab)normal values may have an im portant role in multiple-fault diagnosis, as can be seen below.

reprl, (resp. ref'(f)) is 1 when the observation of a (

  resp. ail the observations) is (resp. are) compatible for sure with the presence off.Let x._ 1 denote the set of all attributes a for which there is a &,awn relation with the presence off. Tuen, if a if_ X..r, nt is 1 on Va. So, from a computational point of view, min can be aE.i'I replaced by min in (1) and (3).

	incom
	patible with the observation on a (resp. an observation);
	• rel" evaluates how relevant to the observations a fault is:
	rel"([)== min rel"( a0!	(3 )
	whcre rel"t evaluates to what extent the relation betwcen the
	observation of a and the presence off holds:
		(4)
	where ➔D denotes Dienes' fuzzy implication: (x ➔Dy) =
	min(l-x,y).

) uEUa cons"[ (resp. cons"([)) is O when the presence off is


MIT, USA Eric Atkinson

Guillaume Baudart

Checking Bounded-Memory Execution for Delayed Sampling on Probabilistic Streams

Probabilistic Programming with Streams. Prior work introduced a probabilistic programming language, ProbZelus, to extend probabilistic programming functionality to unbounded streams of data [

Bounded-Memory Delayed Sampling. The challenge in adapting delayed sampling to computations on streams is that such computations run for indefinite periods of time and are often subject to stringent limits on resources, such as memory. [START_REF] Baudart | Reactive Probabilistic Programming[END_REF] showed that, in many cases, only a finite number of nodes in delayed sampling's graph data structures were reachable at any given time, and the rest could not influence the computation in the future and could be removed from memory. However, this behavior depends on the probabilistic model under consideration; delayed sampling is not guaranteed to maintain a bounded amount of memory for all programs. The result is then that though probabilistic programming languages are designed to hide the complexities of developing probabilistic inference algorithms, certain compositions of a model and the inference algorithm will result in undesirable behaviors that the developer did not anticipate. Moreover, the developer has no means to reason about these behaviors except by inspecting the implementation of the inference algorithm.

Our goal is to identify the semantic conditions under which applying delayed sampling to probabilistic programs that operate on streams will execute in bounded memory and to define a static analysis to enforce them.

In the following we present on an example the programming model of ProbZelus (Section 2) and how we can statically check that the delayed sampling inference algorithm can execute in bounded memory (Section 3).

PROBABILISTIC PROGRAMMING WITH STREAMS

The stream function hmm presented in Figure 1 specifies a hidden Markov model [START_REF] Baum | Statistical Inference for Probabilistic Functions of Finite State Markov Chains[END_REF], a common probabilistic model for tracking applications in which the goal is to estimate the trajectory of an object given noisy measurements of the object's position. We present the example in 𝜇𝐹 , a purely functional calculus for probabilistic programming with streams. The program is a set of stream function definitions that each consist of (1) an initializer, and (2) a step function that given the previous state and an input value produces an output value and a new state [START_REF] Mealy | A method for synthesizing sequential circuits[END_REF]]. The stream function's state consists of a latent random variable, pre_x, that denotes the position at the previous iteration. The state is latent in that it is not possible to directly observe the current position; instead we must leverage a noisy measurement or observation of the position to infer a probability distribution over potential states.

Inside the definition of hmm, the program models the latent nature of x by sampling the current position from a Gaussian distribution centered around its previous position pre_x (L.4). The program models the observation by taking the observed sensor value as input, obs, and supplying it as an input to the observe operator. In this example, the observe specifies that obs is an observation from a Gaussian distribution centered around the position x. The observe operator conditions the program's execution on the observed value (L.5) in that it adjusts the distribution that will be inferred for x.

The sequence of diagrams in Figure 2 illustrates the evolution of a representation of the hidden Markov model over the first four iterations of the program. Each light grey node denotes a latent random variable for pre_x or x at a given iteration. Each dark grey node denotes an observation at the given iteration. Each solid black arrow signifies a dependence between random variables as in a traditional Bayesian network representation of probabilistic graphical model [START_REF] Koller | Probabilistic Graphical Models -Principles and Techniques[END_REF]. Of note, each observation at each iteration depends on the current position and the state at a given iteration depends only on the position at the previous iteration.

INFERENCE WITH DELAYED SAMPLING

The hmm probabilistic model alone is not enough to be able to reason about the estimated position. Instead, we must perform inference on the model to compute a posterior distribution of x conditioned on observations. In this paper, we study delayed sampling [START_REF] Baudart | Reactive Probabilistic Programming[END_REF][START_REF] Murray | Delayed Sampling and Automatic Rao-Blackwellization of Probabilistic Programs[END_REF] as the algorithmic implementation of the inference.

Delayed sampling leverages symbolic execution to reason about the relationship between random values to perform exact inference in the form of returning, if possible, a closed-formed distribution. Delayed sampling operates by dynamically maintaining a graph -i.e., a Bayesian network -that records the dependence relationships between the random variables in the program (Figure 2). The key idea is that rather than sample a concrete value for each random variable in the program (e.g., x), delayed sampling instead returns a reference to a node in the graph. This node contains a closedform representation of the distribution that the sample operator sampled from, along with the distribution's dependence on other random variables in the program.

Bounded-Memory Delayed Sampling

A key concern when applying delayed sampling to streams, which may execute for an indefinite number of iterations, is if the size of the delayed sampling graph is bounded from above by a fixed constant for all iterations of the program. If not, then the delayed sampling graph may not consume bounded memory and, hence, the program may exhaust its resources if permitted to execute indefinitely as streams are often permitted to do. [START_REF] Baudart | Reactive Probabilistic Programming[END_REF]. Each node denotes either a value (dark gray) or a distribution (light gray). A plain arrow denotes a dependency in the underlying Bayesian network. A dotted arrow denotes a pointer in the implementation of the delayed sampling graph. Each label indicates the program variable that correponds to a node. An ✗ on a node denotes that the node is not reachable from the program state.

val hmm_first = stream { init = (true, 0.0, 0.0); step ((first, i, pre_x), obs) = let (i, pre_x) = if first then (let i = sample (gaussian (0.0, 1.0)) in (i, i)) else (i, pre_x) in let x = sample (gaussian (pre_x, 1.0)) in let () = observe (gaussian (x, 1.0), obs) in (x, (false, i, x)) } Fig. 3. Model with unbounded memory consumption.

In general, bounding memory use is challenging because the underlying Bayesian network can in fact be unbounded. Nevertheless, a delayed sampling implementation can maintain bounded memory for some programs, depending on the operation of said programs.

Bounded-Memory Example. Figure 2 shows how delayed sampling maintains bounded memory for the program in Figure 1. The delayed sampling implementation must keep in memory all the nodes that are reachable from any node referenced in the program state. The dashed lines in Figure 2 visualize the reachability relation: the node each line points to is reachable from the node the line points from. As the program evolves its state and changes the variables the state contains, nodes in the delayed sampling graph may become unreachable. Nodes marked with an ✗ are unreachable.

Figure 2a shows the delayed sampling graph after the first iteration. The graph consists of two nodes: one introduced by sampling the variable x, and one introduced by the observation of obs. At the end of the step, only the variable x is in the program state and reachable. The node introduced by the observation represents a concrete value that is no longer reachable.

Figure 2b shows the delayed sampling graph after the second iteration. The program has added two nodes to the graph for sampling x and observing obs. Again, only the variable x is in the program state and reachable. The nodes left over from the first iteration and the node introduced for the observation are still in the graph, but are no longer reachable.

Figures 2c and2d show the delayed sampling graph at iterations 3 and 4, respectively. In each case the most recently introduced node for x is reachable, and the nodes from the previous iterations and the observation obs are unreachable. This pattern continues at future iterations. In general, the program ensures that at any iteration, the most recently introduced node is reachable, and the rest are unreachable. Because the set of reachable nodes is at most one for all iterations, inference executes in bounded memory.

Unbounded-Memory Example. Figure 3 presents an example of a program that does not execute in bounded memory. This is a modified version of hmm from Figure 1 that samples an initial latent The trace is a ::-separated list of primitive operations, where each primitive operation is a sampling operation or an observation operation obs. In this diagram, we use x 𝑛 and y 𝑛 to refer to the random variables introduced at iteration 𝑛 by, respectively, sampling x and observing obs in Figure 1.

𝜏 2 = i nil :: x 1 i :: y 1 x 1 :: obs y 1 :: x 2 x 1 :: y 2 x 2 :: obs y 2 𝑣 2 = (false, i, x 2 ) Fig. 6. A depiction of a trace of the program in Figure 3. The figure depicts the trace 𝜏 2 and the value of the program state 𝑣 2 at the end of iteration 2. In this diagram, we use i, x 𝑛 , y 𝑛 , respectively, to refer to the random variable introduced by sampling i, the variable introduced at iteration 𝑛 by sampling x, and the variable introduced at iteration 𝑛 by observing obs in Figure 3. We have highlighted the elements of the unseparated paths between i and x 2 in green.

position i from a Gaussian distribution and keeps a reference to this random variable in the state.

Figure 4 shows how the program in Figure 3 fails to maintain bounded memory.

Figure 4a shows the delayed sampling graph after the first iteration. The graph consists of three nodes introduced by sampling the variable i, sampling the variable x, and by the observation of obs. The program state comprises variables i and x, and their associated nodes are reachable.

Figure 4b shows the delayed sampling graph after the second iteration. The program has added two nodes to the graph for sampling x and observing obs. Since the variable i is in the program state, the node between i and x is reachable.

Figures 4c and4d show that in the next iterations two new nodes are introduced at each step and one remains reachable. The primary observation to note is that the number of introduced nodes increases at every iteration. Therefore, there is no bound on the size of the delayed sampling graph and, hence, the program does not execute in bounded memory.

Analyzing Delayed Sampling

We propose an analysis that can show that the program in Figure 1 maintains bounded memory while the program in Figure 3 does not. For that, we define two dataflow properties that encode whether a program execution achieves bounded memory: the unseparated path property and the 𝑚-consumed property. We then show how these dataflow properties can be verified using a static analysis.

𝜏 2 = i nil :: x 1 i :: y 1 x 1 :: obs y 1 :: x 2 x 1 :: y 2 x 2 :: obs y 2 𝑚-consumed i x 1 y 1 x 2 y 2 unseparated paths (i, i), 0 (i, x 1 ), 1 (i, y 1 ), 2 (i, x 1 ), 1 (i, x 2 ), 2 (i, y 2 ), 3 (i, x 2 ), 2 Fig. 7. A depiction of the abstract graphs of the program in Figure 3, with the same trace as Figure 6. At each operation, we depict the 𝑚-consumed abstract graph, which is a set of nodes that have been introduced but not consumed. Because this set is empty at the end of any iteration, the program satisfies the 𝑚-consumed semantic property. The unseparated paths abstract graph is a mapping, for each unseparated path in the graph, from its pair of endpoints to its length. We depict the longest path residing in the mapping. After each iteration, this longest path continues to lengthen, so the program does not satisfy the unseparated paths semantic property.

Traces. We formalize the dataflow properties as properties of traces. A trace is a recording of the important features of a program execution. In our case, a trace records all sampling and observation operations that the program has executed, as well as the variables that were involved in these operations. Figure 5 illustrates a trace of the execution of the program in Figure 1. Unseparated Paths. An unseparated path in a trace is a sequence of variables {𝑥 𝑖 }, where the trace specifies that each variable 𝑥 𝑖 was sampled from its predecessor 𝑥 𝑖-1 and no 𝑥 𝑖 is observed. The unseparated path property states that there is a uniform bound 𝑐 so that for all iterations no variable in the program state starts an unseparated path with more than 𝑐 variables in it.

Figure 6 illustrates the trace for the program in Figure 3. This program carries the variable i in the program state, and because the trace specifies that x 1 was sampled from i, and x 2 was sampled from x 1 , the sequence i, x 1 , x 2 is an unseparated path with 3 variables. In general, at iteration 𝑛, the program in Figure 3 maintains that i is in the program state and starts an unseparated path with length 𝑛 + 2. Because no bound can exist on the length of this path for an arbitrary number of iterations, this program fails the unseparated path property.

𝑚-consumed.

A variable is 𝑚-consumed if it is no more than 𝑚 sampling operations away from a variable that is consumed by an observe statement. The 𝑚-consumed property states that there is a uniform bound 𝑚 such every variable introduced by a sampling operation is 𝑚-consumed for some 𝑚 ≤ 𝑚. We note that the traces in Figures 5 and6 satisfy the 𝑚-consumed property, because every variable is at most 2-consumed. For all 𝑡, y 𝑡 is 0-consumed because it is directly observed, and x 𝑡 is 1-consumed because y 𝑡 is sampled from x 𝑡 and y 𝑡 is 0-consumed. The variable i is 2-consumed because x 1 is sampled from i, and x 1 is 1-consumed.

Analysis. Our goal is ultimately to analyze whether a given program will execute in bounded memory. This property is satisfied iff it satisfies both the unseparated path and 𝑚-consumed properties. This reduces the problem of analyzing the bounded-memory behavior of a program to analyzing these dataflow properties. Our analysis utilizes an abstract delayed sampling graph with the key aspects of these properties. For the 𝑚-consumed property, the abstract graph maintains a set of variables that have been introduced but not yet consumed, and for unseparated paths, it maintains an upper bound on their length. For example, the abstract graphs for the trace in Figure 6 are given in Figure 7.

CONCLUSION

Probabilistic programming has been augmented by constructs that perform inference over unbounded iterations on streams of data. Underlying this programming model is delayed sampling, which combines the benefits of exact inference and the flexibility of sampling.

We introduce the 𝑚-consumed and unseparated path semantic properties, which show that delayed sampling can execute in bounded memory for reactive probabilistic programs. To the best of our knowledge, our work is the first to develop a resource analysis for a probabilistic program in relation to its probabilistic programming system's underlying inference algorithm. We hope this work will enable automatic inference mechanisms whose performance is better understood by model developers in probabilistic programming languages.

  Fig. 1. A streaming probabilistic model (HMM) in 𝜇𝐹 .

Fig. 2 .

 2 Fig.2. The evolution of the delayed sampling graph for the hidden Markov model in Figure1(hmm) as implemented by[START_REF] Baudart | Reactive Probabilistic Programming[END_REF]. Each node denotes either a value (dark gray) or a distribution (light gray). A plain arrow denotes a dependency in the underlying Bayesian network. A dotted arrow denotes a pointer in the implementation of the delayed sampling graph. Each label indicates the program variable that

Fig. 4 .

 4 Fig. 4. The evolution of the delayed sampling graph for the variant of a HMM probabilistic model in Figure3. Nodes and edges have the same meaning as in Figure2.𝜏 2 = x 1 nil :: y 1 x 0 :: obs y 0 :: x 2 x 1 :: y 2 x 2 :: obs y 2


Julien Christmann

Jean-Luc Gardette

Gérard Pichon

Bruno Bouchut

Sandrine Therias
email: sandrine.therias@uca.fr

Photostabilization of polyethylene by a hindered amine light stabilizer in blooming conditions and impact of MDO processing

Keywords: Polyethylene, HALS, Tinuvin® 770, Machine Direction Orientation, Photooxidation

The effect of Machine Direction Orientation (MDO) processing on the photostabilization of LLDPE samples containing Tinuvin® 770 was studied. The amount of Tinuvin® 770 in the processed samples was deliberately chosen at a level higher than the reported solubility at ambient temperature for two main reasons: the first reason was to assess the equilibrium between the soluble and nonsoluble forms of this additive and the impact of the MDO process, and the second was to demonstrate the role of a higher temperature in accelerated artificial photoaging on the protective role played by the nonsoluble part. The results reported in this article show that at room temperature, a nonnegligible amount of Tinuvin® 770 precipitates at the surface, which is commonly termed blooming. However, the additive that bloomed at the surface can be dissolved within the polymer with increasing temperature, which is the case in the conditions of accelerated artificial photoaging but not the case of natural outdoor weathering. Such physical phenomena are responsible for a possible bias brought by temperature during accelerated photoaging, which can affect aging tests. This questions the representativeness of accelerated aging at higher temperatures for polymers containing a blooming stabilizer.

Introduction

It is well established that polymers, such as polyethylene, are prone to degradation through an oxidative chain-radical mechanism upon exposure to external stress factors such as heat or UV light [START_REF] Rivaton | Basic aspects of polymer degradation[END_REF][START_REF] Rabek | Polymer Photodegradation[END_REF][START_REF] Rabek | Photodegradation of Polymers[END_REF]. This leads to the production of oxidation products, such as ketones (further reacting by Norrish mechanisms), carboxylic acids, and aldehydes, and causes the modification of the polymer properties (especially mechanical and aesthetic properties) [START_REF] Rabek | Photodegradation of Polymers[END_REF][START_REF] Gardette | Photo-and thermal-oxidation of polyethylene: Comparison of mechanisms and influence of unsaturation content[END_REF]. To prevent photodegradation during processing and subsequent service life, stabilizers (or antioxidants) and/or UV absorbers are generally added to the polymer formulation at concentrations of 0.01 to 1.0 wt%. Stabilizers are organic compounds readily able to slow oxidation reactions by reacting with key intermediates such as hydroperoxides and radicals. UV absorbers aim to protect the polymer from UV absorption and/or the stabilizers from direct photodegradation, which dramatically reduces their protective effect. To act efficiently, an additive must be incorporated in an active and soluble form.

The solubility of an additive within a polymer is actually governed by several factors [START_REF] Billingham | Solubility of phenolic antioxidants in polyolefins[END_REF][START_REF] Billingham | The solubility of stabilizing additives in polypropylene[END_REF]: the temperature, the intrinsic nature of the additive (its heat of fusion and melting point) and its interactions with the polymer (also referred to as their compatibility). It is also well documented that soluble additives are only present in the amorphous phase of polyolefins, as the crystalline areas are too densely packed for the additive to enter [START_REF] Frank | Distribution of ultraviolet stabilizers in crystalline polypropylene[END_REF][START_REF] Billingham | Applications of ultraviolet microscopy to polymers[END_REF]. Because they are mainly based on functional groups such as phenols, amines, phosphites or sulfides, common stabilizers are polar molecules whose solubility in apolar polyolefins, such as polyethylene, is poor at ambient temperature [START_REF] Billingham | Solubility of phenolic antioxidants in polyolefins[END_REF][START_REF] Malík | Diffusion of hindered amine light stabilizers in low density polyethylene and isotactlc polypropylene[END_REF].

Additives are generally introduced at concentrations far higher than room-temperature equilibrium concentrations. They are nevertheless fully soluble during polymer processing at high temperature due to the beneficial effect of temperature on the solubility. This results in supersaturation upon cooling and storage, which is an important factor in determining their service lifetime and has to be considered in accelerated aging tests [START_REF] Billingham | The solubility of stabilizing additives in polypropylene[END_REF]. Two phenomena can then be encountered: i. microprecipitation (formation of stabilizer nodules within the polymer, leading to microphase separation) and ii. blooming (surface segregation of the stabilizer through exudation). Low-weight (molecular) antioxidants are prone to blooming, as their diffusion coefficient within the polymer is higher [START_REF] Malík | Diffusion of hindered amine light stabilizers in low density polyethylene and isotactlc polypropylene[END_REF]. In a recent paper, Xu et al. [START_REF] Xu | Physico-chemical characterization of the blooming of Irganox 1076® antioxidant onto the surface of a silane-crosslinked polyethylene[END_REF][START_REF] Xu | Physico-chemical analysis of a silane-grafted polyethylene stabilized with an excess of Iragnox 1076®. Proposal of a microstructural model[END_REF] studied the blooming of the molecular phenolic antioxidant Irganox 1076® at the surface of silane-crosslinked polyethylene.

Tinuvin® 770 is a low-weight Hindered Amine Light Stabilizer (HALS) used in polyolefins [START_REF] Malík | Diffusion of hindered amine light stabilizers in low density polyethylene and isotactlc polypropylene[END_REF][START_REF] Zehnacker | IR spectrometric determination of concentrations of soluble and insoluble fractions of bis(2,2,6,6-tetramethyl-4-piperidinyl) sebacate in polymers: application to LDPE films[END_REF][START_REF] Qing | Photoinitiated crosslinking of low density polyethylene. III: Degradation and stabilization of photocrosslinked polyethylene[END_REF][START_REF] Allen | A comparative study of structure and polymer effects on the photo-stabilising activity of hindered piperidine light stabilisers[END_REF]. It contains two piperidyl rings separated by a sebacate ester functional group (Figure 1). Its solubility in LDPE has been estimated to be 0.1 wt% at room temperature [START_REF] Malík | Diffusion of hindered amine light stabilizers in low density polyethylene and isotactlc polypropylene[END_REF][START_REF] Zehnacker | IR spectrometric determination of concentrations of soluble and insoluble fractions of bis(2,2,6,6-tetramethyl-4-piperidinyl) sebacate in polymers: application to LDPE films[END_REF]. Progressive blooming from this polymer after high-temperature processing was studied by Zehnacker et al. [START_REF] Zehnacker | IR spectrometric determination of concentrations of soluble and insoluble fractions of bis(2,2,6,6-tetramethyl-4-piperidinyl) sebacate in polymers: application to LDPE films[END_REF]. Machine direction orientation (MDO) is a technology essentially employed for polyethylene (PE) and consists of uniaxially stretching a polymer film between consecutive hot and cold rolls to improve its properties while reducing the amount of matter necessary to reach them. More specifically, the process can be decomposed into four steps: preheating, stretching, annealing, and cooling [START_REF] Breese | The Wiley Encyclopedia of Packaging Technology (3 rd Edition)[END_REF][START_REF] Hatfield | Multilayer Flexible Packaging (2 nd Edition)[END_REF]. First, heated rolls uniformly bring the polymer film to the desired temperature for stretching. Then, it is drawn between two rolls inversely rotating at different speeds (namely, the "slow" and "fast" rolls). The drawn film goes between heated rolls for thermal stabilization, especially for shrinkage ability. Finally, it is cooled at ambient temperature before winding. The draw ratio (DR or ) corresponds to the ratio between the initial and final thicknesses of the film, which is mainly governed by the ratio of the speeds of the two drawing rolls. As the drawing temperature is below the melting point of the polymer, this process is referred to as a "cold-drawing" process. At the macromolecular scale, it causes the initial spherulitic morphology to progressively transform into a fibrillar morphology oriented along the machine direction (MD) as the drawing rate is increased [START_REF] Bafna | Integrated mechanism for the morphological structure development in HDPE melt-blown and Machine-Direction-Oriented films[END_REF]. Accordingly, the polymer optical (increase in gloss and transparency, decrease in haze) [START_REF] Hatfield | New MDO medium molecular weight high density polyethylene films[END_REF][START_REF] Tabatabaei | Effect of Machine Direction Orientation conditions on properties of HDPE films[END_REF][START_REF] Chatterjee | Machine Direction Orientation of high density polyethylene (HDPE): Barrier and optical properties[END_REF], mechanical (increase in the Young's modulus along the machine and transverse directions, as well as stress at break along the MD) [START_REF] Hatfield | New MDO medium molecular weight high density polyethylene films[END_REF][START_REF] Tabatabaei | Effect of Machine Direction Orientation conditions on properties of HDPE films[END_REF][START_REF] Chatterjee | Machine Direction Orientation of high density polyethylene (HDPE): Barrier and optical properties[END_REF][START_REF] Srinivas | Structure and properties of oriented polyethylene films[END_REF] and barrier (gas, water vapor) [START_REF] Hatfield | New MDO medium molecular weight high density polyethylene films[END_REF][START_REF] Srinivas | Structure and properties of oriented polyethylene films[END_REF] properties are enhanced. MDO technology was developed 50 years ago [START_REF] Schut | MDO films: Lots of promise, big challenges[END_REF] and is widely used at the industrial scale. However, its effect on polymer properties remains confidential in the scientific literature [START_REF] Bafna | Integrated mechanism for the morphological structure development in HDPE melt-blown and Machine-Direction-Oriented films[END_REF][START_REF] Hatfield | New MDO medium molecular weight high density polyethylene films[END_REF][START_REF] Tabatabaei | Effect of Machine Direction Orientation conditions on properties of HDPE films[END_REF][START_REF] Chatterjee | Machine Direction Orientation of high density polyethylene (HDPE): Barrier and optical properties[END_REF][START_REF] Srinivas | Structure and properties of oriented polyethylene films[END_REF][START_REF] Ratta | Structure-property-processing investigations of the tenter-frame process for making biaxially oriented HDPE film. I. Base sheet and draw along the MD[END_REF][START_REF] Breese | Modelling the mechanical properties of highly oriented polymer films: A fibre/gel composite theory approach[END_REF], and its consequences on stabilizer solubility and polymer photoaging have never been considered thus far.

From a general point of view, the effect of polymer orientation (regardless of its origin) on additive solubility has scarcely been considered. Moisan studied the evolution of Irganox 1076® diffusion and solubility in oriented LDPE films [START_REF] Moisan | Diffusion des additifs du polyethylène -III. Influence de l'orientation[END_REF]. Following an increase at a low draw ratio, the solubility was reported to decrease as the orientation increased. However, all the experiments were performed at temperatures higher than the melting point of the additive. This paper aims to investigate the impact of the MDO process on the photostability of PE films containing Tinuvin® 770, with special concern for the protective role of the stabilizer precipitated at the surface (blooming) in accelerated photoaging tests. For this purpose, LLDPE films were deliberately processed at 0.3 wt% HALS, which is a concentration that can lead to supersaturation of the additive at room temperature. Tinuvin® 770 was first individually characterized, and its presence in the films was assessed by infrared spectroscopy. Reversible blooming of the stabilizer was evidenced by washing and thermal treatment, and quantification of the amount of soluble HALS was performed. One may question how much of the additive can be present in the polymer at equilibrium and how physical phenomena affect aging tests. Undrawn and MDO-stretched samples were compared with respect to their Tinuvin® 770 content. Finally, photoaging kinetics in conditions of accelerated artificial aging were obtained and discussed based on the effect of the MDO process on the stabilizer total content.

Experimental

Materials

The linear low-density polyethylene (hereafter denoted "PE-") was LLDPE BorShape™ FX 1001 grade supplied by Borealis [START_REF]BorShape™ FX 1001 product data sheet[END_REF]. Tinuvin® 770 (bis(2,2,6,6-tetramethyl-4-piperidyl) sebacate, m.p. = 81-83 °C [START_REF] Billingham | The solubility of stabilizing additives in polypropylene[END_REF][START_REF] Malík | Diffusion of hindered amine light stabilizers in low density polyethylene and isotactlc polypropylene[END_REF][START_REF] Zehnacker | IR spectrometric determination of concentrations of soluble and insoluble fractions of bis(2,2,6,6-tetramethyl-4-piperidinyl) sebacate in polymers: application to LDPE films[END_REF]) was supplied by BASF. The ethanol (EtOH, 96 %) used for washing was purchased from VWR Chemicals. All products were used as received.

MDO processing

All PE films were provided by the Barbier group (Sainte-Sigolène, France). Reference films without Tinuvin® 770 and films stabilized with 0.3 wt% Tinuvin® 770 ("PE-770-") were prepared. Undrawn films (denoted by "-1") were processed by cast extrusion to a final thickness of 50 µm. Extrusion and MDO processing was performed at Omya (Oftringen, Switzerland) from cast-extruded films of 150 µm thickness. The heating step was performed at 90 °C, with subsequent annealing performed at 60 °C before cooling at room temperature on water-cooled rolls. A final thickness of approximately 50 µm was obtained, corresponding to an MDO draw ratio of 3 (samples denoted by "-3"). A list of the PE samples is given in Table 1.

Irradiation

Photodegradation experiments were carried out in a SEPAP 12/24 unit [START_REF] Lemaire | Vieillissement des polymères: empirisme ou science? Caoutchoucs et Plastiques[END_REF] in the presence of air. This accelerated weathering device was equipped with 4 medium-pressure mercury lamps (400 W) emitting polychromatic light whose wavelengths lower than 300 nm are filtered by a borosilicate envelope. The delivered light intensity was 90 Wm -2 in the wavelength range of 300-420 nm, and the chamber temperature was measured at 52 °C. For transparent PE samples, it is assumed that the sample temperature is close to this value.

Heating

Films were submitted to thermal treatment at 52 °C (same temperature as that of the irradiation device) in a forced convection oven from Binder.

Infrared spectroscopy

Infrared spectra were recorded between 4000 and 400 cm -1 in the transmission mode with a Nicolet 6700 FTIR spectrometer (ThermoFisher) equipped with a DTGS-KBr detector and operated with OMNIC software. The analysis chamber was maintained under a dry atmosphere by the continuous purging of dry air. Each spectrum was obtained by the accumulation of 32 scans with a 2 cm -2 resolution. When necessary, FTIR spectra can be normalized by the absorbance of the 1378 cm -1 band (symmetric bending of end-of-chain -CH 3 groups).

FTIR spectroscopy was also performed in single reflection Attenuated Total Reflection (ATR) mode between 4000 and 600 cm -1 with two types of crystals (Ge and diamond). For the former, a Nicolet IS10 spectrometer (ThermoFisher) equipped with a Smart OMNI-Sampler accessory (ThermoFisher) was used (64 scans, 4 cm -1 resolution). For the latter, a Nicolet 380 apparatus coupled with a Golden Gate ATR modulus (Specac) was employed (64 scans, 4 cm -1 resolution).

In ATR mode, the penetration depth D p of the IR beam depends on its wavelength according to Eq. 1 [START_REF] Skoog | Principles of Instrumental Analysis (6 th Edition)[END_REF].

Eq. 1

In this equation,  is the wavelength, n 1 is the ATR crystal refractive index (4.0 for Ge, 2.4 for diamond),  is the angle of incidence of the IR beam (45° for the apparatus used), and n 2 is the sample refractive index (1.51 for PE [START_REF] Brandrup | Polymer Handbook (4 th Edition)[END_REF]). For instance, the penetration depth in the 1700 and 1800 cm -1 region ranges between 0.37 and 0.39 µm for a Ge crystal and is 1.19-1.26 µm for a diamond crystal. Spectra were corrected for the variation in D p with wavelength, and the shifting of absorption bands was corrected with the advanced ATR correction function of OMNIC software.

FTIR mapping was realized with a Spectrum 100 spectrometer (PerkinElmer) coupled with a Spotlight 400 imaging system (PerkinElmer). In transmission mode, images of 350×350 µm dimensions (56×56 pixels, 6.25×6.25 µm per pixel) were recorded between 4000 and 650 cm -1 (128 scans per pixel, 4 cm - 1 resolution). ATR mapping was performed with a Ge crystal to obtain images of 300×300 µm dimensions (192×192 pixels, 1.56×1.56 µm per pixel) between 4000 and 720 cm -1 (8 scans per pixel, 4 cm -1 resolution).

UV-visible spectroscopy

UV-visible spectra of polymer films were recorded with a Shimadzu UV-2600 scanning spectrophotometer equipped with an ISR-2600 Plus integration sphere unit and operated with UVProbe software. Acquisition was performed at a scan speed of 500 nm/min ("slow" mode) with a 5 nm slit width and 2 nm data interval.

SEM imaging

SEM images were recorded with an SEM-FEG Supra 55VP microscope from Zeiss. Samples were metallized with Au prior to secondary electron imaging in a high vacuum (7×10 -7 Torr) at 10 kV.

Thermal analysis

Differential Scanning Calorimetry (DSC) was performed with an 822e apparatus from Mettler Toledo. Samples (between 11 and 18 mg) were introduced into an aluminium crucible and subjected to a heating-cooling-heating cycle from -20 to 160 °C at a rate of 10 °C/min under a dry air atmosphere. The crystallinity rate was estimated from the ratio of the melting enthalpy of the sample to that of a 100 % crystalline PE sample (293 Jg -1 [START_REF] Wunderlich | Thermal Analysis of Polymeric Materials[END_REF]). Data from the first heating step were used to consider the thermal history of the sample.

Results and discussion

Characterization of Tinuvin® 770 in the PE films

FTIR spectra of pure Tinuvin® 770 were recorded in transmission (dispersed in a KBr tablet) and ATR modes (diamond and Ge, Figure S1). Several absorption bands characteristic of the functional groups of Tinuvin® 770 can be identified at 3320 cm -1 (N-H symmetric stretching), 1720 cm -1 (C=O symmetric stretching), 1242 cm -1 and 1218 cm -1 (superposition of several vibrations, especially N-H out-of-plane bending, C-N stretching and CH 2 deformation) and 1168 cm -1 (piperidyl ring asymmetric stretching and sebacate ester C-O asymmetric stretching) [START_REF] Zehnacker | IR spectrometric determination of concentrations of soluble and insoluble fractions of bis(2,2,6,6-tetramethyl-4-piperidinyl) sebacate in polymers: application to LDPE films[END_REF][START_REF] Socrates | Infrared and Raman Characteristic Group Frequencies. Tables and Charts[END_REF][START_REF] Vedal | The vibrational spectra of piperidine and morpholine and their N-deuterated analogues[END_REF][START_REF] Górnicka | Epxerimental (FT-IR and Raman) and theoretical (DFT) studies on the vibrational dynamics in cytisine[END_REF].

FTIR transmission spectra of PE films stabilized with Tinuvin® 770 were recorded for the undrawn PE-770-1 (Figure 2a) and MDO-drawn PE-770-3 films (Figure 2b). For comparison, the spectra of the pristine PE films (PE-1 and PE-3) are superimposed. It can be observed that additional bands corresponding to those reported above for the stabilizer are present. Enlarging the spectra between 1900 and 1500 cm -1 (Figure 3) reveals the presence of an extra signal located at 1738 cm -1 , either as a shoulder for PE-770-1 or a band for PE-770-3. This has been attributed by Zehnacker et al. [START_REF] Zehnacker | IR spectrometric determination of concentrations of soluble and insoluble fractions of bis(2,2,6,6-tetramethyl-4-piperidinyl) sebacate in polymers: application to LDPE films[END_REF] to the C=O symmetric stretching of Tinuvin® 770 dissolved in LDPE. According to the spectra of pure Tinuvin® 770, the 1720 cm -1 band is associated with its undissolved form (vide infra). The presence of two bands suggests that there is an equilibrium between dissolved and undissolved Tinuvin® 770 in the PE samples. This is not surprising, as the amount of stabilizer initially introduced into the polymer formulation during film processing (0.3 wt%) is higher than its reported solubility in LDPE at ambient temperature (0.1 wt%) [START_REF] Malík | Diffusion of hindered amine light stabilizers in low density polyethylene and isotactlc polypropylene[END_REF][START_REF] Zehnacker | IR spectrometric determination of concentrations of soluble and insoluble fractions of bis(2,2,6,6-tetramethyl-4-piperidinyl) sebacate in polymers: application to LDPE films[END_REF]. The insoluble part of Tinuvin® 770 would then progressively bloom at the surface of the films during cooling and storage.

Although Tinuvin® 770 is not the best commercially available HALS for PE, it can be used as an efficient probe for studying blooming and for highlighting solubility differences upon MDO processing. Indeed, it can be observed in Figure 3 that the equilibrium between soluble and nonsoluble HALS is affected by MDO processing, which will be discussed later.

Evidence of Tinuvin® 770 blooming at the PE surface

An easy way to experimentally demonstrate blooming consists of washing the sample films with a solvent of the stabilizer that is not a solvent of the polymer. Ethanol was chosen, and PE-770-1 and PE-770-3 films were briefly washed by spraying and then dried overnight. FTIR transmission spectra were recorded before and after washing (Figure 4 for PE-770-1, Figure S2 for PE-770-3). They show a general decrease in the absorbance of the Tinuvin® 770 characteristic bands. It is worthy to note a total disappearance of the 1720 cm -1 band, while the band at 1738 cm -1 remains unaltered. The complete disappearance of the band at 1720 cm -1 indicates that Tinuvin® 770 located at the surface of the films was removed by washing. To further assess the presence of bloomed Tinuvin® 770 at the surface of the samples, FTIR ATR mapping was performed for a PE-770-1 film and plotted at 1720 cm -1 (Figure 5, left). Particles of tens of microns in size with a strong absorbance at this wavenumber are distributed at the film surface (according to Eq. 1, the penetration depth at 1720 cm -1 is less than 0.5 µm). Spectra recorded at the location of the absorbing particles (Figure 5, right) show the presence of the other bands reported for Tinuvin® 770 (vide supra), and these particles can then be confidently identified as bloomed Tinuvin® 770 heterogeneously distributed at the film surface. This observation was confirmed by comparing SEM images recorded for unwashed and EtOH-washed PE-770-1 films (Figure 6). Several micrometre-sized particles are distributed at the surface of the unwashed sample but are totally absent from the washed sample. These experiments unambiguously confirm that a nonnegligible amount of Tinuvin® 770 blooms at the surface of the PE samples. PE-770-1 films were then submitted to thermal treatment at 52 °C for 20 h, and transmission FTIR spectra were recorded at different time intervals (Figure 7). Such a temperature was selected because it enables smooth heating (below the Tinuvin® 770 melting point) and corresponds to the actual temperature within the accelerated photoaging setup thereafter used (vide infra). The 1720 cm -1 band progressively decreases in intensity for the benefit of the 1738 cm -1 band, which is interpreted as a progressive solubilization of the bloomed HALS. This was confirmed by the disappearance of the micron-sized bloomed Tinuvin® 770 particles upon the same thermal treatment, as shown in the SEM images (Figure S3). A similar experiment performed on pristine PE-1 showed that no thermooxidation of PE was observed within this range of time. From the variations in the absorbance at 1738 cm -1 , it is possible to estimate the amount of bloomed Tinuvin® 770, which is thermally re-solubilized (Figure S4). More than 90 % of bloomed Tinuvin® 770 can be re-solubilized within the first 4 hours. This confirms former results reporting that the blooming of Tinuvin® 770 is thermally reversible [START_REF] Zehnacker | IR spectrometric determination of concentrations of soluble and insoluble fractions of bis(2,2,6,6-tetramethyl-4-piperidinyl) sebacate in polymers: application to LDPE films[END_REF]. Experiments run at higher temperatures (namely, 60 and 100 °C) showed a similar but faster process. Note that similar observations were made for PE-770-3 films after EtOH washing and thermal treatment.

No washing EtOH washing

Figure 7: FTIR transmission spectra of PE-770-1 before (black) and after various durations of heating at 52 °C. The arrows indicate the evolution with increasing heating duration.

FTIR transmission mapping of PE-770-1 was performed at 1720 and 1738 cm -1 before and after thermal treatment to assess the distribution of bloomed and solubilized Tinuvin® 770 respectively (Figure 8). The distribution of the stabilizer within the depth was not considered here. The mapping image at 1720 cm -1 shows that bloomed particles are heterogeneously distributed before heating and totally disappear after it (Figure 8a), in accordance with the previous reported observations. In addition, a homogeneous distribution of HALS is observed in the image at 1738 cm -1 after solubilization of the bloomed particles (Figure 8b). Solubilized Tinuvin® 770 is then homogenously distributed within the polymer rather than preferentially located close to the original position of the bloomed particles, which indicates that it can diffuse into the polymer.

Upon storage at ambient temperature, the opposite effect is observed (Figure S5), with a progressive blooming of the stabilizer at the film surface. However, note that the associated kinetics are far slower. The reported results clearly show that the solubilization/blooming of Tinuvin® 770 is a thermally activated and reversible process. In addition, the easy experimental distinction between its soluble and bloomed forms from the 1738/1720 cm -1 bands observed by FTIR spectroscopy makes Tinuvin® 770 a good probe for the HALS blooming/re-solubilization phenomenon and its consequences during accelerated photoaging.

Quantification of soluble Tinuvin® 770 in samples

Quantification of soluble Tinuvin® 770 from FTIR spectra can be performed using Eq. 2:

Eq. 2

In this equation, A 1738 and  1738 are the absorbance and the molar absorption coefficient of Tinuvin® 770 at 1738 cm -1 , respectively, M 770 is the molar mass of Tinuvin® 770 (expressed in kgmol -1 ), l is the thickness of the polymer film, and  PE is the density of the studied PE film (0.931 kgL -1 [START_REF]BorShape™ FX 1001 product data sheet[END_REF]). Factor 2 in the equations considers the fact that there are 2 ester functional groups per molecule of Tinuvin® 770.

The literature indicates a value of 580 L mol -1 cm -1 for the molar absorption coefficient at 1738 cm -1 of Tinuvin® 770 [START_REF] Malík | Diffusion of hindered amine light stabilizers in low density polyethylene and isotactlc polypropylene[END_REF]. The amount of Tinuvin® 770 was calculated after complete thermal solubilization in PE-770-1 samples (vide supra). An average value of 0.28±0.1 wt% was obtained from experiments performed for 8 samples. This value reflects the quantity initially introduced during the processing of the samples (0.3 wt%). Our results indicate that no (or at least only a negligible part of) Tinuvin® 770 is lost through evaporation during re-solubilization, which seems reasonable considering the low temperature used (52 °C) and short heating time (20 h).

The same calculation was made to obtain the amount of soluble Tinuvin® 770 at ambient temperature after EtOH washing. Values of 0.05 and 0.08 wt% were found for PE-770-1 (8 samples) and PE-770-3 (10 samples), respectively. This suggests that the amount of soluble Tinuvin® 770 in both films is lower than the reported solubility (0.1 wt%) [START_REF] Malík | Diffusion of hindered amine light stabilizers in low density polyethylene and isotactlc polypropylene[END_REF][START_REF] Zehnacker | IR spectrometric determination of concentrations of soluble and insoluble fractions of bis(2,2,6,6-tetramethyl-4-piperidinyl) sebacate in polymers: application to LDPE films[END_REF] and is also affected by MDO processing.

Impact of the MDO process on Tinuvin® 770 solubility in PE

As seen in Figure 3, there are strong differences in the 1720/1738 cm -1 ratios of the undrawn and MDO-drawn samples. Indeed, the ratio strongly favours the 1720 cm -1 band (bloomed form) for PE-770-1, while it is more equilibrated for PE-770-3. In addition, PE-770-1 appears to contain far more bloomed HALS, while PE-770-3 contains more of the soluble form (vide supra). The overall content of Tinuvin® 770 is then finally higher for the undrawn sample.

DSC experiments reveal that the crystallinity rates of the original and drawn samples are similar (approximately 47 % for both of them). The increase in the soluble content observed for the MDOprocessed samples is thus independent of the crystallinity rate. It would rather originate from the MDO process itself, which also leads to a strong decrease in the bloomed amount at the surface. These effects might then affect the photostability of the drawn samples.

Accelerated photoaging of stabilized samples

To study the effect of the MDO process on the photostability of the PE-770-1 and PE-770-3 samples, accelerated photoaging experiments were performed in a SEPAP 12/24 device at 52 °C. The extent of photooxidation was assessed by FTIR transmission (Figure 9a and Figure 10a for the 1900-1500 cm -1 range; see also Figures S6-9 for 3800-3100 and 1100-900 cm -1 ) and UV-visible spectroscopies (Figure 9b and Figure 10b).

As seen from the FTIR results of both samples, the temperature at which the experiments were carried out (52 °C) caused a re-solubilization of bloomed Tinuvin® 770 within the first hours of aging, as previously observed in section 3.3. On a longer timescale, the extent of polyethylene photooxidation can be followed through an increase in the intensity of absorption bands due to the appearance of oxidation products such as carboxylic acids (C=O stretching at 1713 cm -1 ), hydrogenbonded alcohols and hydroperoxides (O-H stretching at 3430 cm -1 ) and vinyl groups (C-H bending at 909 cm -1 ) [START_REF] Rabek | Photodegradation of Polymers[END_REF][START_REF] Gardette | Photo-and thermal-oxidation of polyethylene: Comparison of mechanisms and influence of unsaturation content[END_REF][START_REF] Socrates | Infrared and Raman Characteristic Group Frequencies. Tables and Charts[END_REF]. The latter are due to the Norrish II reactions occurring for ketones upon UV irradiation. UV-visible spectra show the progressive disappearance of bands located at 230 and 275 nm, which are characteristic of the phenolic antioxidant in the polymer used for processing (also observed for pristine PE samples). No signal from Tinuvin® 770 was observed by UV-visible spectroscopy. Later, a growing absorption front in the UV region was observed, which was associated with the formation of oxidation products [START_REF] Gardette | Photo-and thermal-oxidation of polyethylene: Comparison of mechanisms and influence of unsaturation content[END_REF]. Usually, photooxidation kinetics are drawn from the absorbance measured by subtracting the spectrum before irradiation from the spectra recorded after UV exposure. However, in the present case, the absorbance value at 1713 cm -1 before UV exposure contains a nonnegligible contribution of the 1720 cm -1 absorption band of bloomed Tinuvin® 770, which has been shown to disappear through re-solubilization in the photoaging unit (vide supra). This will then distort the associated a.

b. PE-770-3 differential spectra. Therefore, A values at 1713 cm -1 were measured by subtracting the absorbance after 100 h of UV exposure, when complete re-solubilization of the HALS stabilizer occurred (Figures S10 andS11). Such an issue was not encountered for the two other bands characteristic of the oxidation products at 3430 and 909 cm -1 (Figures S12-15). The evolution of A at 1713 cm -1 is plotted for PE-770-1 and PE-770-3 in Figure 11 (see data for 3430 and 909 cm -1 in Figures S16 andS17). Photodegradation is faster for the drawn sample than for the undrawn sample. For instance, a A value of 0.25 is reached in 2000 h for PE-770-3 vs. 2250 h for PE-770-1. A similar study was performed for undrawn (PE-1) and drawn (PE-3) pristine polyethylene (Figure S18). This MDO process alone (with the morphological evolution it induces at the macromolecular scale) does not have such an effect on the photoaging kinetics of the unstabilized samples. As a consequence, the faster photodegradation observed for PE-770-3 is due to the loss of a large part of the bloomed stabilizer during MDO processing, which is not able to re-solubilize and prevent photooxidation. Because of the fast re-solubilization induced by the conditions of the accelerated photoaging experiment, the sample that contains the higher total (soluble and bloomed) amount of HALS is consistently better protected against oxidation (PE-770-1 in that case).

To illustrate the effect of re-solubilization on the photooxidation kinetics, a similar experiment was performed on a washed PE-770-1 sample (Figure 11). The rate of formation of oxidation products is actually higher than that for the unwashed sample (A = 0.25 after 1200 h). This can be explained by the absence of the bloomed part of Tinuvin® 770 after washing, which cannot re-solubilize during the first hours of the photoaging experiment and prevent bulk oxidation of the polymer. This is an important result that unambiguously indicates that the re-solubilized form of Tinuvin® 770 is actually active for photostabilization.

Conclusion

Undrawn and MDO-drawn polyethylene samples stabilized by Tinuvin® 770 in conditions of supersaturation were studied. At room temperature, the stabilizer is present either in a soluble form within the polymer or in a precipitated form at the film surface as bloomed particles. This was confirmed by washing and heating experiments. For MDO-drawn samples with a draw ratio of 3, the total amount of Tinuvin® 770 was lower than that in the undrawn samples, whereas the amount of Tinuvin® 770 dissolved in the polymer was slightly higher. This means that the MDO process leads to a loss of bloomed Tinuvin® 770. Undrawn samples degrade faster than MDO-drawn samples during accelerated photoaging experiments performed at 52 °C in a SEPAP 12/24 unit. The bloomed stabilizer can be dissolved in the polymer by increasing the temperature, which can explain the larger total amount (soluble and bloomed) of Tinuvin® 770 in the undrawn samples. They are then better protected against oxidation during accelerated photoaging at higher temperatures. This was confirmed by the observation of faster oxidation kinetics for a sample washed prior to photooxidation to remove the bloomed HALS.

The reported results indicate that the bloomed stabilizer can be efficient and play a role as a stabilizer in the case of aging tests performed at temperatures higher than those encountered in natural weathering. The complete re-solubilization of bloomed Tinuvin® 770 at the temperature used in accelerated photoaging conditions (52 °C in the SEPAP unit) points out a possible bias brought by increasing temperature when testing polymers with blooming stabilizers, in addition to the bias already reported (day and night cycles, seasonal variations, atmospheric pollution, etc.) [START_REF] Rabek | Photodegradation of Polymers[END_REF]. Indeed, re-solubilization due to increasing temperature enhances the concentration of stabilizer within the polymer compared to its actual solubility at ambient temperature, which might affect the representativeness of accelerated photoaging compared to natural weathering. Moreover, the impact of rain during exposure to natural weathering could reduce the amount of bloomed stabilizer, as it can be progressively washed away from the surface. All these results emphasize that great care must be taken when dealing with blooming stabilizers in accelerated photoaging systems. As blooming HALS are generally used in association with non-migrating (i.e. polymeric) stabilizers, the effect of MDO processing on the solubility of such high molecular weight compounds and mixture of both kinds of additives would also be of great interest.

Figure 1 :

 1 Figure 1: Chemical structure of Tinuvin® 770.

Figure 2 :

 2 Figure 2: FTIR transmission spectra of a. undrawn samples PE-1 (light blue) and PE-770-1 (blue) and b. MDO-drawn samples PE-3 (light red) and PE-770-3 (red).

Figure 3 :

 3 Figure 3: FTIR transmission spectra of PE-770-1 (blue) and PE-770-3 (red) in the 1900-1500 cm -1region.

Figure 4 :

 4 Figure 4: FTIR transmission spectra of PE-770-1 before (light blue) and after (blue) washing with EtOH and overnight drying.

Figure 5 :

 5 Figure 5: FTIR ATR mapping image (300×300 µm) of PE-770-1 recorded at 1720 cm -1 (left) and spectra taken in the A and B zones identified in the mapping image (right). Each pixel is of 1.56×1.56 µm dimensions.

Figure 6 :

 6 Figure 6: SEM images (× 400) of unwashed (left) and EtOH-washed (right) PE-770-1 films.

1 Figure 8 :

 18 Figure 8: FTIR transmission mapping image (350×350 µm) of PE-770-1 recorded at a. 1720 cm -1 before (left) and after (right) heating and b. at 1738 cm -1 before (left) and after (right) heating. Each pixel is of 6.25×6.25 µm dimensions.

Figure 9 :

 9 Figure 9: a. FTIR transmission spectra and b. UV-visible spectra of PE-770-1 after various photoaging durations. The arrows indicate the evolution with increasing UV exposure time.

Figure 10 :

 10 Figure 10: a. FTIR transmission spectra in the 1900-1500 cm -1 region and b. UV-visible spectra of PE-770-3 after various photoaging durations. The arrows indicate the evolution with increasing UV exposure time.

Figure 10 also

 10 shows the kinetic curve for a PE-770-1 sample washed with EtOH before photooxidation (vide infra).

Figure 11 :

 11 Figure 11: Photoaging kinetics at 1713 cm -1 of PE-770-1 (blue circle), EtOH-washed PE-770-1 (green square) and PE-770-3 (red triangle). The dashed line indicates a value of A = 0.25.

Figure 1 :Figure 2 :

 12 Figure 1: Chemical structure of Tinuvin® 770 Figure 2: FTIR transmission spectra of a. undrawn samples PE-1 (light blue) and PE-770-1 (blue) and b. MDO-drawn samples PE-3 (light red) and PE-770-3 (red).

Figure 3 :

 3 Figure 3: FTIR transmission spectra of PE-770-1 (blue) and PE-770-3 (red) in the 1900-1500 cm -1 region.

Figure 4 :

 4 Figure 4: FTIR transmission spectra of PE-770-1 before (light blue) and after (blue) washing with EtOH and overnight drying.

Figure 5 :

 5 Figure 5: FTIR ATR mapping image (300×300 µm) of PE-770-1 recorded at 1720 cm -1 (left) and spectra taken in the A and B zones identified in the mapping image (right). Each pixel is of 1.56×1.56 µm dimensions.

Figure 6 :

 6 Figure 6: SEM images (× 400) of unwashed (left) and EtOH-washed (right) PE-770-1 films.

Figure 7 :

 7 Figure 7: FTIR transmission spectra of PE-770-1 before (black) and after various durations of heating at 52 °C. The arrows indicate the evolution with increasing heating duration.

Figure 8 :

 8 Figure 8: FTIR transmission mapping image (350×350 µm) of PE-770-1 recorded at a. 1720 cm -1 before (left) and after (right) heating and b. at 1738 cm -1 before (left) and after (right) heating. Each pixel is of 6.25×6.25 µm dimensions.

Figure 9 :

 9 Figure 9: a. FTIR transmission spectra and b. UV-visible spectra of PE-770-1 after various photoaging durations. The arrows indicate the evolution with increasing UV exposure time.

Figure 10 :

 10 Figure 10: a. FTIR transmission spectra in the 1900-1500 cm -1 region and b. UV-visible spectra of PE-770-3 after various photoaging durations. The arrows indicate the evolution with increasing UV exposure time.

Figure 11 :

 11 Figure 11: Photoaging kinetics at 1713 cm -1 of PE-770-1 (blue circle), EtOH-washed PE-770-1 (green square) and PE-770-3 (red triangle). The dashed line indicates a value of A = 0.25.

Table 1 :

 1 Tinuvin® 770 content and MDO draw ratio of PE samples.

	Sample	wt% Tinuvin® 770	MDO draw ratio
	PE-1	0	1 (undrawn)
	PE-3	0	3
	PE-770-1	0.3	1 (undrawn)
	PE-770-3	0.3	3

Table 1 :

 1 Tinuvin® 770 content and MDO draw ratio of PE samples.

Acknowledgements

The authors wish to thank ANR for financial support within the framework of the ANR-Labcom POPBA project. SEM imaging was performed at 2MAtech, and FTIR mapping at CNEP.