Pierre-Olivier Goffard
email: goffard@unistra.fr.

Pierrick Piette
email: pierrick.piette@gmail.com.

Gareth W Peters
email: garethpeters@ucsb.edu.

Market-based insurance ratemaking

Keywords: MSC 2010: 62P05, 91G70, 62F15 Insurance Pricing, Bayesian Inference, Approximate Bayesian Computation, Isotonic Regression

This paper introduces a novel method for pricing insurance policies using market data. The approach is designed for scenarios in which the insurance company seeks to enter a new market lacking historical data.

The methodology involves an iterative two-step process. First, a suitable parameter is proposed to characterize the underlying risk. Second, the resulting pure premium is linked to the observed commercial premium using an isotonic regression model. To validate the method, comprehensive testing is conducted on synthetic data, followed by its application to a dataset of actual pet insurance rates. To facilitate practical implementation, we have developed an R package called IsoPriceR. By addressing the challenge of pricing insurance policies in the absence of historical data, this method contributes to enhancing pricing strategies in emerging markets.

Introduction

Modern insurance pricing relies on predictive modeling methods to ensure that premiums reflect, as accurately as possible, the average cost of claims. To achieve this, insurers rely on historical data to train statistical learning models and calculate what is called the pure premium. While the foundation of standard actuarial practice often rests on generalized linear models (GLM), see Renshaw [START_REF] Renshaw | Modelling the claims process in the presence of covariates[END_REF], the relentless evolution of data science has ushered in a new era where more sophisticated machine learning algorithms are also coming into play, see Blier-Wong et al. [START_REF] Blier-Wong | Machine learning in p&c insurance: A review for pricing and reserving[END_REF] and the reference therein. However, a challenge arises when an insurance company enters a new market, lacking historical data on the risks it aims to cover. In this context, conventional predictive modeling tools are failing, leaving insurers at a crossroads, looking for innovative solutions to navigate uncharted territory.

Although an insurance company may lack historical data in a new market, there is an attractive alternative:

the ability to observe and analyze market data consisting of rates offered by competitors for similar insurance policies. Our approach leverages this market data to provide insights into the underwritten risk leading to the calculation of insurance premiums.

Consider a scenario in which the underlying risk is represented by a positive random variable, denoted as X.

This variable represents the total amount of the claim over the period covered by the insurance policy and is linked to a vector of parameters, θ ∈ R d . Our main objective is to determine the parameter values which best explains the market data, denoted as D. Market data comes from various sources. It can be collected from a multitude of quotes from competitors on the market or extracted from insurance aggregator websites.

Our approach uses this market data to derive parameter values, thereby providing a market-aligned pricing strategy.

In order to make inference on the risk profile of the loss process, given by θ, we adopt a Bayesian approach in which the parameter is a random variable having a prior distribution p(θ) that we update using the market data to yield the posterior distribution p(θ|D). Standard Bayesian techniques are not suitable here, for we cannot write the likelihood function of our data. We do not have actual data points drawn from the random variable X, since it is assumed we are in a setting in which we don't have any historical records or that they are insufficient for practical use. Instead, we will assume that what is observed and available for inference is data that consists of commercial rates offered to customers in the market for equivalent or highly similar insurance products. The parameter θ allows us to compute a pure premium p depending on the coverage brought by the insurance policy. Under the expectation principle, the commercial premium π results from applying a loading f to the pure premium p such that π = f (p) > p.

The loading function f : R + → R + is unknown and is estimated via an isotonic regression model. We chose this method because the monotonic relationship between pure and commercial premiums is desirable. Also, market data is inherently noisy and isotonic regression is more robust to outliers than a simple linear regression for example.

The procedure may be summarized as follows After iterating the above steps, we get a sequence of parameter-loading function pairs: (θ * 1 , f * 1 ), (θ * 2 , f * 2 ), . . .. This sequence equips us with the means to price our own insurance policies. The problem we tackle is an inverse problem and our solution is inspired from indirect inference methodologies pionneered by Gourieroux et al. [START_REF] Gourieroux | Indirect inference[END_REF]. The proposed algorithm resembles Approximate Bayesian Computation (ABC) algorithms described in the book of Sisson et al. [START_REF] Scott A Sisson | Handbook of Approximate Bayesian Computation[END_REF]. ABC algorithms have found successful applications in a range of actuarial science and risk management problems. We refer the readers to the works of Peters et al. [START_REF] Gareth W Peters | Chain ladder method: Bayesian bootstrap versus classical bootstrap[END_REF], Dean et al. [START_REF] Thomas A Dean | Parameter estimation for hidden markov models with intractable likelihoods[END_REF], Peters and Sisson [START_REF] Peters | Bayesian inference, monte carlo sampling and operational risk[END_REF] and Goffard and Laub [START_REF] Goffard | Approximate bayesian computations to fit and compare insurance loss models[END_REF] for further insights. Isotonic regression, a well-established statistical methodology, see for instance Barlow et al. [START_REF] Barlow | Statistical inference under order restrictions[END_REF], plays a central role in our approach. Its recent application in actuarial science, as demonstrated by Wüthrich and Ziegel [START_REF] Wüthrich | Isotonic recalibration under a low signal-to-noise ratio[END_REF], addresses the autocalibration challenges that can arise when pricing insurance contracts using machine learning algorithms.

The rest of the paper is organized as follows. Section 2 describes the risk model used in this study and discusses insurance pricing principles. Section 3 provides a detailed account of the algorithmic procedure. Our method is presented as an Approximate Bayesian Computation (ABC)-type optimization algorithm, which incorporates a simple isotonic regression model. Section 4 presents the results of a simulation study designed to showcase the performance of our method in a controlled environment. Lastly, we apply our algorithm on a dataset made of real-world pet insurance rates in Section 5.

An individual seeks to hedge against a risk X modeled by a positive random variable, over a given period of time, say one year. A common model used for X in property and casualty insurance is given by a compound loss variable

X = N k=1 U k ,
where N is a counting random variable and the U k 's are independent and identically distributed (iid) positive random variables independent from N . The random variable N is the number of occurrences of an event over a given time period (annually), each of these events is associated to a compensation U k .

Pure premium computation

An insurance company offers to bear part of this risk g(X) ≤ X in exchange for a premium which should compensate the average cost of claim given by

p = E [g(X)] ,
referred to as the pure premium. We consider in this work a function g defined as

g(x) = min(max(r • x -d, 0), l),
where r ∈ (0, 1] is the coverage rate, d > 0 is the deductible and l > 0 is the limit. Let us consider a scenario where the risk has a Poisson-lognormal distribution X ∼ Poisson(λ = 3) -LogNorm(µ = 0, σ = 1) and that n = 100 insurance coverages are proposed. These are characterized by a rate, a deductible and a limit, set randomly as

r i ∼ Unif([0.5, 1]), d i ∼ Unif([0.5, 6
]), and l = ∞, for i = 1, . . . , 100.

Figure 1 shows the pure premiums

p i = E(g i (X)) = E(min(max(r i • x -d i , 0), l i )), i = 1, . . . , 100. (1)
as a function of the rates and deductibles.

The pure premium are increasing in the rates and decreasing in the deductible. Note that the pure premiums were estimated via a Crude Monte Carlo simulation method to overcome the lack of explicit formula for the distribution functions of X. In practice, the rate offered to policyholders include a loading to compensate for the variability of the risk and cover the management costs. We describe this loading in the next section. 3) -LogNorm(µ = 0, σ = 1) risk.

From pure premiums to commercial premiums

Let f : R + → R + be a nondecreasing function, such that

π = f (p) ≥ p.
The function f is referred to as the loading function. As the commercial premium is a function of the pure premium then we are applying the expectation premium principle. Other premium principle are also possible like the standard deviation principle discussed in Remark 2.2. A simple loading function is linear in the pure premium as

f (x) = (1 + η)x,
where η > 0. The loading function used by insurance companies is unknown to us. For instance, take the pure premiums in (1) and apply the following linear loadings

η i ∼ Unif([0.5, 2]), for i = 1, . . . , n.
The commercial premium then relates to the pure premium as

π i = (1 + η i )p i , for i = 1, . . . , n. ( 2
)
Figure 2 displays the commercial premium as a function of the pure premium. In our problem we only observe the commercial premium. When considering a candidate parameter θ of the risk, we can compute the pure premiums but we need to link them to commercial premiums. For that we use isotonic regression which we describe hereafter. Pure premium Commercial premium

Isotonic regression

Isotonic regression is a statistical technique used for fitting a non-decreasing function to a set of data points.

Our datapoints are pairs of pure and commercial premiums (p i , π i ) i=1,...,n . Assume that the pure premium have been ordered such that p i ≤ p j for i ≤ j, isotonic regression seeks a least square fit π i for the π i 's such that π i ≤ π j for p i ≤ p j . It reduces to find π 1 , . . . , π n that minimize n i=1 ( π iπ i ) 2 , subject to π i ≤ π j whenever p i ≤ p j .

Since the p i 's fall in a totally ordered space, a simple iterative procedure called the Pool Adjacent Violator Algorithm (PAVA) can be used. Here's a high-level overview of how it works:

1. Initialize the sequence of values to be the same as the data points π * i = π i .

2. Iterate through the sequence and identify "violations," which occur when the current value is greater than the next value, that is

π * i > π * i+1 for some i = 1, . . . , n.
When a violation is found, adjust the values in the associated segment of the sequence to be the average of the values,

π * i ← (π * i + π * i+1 )/2,
ensuring monotonicity.

Repeat

Step 2 until no violations are left.

We use the isoreg function from R to get the fitted values π i , i = 1, . . . , n. To complete the isotonic regression task we shall find a function f such that f (p i ) = π i . A common choice is a piece-wise constant function that interpolates the π i 's. The isotonic fit of the data of Subsections 2.1 and 2.2 is provided on Figure 3.

Pure premium Commercial premium

This impression is partly due to the linear link between pure and commercial premium in (2). Isotonic regression is a non-parametric approach, meaning it doesn't make strong assumptions about the underlying distribution or functional form of the relationship between variables. This can be advantageous when the true relationship is not well

represented by a linear model. Furthermore, Isotonic regression is generally more resistant to outliers compared to linear regression due to its piecewise constant nature and the way it enforces monotonicity.

Remark 2.2. Our presentation up to now focuses on the expectation premium principle as we try to inform the link

f between the commercial premium π and the pure premium p = E[g(X)]. Other premium principles such as the standard deviation principle can be considered by slightly adapting the method. Under such principle we have

π = E[g(X)] + η • V [g(X)].
(3)

Dividing by E[g(X)] on both sides yields π

E[g(X)] = 1 + η • CV[g(X)],
where CV is the coefficient of variation. The ratio π/E[g(X)] is the inverse of the so called loss ratio in practice. It is a widely used indicator of the profitability of an insurance portfolio. The isotonic regression in this context is used to learn the link f between the inverse loss ratio and the coefficient of variation as

LR -1 = f {CV[g(X)]}.
More sophisticated premium principles such as the Escher principle or the utility indifference principle are also possible. Premium principles are described at length in actuarial science textbooks such as Dickson [START_REF] Dickson | Principles of premium calculation[END_REF]. Considering a premium principle instead of another will change the estimation of the risk parameter. It can be cast as a specific instance of model misspecification.

We are now equipped to go over the algorithm to find the parameters θ of the risk X consistent with our commercial rates π 1 , . . . , π n of which the isotonic regression is a key ingredient.

Market derived insurance ratemaking

The data at hand is a collection of insurance rates D = {π 1 , . . . , π n }. We assume that these rates were obtained through the following formula

π i = f i {E θ [g i (X)]} , i = 1, . . . , n,
where the loading function f i is unknown, the insurance coverage g i for policy i is known and the risk X is parametrized by an unknown parameter θ ∈ Θ ⊂ R d . Our solution alternates between proposing parameter values for the risk and approximating the f i 's using isotonic regression. The learning algorithm is similar to that of ABC algorithms, we simply refine the procedure laid out in the introduction to get an approximation of the posterior distribution p(θ|D).

We start by setting a prior distribution p(θ) over the parameter space that we sequentially improve through intermediate distributions characterized by a sequence of tolerance levels (ϵ g ) g≥0 that decreases gradually as The algorithm is initialized by setting ϵ 0 = ∞ and p ϵ 0 (θ|D) = p(θ). For generation g ≥ 1, we hold an intermediate distribution p ϵ g-1 (θ|D) from which we can sample particles as

∞ = ϵ 0 > ϵ 1 > ϵ 2 > . . . > 0.
θ * ∼ p ϵ g-1 (θ|D).
We compute the associated pure premium

p i = E θ * [g i (X)] , for i = 1, . . . , n.
We wish to ensure that the loss ratios defined by

LR i = p i /π i for i = 1, . . . , n, fall in a specific range LR i ∈ [LR low , LR high ].
We therefore define the distances

d * 1 = n i=1 p i • LR -1 low -π i 2 +
, and

d * 2 = n i=1 π i -p i • LR -1 high 2 + , (4)
where [x] + = max(x, 0) denotes the positive part of x. We then fit the isotonic regression model

π i = f (p i ) + e i , for i = 1, . . . , n,
where e i is an error term that captures the mismatch between the true value of the pure premium and its empirical counterpart estimated by the competitor insurance company using its historical data and the company specific loading function. The commercial premiums based on θ * and predicted by the isotonic regression model

π i = f (p i ) , for i = 1, . . . , n,
are compared to the observed rate via the distance

d * 3 = n i=1 ( π i,j -π i ) 2 .
that we combine to the distances in (4) to get

d * = d * 1 + d * 2 + d * 3 .
If the distance satisfies d * < ϵ g-1 then we keep the associated particle θ * . New particles are proposed until we reach N accepted particles denoted by θ g 1 , . . . , θ g K . We also store the distances d g 1 , . . . , d g K . We need to set the next tolerance threshold ϵ g which is used to calculate the particle weights

w g j ∝ p(θ g j ) p ϵ g-1 (θ) I d g j <ϵ g-1 , j = 1, . . . , K.
The tolerance threshold is chosen so as to maintain a specified effective sample size (ess) of K/2 as in Del Moral et al. [START_REF] Del Moral | An adaptive sequential monte carlo method for approximate bayesian computation[END_REF]. Following Kong et al. [START_REF] Kong | Sequential imputations and Bayesian missing data problems[END_REF], the ess is estimated by 1/ K j=1 (w g j ) 2 . This weighted sample then allow us to update the intermediate distribution as

p ϵ g (θ|D) = K j=1 w g j K H (θ -θ g j ),
where K H is a multivariate kde with smoothing matrix H. A common choce for the kde is the multivariate Gaussian kernel with a smoothing marix set to twice the empirical covariance matrix of the cloud of particles {θ g j , w g j } as in Beaumont et al. [START_REF] Mark | Adaptive approximate Bayesian computation[END_REF]. The procedure is summarized in Algorithm 1. The algorithm's performance is influenced by two key parameters: the number of generations, denoted as G, and the population size, denoted as K. As one would expect, the computational time for the algorithm increases with higher values of both these parameters. Therefore, the choice of suitable values for G and K can be made in consideration of a predetermined computational time budget. An alternative approach to determine the number of generations is to halt the algorithm when the difference between two consecutive tolerance levels falls below a user-defined threshold ∆ ϵ . This adaptive stopping criterion ensures that the algorithm terminates when it reaches a desired level of accuracy. The pure premiums are computed via Monte Carlo simulation. The accuracy depends on the number R of copies of X involved in the Monte Carlo estimations.

In summary, the user must configure several aspects of the algorithm. This includes setting the population size K, determining the number of generations G (or, alternatively, setting ∆ ϵ as the stopping criterion), specifying prior assumptions p(θ), a corridor of loss ratio to premium [LR low , LR high ] (which can be guided by expert opinions) and selecting the number of Monte Carlo replications R. This last choice has a direct impact on the precision of the calculation of pure premiums.

After the algorithm terminates, it is customary to focus on the last generations of particles for inference.

for j = 1 → K do 4: repeat 5: generate θ * ∼ π ϵ g-1 (θ | x) 6: compute p * i = E θ * [g i (X)] , for i = 1, . . . , n 7: compute d * 1 = n i=1 p i • LR -1 low -π i 2 +
, and

d * 2 = n i=1 π i -p i • LR -1 high 2 + 8:
fit the isotonic regression model find ϵ g ≤ ϵ g-1 so that ess = K j=1 (w

π i = f (p * i ) + e i , for i = 1, . . . , n 9: set π * i = f (p * i ), for i = 1, . . . , n 10: compute d * 3 = n i=1 (π * i -π i ) 2 11: compute d * = d * 1 + d * 2 + d *
g j ) 2 -1 ≈ K/2, where w g j ∝ p(θ g j ) p ϵ g-1 (θ g j | D) I d j <ϵ g , k = 1, . . . , K 16: compute p ϵ g (θ | D) = K j=1 w g j K H (θ -θ g j
) 17: end for study, discussed in the following section, is designed to investigate the convergence behavior and to compare the characteristics of the map and mode estimators.

Methodology Assessment via Simulation

In this section, we embark on an empirical exploration, seeking to understand how the posterior distribution of the parameters behaves as the sample size n increases. From a theoretical standpoint, it's important to recognize that commercial rates may not capture all the nuances of the underlying claim data. This divergence between the information contained in the commercial rates and the comprehensive claim data implies that the Approximate Bayesian Computation (ABC) posterior does not necessarily align with the true posterior distribution.

This experimentation has been designed to resemble as much as possible the real data situation considered in Section 5. We assume that the risk is given by

X = N i=1 U i ,
where

N ∼ Poisson(λ = 0.58), (5)
and

U i ∼ LogNorm(µ = 5.75, σ = 1), i = 1, . . . , N . (6)
The U i 's are iid and independent from N . We suppose that we know the variance parameter σ and we try to draw inference on λ and µ. The parameter values of the claim frequency and severity in ( 5) and ( 6) respectively are those infered in Section 5 for the Poisson-LogNorm model using the map estimator. The prior distributions are set to independent uniforms for λ and µ as λ ∼ Unif([0, 10]), and µ ∼ Unif([-10, 10]).

We generate artificial synthetic commercial premiums for this case study according to

π i = (1 + η i )E[g i (X)] = (1 + η i )E{min[max(r i • X -d i , 0), l i ]}, i = 1, . . . , n,
where the premium parameters r, d and l are sampled from that of the real data considered in Section 5, so that the simulated data is as close as possible to the real data. The η i 's are iid from

η i ∼ Unif([1.43, 2.5]).
which corresponds to loss ratios between 40% and 70%. We further set LR low = 40% and LR high = 70%. We consider sample of sizes 50, 100, 250 and 500. We configure the algorithm with a population size of 1, 000 and use 2, 000 Monte Carlo replications. To ensure the algorithm's efficiency, we set a stopping threshold, requiring that the difference between two consecutive tolerance levels is smaller than 1 for the algorithm to halt. These settings are kept for the analysis of real-world data, as they strike a balanced compromise between accuracy and computing time. We first run our procedure on a single sample of fake data, Figure 4 shows the resulting posterior distributions of λ and µ. As the sample size increases, the posterior distributions tend to approach the targeted parameter values more closely. Once the parameters have been identified, actuaries often focus on assessing key features of the risk distribution, such as the average claim severity and the probability of no claims. Figure 5 shows the predictive posterior distributions of several metrics, including the average claim amount, the average claim frequency, the probability of no reported claims, the average total claim amount, and the average loss ratio, defined as

LR = 1 n n i=1 p i π i ,
for sample of sizes 50, 100, 250, and 500.

Irrespective of the sample size, the posterior predictive distribution always include the true value. Note that the true loss ratio is set to the mean of 1/η where η ∼ Unif([1.43, 2.5]).

We now repeat 100 identical runs of the experiment. Our goal is to compare the result obtained using our two pointwise estimators: the mean a posteriori map and the mode a posteriori mode. The estimators of the parameters λ and µ are given on Figure 6.

Both of the point-wise estimators seem to converge toward the parameter values that generated the data. The map exhibits a better behavior than the mode as its variability decreases in a notable way as the sample size

increases. An increase in the number of particles in the cloud (set to 1, 000) would improve the reliability of the posterior distribution and provide a more accurate estimation of the mode

In Figure 7, we present a comparison of key metrics, including the average claim amount, the average claim frequency, the probability of no reported claims, the average total claim amount, and the average loss ratio. Both estimation methods yield satisfactory results in recovering the characteristics of the loss distribution but the use of the map yields more reliable estimations.

5 Application to the pet insurance market

Evolution and growth of the pet insurance market

Pet insurance is a product designed to cover the costs of veterinary care for pets. It operates on a similar principle to human health insurance, providing a way for pet owners to manage the financial risks associated with unexpected medical expenses for their animals. Usually the expenses are covered in case of an accident or a disease. Pet owners can choose from different policy options based on their budget and coverage needs.

Policies may vary in terms of deductibles (d), coverage limits (l), and reimbursement percentages (r). The cost of premiums can depend on various factors, including the pet's age, breed, health condition, and the level of The pet insurance market has been witnessing significant growth globally, driven by increasing pet ownership (especially with so-called pandemic pets, i.e. animals adopted during 2020 lockdowns), rising veterinary costs and the changing role that a pet plays in a families social structure. This latter factor is also influenced by changing societal views and the increased awareness of the importance of health and welfare of pets, which in turn comes with increased consideration of regular veterinary health checks . In order to offset the cost associated with such expenditures, there has begun to be a broader interest in households purchasing pet insurance.

To date, the adoption and acceptance of pet insurance still varies significantly across regions of the world.

Nordic countries, such as Sweden, have historically had a very high penetration rate with around 70% of pets insured. Some Anglo-Saxon countries (UK and Germany mostly) have seen significant growth in the pet insurance market during the last decades, leading to 30% of penetration rate. Other developed countries, like

France, have significantly lower market sizes, with less than 10% of pets that are insured, which suggests high growth potential. The market place for pet insurance in the USA is currently also experiencing sustained growth. According to, MarketWatch guides annual insurance surveys1 , about 44.6%, of pet owners stated they currently have pet insurance in the nationwide survey. Furthermore, the North American Pet Health Insurance Association (NAPHIA) undertook a survey in 2022 on the "State of the Industry Report" and found that more than 4.41 million pets were insured in North America in 2021, up from 3.45 million in 2020. The report also found that $2.84 billion of pet insurance premiums were in force in 2021, a 30.5% increase from 2020.

Primarily, most companies offer pet insurance plans designed to protect dogs and cats. As dog and cat ownership has increased over the last few years, so has the need for pet insurance. As an example in the USA

The American Pet Products Association (APPA) found that the increase in pet ownership of dogs and cats between 2017 and 2021 produced and increase in both cat and dog ownership, which further supports the subsequent trends identified by NAPHIA in the following trends in pet insurance uptake demonstrated over This growth continues to spur increases in the capital investments associated with such an insurance line of business:

• in Sweden, Lassie has raised 11m euros in 2022;

• in the UK, ManyPets has raised $350m at a valuation higher than $2bn in 2021;

• in France, Dalma has raised 15m euros in 2022.

Hence, the pet insurance market is becoming more competitive with an increasing number of insurance companies or brokers offering pet insurance policies. To gain new market shares as a new agent, there is a need to propose differentiated products such as new cover mixes without deductible and higher limits.

Data description

We have collected data on 89 pet health insurance plans offered by various insurers in the French market.

These insurance plans are each characterized by specific coverage parameters, including the coverage rate r, the deductible d, and the coverage limit l. The compensation for an annual expense of amount X is calculated

as min [max(r • X -d, 0), l].
Figure 8 provides a visual overview of the range of insurance coverage options available in the pet insurance market.

We note on Figure 8b that the majority of the insurance coverages do not feature a deductible. We can have a look at the commercial rates offered when d = 0 on Figure 9.

As expected, The commercial premiums increase in both the limit and rate of coverage.

Model fits

We consider three claim frequency distributions including Poisson(λ), Bin (12, p) and Geom(p). The choice of setting the number of trials in the Binomial distribution to 12 aligns with our focus on annual expenses, making it a suitable choice to capture the monthly probability of a claim occurrence. The prior settings for the parameters are as follows:

λ ∼ Unif([0, 10]), p ∼ Unif([0, 1]). (7)
We consider three claim severities distributions including LogNorm(µ = 0, σ ), LogNorm(µ, σ = 1) and Gamma(α, β = 1). The prior settings over the parameters of the claim sizes distributions are as follows:

µ ∼ Unif([-10, 10
Combining the distributions for the claim frequency and severities makes in total 9 loss models. The population size in the abc algorithm is set to N = 1, 000. The pure premiums are computed using 2, 000 Monte Carlo replications. The algorithms stops whenever the difference between two consecutive tolerance levels is lower than ∆ ϵ = 1. The bounds for the loss ratio corridor are set to LR low = 40% and LR high = 70%. The posterior distributions of the parameters for each model models are provided on Figure 10. For all the models, the algorithm updates the prior distribution in an informative way. We note the bimodal posterior distributions for the Poisson(λ) -Gamma(α, β = 1) and Bin(12, p) -Gamma(α, β = 1) models, see Figure 10g and Figure 10h, which may be problematic if using the map as a pointwise estimator. Table 2 provides the tolerance levels (ranked in incresaing order) during the last iteration of the abc algorithm for the loss models.

The final tolerance level for almost the models lies between 215 and 221 which is higher than the tolerance obtained in the simulation study which was around 55 for 50 data points and 66 for 100 data points. The lack of fit of the Geom(p) -LogNorm(µ = 0, σ ) is noticeably higher than that of the rest of the models. What we observe here underscores the significant influence of model misspecifications on the accuracy of our analysis. Table 2: Tolerance level during the last iteration of the abc algorithm fo each loss model adhering to the expectation principle for premium calculation and the models employed for claim frequency and claim amounts. Table 3 reports the estimations of the parameters of all the model using the map and the mode.

The Table 3: map and mode estimator for the parameters of the loss models.

that the mode estimation of α in the Poisson(λ) -Gamma(α, β = 1) is lower than one would expect when considering the overall shape of the posterior distribution in Figure 10g.

The predictive posterior distributions of the average claim amount, the average claim frequency, the probability of no reported claims, the average total claim amount and the average loss ratio are provided on Figure 11.

Across these models, we note the consistency when estimating the total claim amount, hovering around €300, and the loss ratio, which centers at approximately 70%. However, distinctive perspectives emerge concerning claim frequency and claim sizes, contingent on the choice of model for the latter.

For instance, the use of the Gamma(α, β = 1) distribution results in fewer claims, yet each claim tends to be more severe on average. On the other hand, the Geom(p) -LogNorm(µ = 0, σ ) model stands out by yielding a higher claim frequency, contributing to a more volatile total claim amount and loss ratio.

These observations hold when examining estimations through both the map and mode estimators in Table 4.

Table 5 reports the estimations of the average total claim amounts and the average loss ratio for all the models offering insights into the pricing strategies adopted by industry participants.

Conclusion

We have developed a robust methodology for risk assessment based on market data. We employ a oneparameter model for the claim frequency and claim size distribution, connecting the pure premium to the commercial premium through an isotonic regression model. This approach optimizes the alignment between commercial and pure premiums while providing a framework for quantifying the associated parameter uncertainty through an Approximate Bayesian Computation algorithm. The methodology's effectiveness and reliability have been validated within a simulation study and a practical application to an actual pet insurance dataset. This methodology is made accessible to the community through our R package, IsoPriceR2 .

While the results are promising, there remain avenues for further research. Future investigations can explore the selection of the most suitable model and consider the integration of historical data when it becomes available. One direction is the development of a credibility framework that combines historical and market data, providing a comprehensive perspective on risk assessment and pricing in emerging markets.

1 . 3 .* i 4 .

 134 Sample a parameter value θ * 2. Compute the pure premiums p * i for each of the insurance policies i = 1, . . . , n Fit an isotonic regression f * to learn the relationship between the commercial premia π i and the pure premia p Build the 'synthetic' market data D * by applying the estimated loading f * to the pure premium f * (p i ) for i = 1, . . . , n 5. If the observed and synthetic market data are close enough then we store the parameter value θ * and the associated loading function f * .

Figure 1 :

 1 Figure 1: Pure premiums as a function of the rate of coverage (r) and the deductible (d) for a Poisson(λ =

Figure 2 :

 2 Figure 2: Pure premium as a function of the commercial premium offered by various insurance companies.

Figure 3 :

 3 Figure 3: Isotonic link between the pure and commercial premiums.

Remark 2 . 1 .

 21 When looking at Figure3, one may object that a simple linear regression model could do the job.

  Pointwise estimators are derived from this final set of particles. Two commonly used estimators include the Mean A Posteriori (map) obtained by averaging the particles in the last cloud and the Mode A Posteriori (mode) which is the mode of the empirical distribution within the final cloud of particles. The forthcoming simulation Algorithm 1 Population Monte Carlo Approximate Bayesian Computation 1: set ϵ 0 = ∞ and p ϵ 0 (θ | D) = π(θ) 2: for g = 1 → G do 3:

Figure 4 :

 4 Figure 4: Posterior distribution of the parameter of the Poisson(λ = 0.58) -LogNorm(µ = 5.75, σ = 1) model based on synthetic market data of sizes 50, 100, 250, and 500.

Figure 5 :

 5 Figure 5: Posterior predictive distributions the average claim amount, the average claim frequency, the probability of no reported claims, the average total claim amount and the average loss ratio for the Poisson(λ = 0.58) -LogNorm(µ = 5.75, σ = 1) model based on synthetic market data of sizes 50, 100, 250, and 500.

  map and bp for µ

Figure 6 :

 6 Figure 6: map and mode estimators of the parameter of the model Poisson(λ = 0.58) -LogNorm(µ = 5.75, σ = 1) based on synthetic market data of sizes 50, 100, 250, and 500.

Figure 7 :

 7 Figure 7: map and mode estimator of the features of the Poisson(λ = 0.58) -LogNorm(µ = 5.75, σ = 1) loss model based on synthetic market data of sizes 50, 100, 250, and 500.

  ]), σ ∼ Unif([0, 10]), and α ∼ Unif([0, 10 5 ]).

Figure 8 :

 8 Figure 8: Overview of the insurance coverage available on the french market.

Figure 9 :

 9 Figure9: Commercial premiums depending on the rate and limit when no deductible is applied.

Figure 10 :

 10 Figure 10: Posterior distribution of the parameters of the loss models when fitted to the pet insurance dataset.

Figure 12 :

 12 Figure 12: Isotonic link between pure and commercial premium for the different loss models.

Figure 13 :

 13 Figure 13: Commercial premium as a function of the pure premium for the Poisson(λ) -LogNorm(µ, σ = 1) depending on the insurance carrier.

  Each intermediate distribution (called a generation) is represented by a cloud of weighted particles (θ j , w j ) j=1,...,K . We aproximate each intermediate posterior distribution using a multivariate kernel density estimator (kde) denoted by p ϵ g (θ|D). The parameters of the algorithm are the number of generations G and the population size K (the number of particles in the cloud).

Table 1 :

 1 Results on pet insurance uptake for USA from the 2022 survey of the North American Pet Health Insurance Association (NAPHIA) "State of the Industry Report".

	Year	Cats Insured Dogs Insured (in millions)
	2017	290, 000	1.5
	2018	348, 000	1.8
	2019	419, 000	2.09
	2020	531, 000	2.57
	2021	727, 000	3.25

time in Table

1

.

https://www.marketwatch.com/guides/pet-insurance/pet-insurance-facts-and-statistics/

see the market_based_insurance_ratemaking Github repository

Acknowledgements

Pierre-O's work is conducted within the Research Chair DIALOG under the aegis of the Risk Foundation, an initiative by CNP Assurances.

Bin(a = 12, p) -LNorm(µ = 0, σ)

Geom(p) -LNorm(µ = 0, σ)

Bin(a = 12, p) -LNorm(µ = 0, σ)

Geom(p) -LNorm(µ = 0, σ)

Pois(λ) -LNorm(µ = 0, σ) 0.00 0.25 0.50 0.75 P(N=0) (c) Posterior distribution of P(N = 0)

Bin(a = 12, p) -LNorm(µ = 0, σ)

Geom(p) -LNorm(µ = 0, σ)

Total Claim Amount (d) Posterior distribution of E(X) for all models when fitted using the map and the mode.

We estimate the pure premium for each model using the map as an estimator of the model parameters and we plot the isotonic regression function for each model to explain the commercial premium on Figure 12.

The Geom(p) -LogNorm(µ = 0, σ ) model stands out by revealing a lower loss ratio and, consequently, a more pronounced upward trend compared to the other models. To highlight the explanatory power of our methodology, let's focus on the Poisson(λ)-LogNorm(µ, σ = 1) loss model. Note that the choice of the loss model is somewhat arbitrary because the information extracted from the data in Figure 12 is relatively consistent across most of the considered models. In Figure 13, we present a plot that illustrates the relationship between The accuracy of the loss model fitting enables us to condense the three-dimensional information of the rate of coverage, deductible, and limit into a single metric: the pure premium. Subsequently, isotonic regression unveils the relationship between commercial and pure premiums, providing a link between the two. The distinctions among various players in the pet insurance market come to light through the color-coded points,


Danilo Carastan-Santos
email: danilo.carastan-dos-santos@inria.fr

Krzysztof Rzadca
email: krzadca@mimuw.edu.pl

Leonel Sousa

Denis Trystram
email: denis-trystram@imag.fr

Social and environmental effects of post-COVID-19 Computer Science virtual conferencing: The Euro-Par case

Conferencing is one of the main pillars of Computer Science research activity regarding career and networking, with conference publications playing a more pronounced role compared to other disciplines. The COVID-19 pandemic forced us to switch to virtual conferencing, and many works have shown the benefits of virtual conferencing in terms of inclusivity and reduction of Green House Gas emissions. We are moving toward the usual conferencing format as it appears that the pandemic is increasingly under control. However, the changes imposed during the period of the pandemic brought many essential lessons regarding conferencing social and environmental effects. A crucial task is to gather these community experiences to give directions on how to keep the learned lessons post-COVID-19. We used the Euro-Par conference to combine these lessons in the Computer Science case. We show practical results that reinforce the marginal emissions of virtual conferencing compared to in-person conference travel. We also open the debate that rethinking the conference utility according to our objectives (scientific and ecologic) and being aware of social/geographical biases are essential factors in participating and organizing post-COVID-19 conferences.

Introduction

Global warming is one of humanity's most critical challenges due to the ever-increasing greenhouse gas (GHG) emissions from human activities, in which aviation contributes up to 3.5% [START_REF] Data | Climate change and flying: what share of global co2 emissions come from aviation?[END_REF] of total global emissions. The Computer Science (CS) scientific community has played its part in these emissions as it has relied on frequent international conferences, often requiring intercontinental travel. Conferences are a traditional part of the CS community since it considers conferences as the main communication channel for research results [START_REF] Bar-Ilan | Web of science with the conference proceedings citation indexes: the case of computer science[END_REF][START_REF] Franceschet | The role of conference publications in cs[END_REF]. In CS conferences, articles pass peer-review and present original research results. Prestigious conferences can reach acceptance rates lower than 20%. CS researchers tend to start their careers, publish more papers, and collaborate with more coauthors in conferences than in journals [START_REF] Kim | Author-based analysis of conference versus journal publication in computer science[END_REF].

The shift to virtual conferencing enforced by the COVID-19 pandemic has thrown the scientific community (and especially the CS community) in a drastic change in the mode of operation, but it also has reintroduced the debate about the ecological impacts of in-person conferencing [START_REF] Van Ewijk | Emission reduction potentials for academic conference travel[END_REF][START_REF] Jäckle | Reducing the carbon footprint of academic conferences by online participation: the case of the 2020 virtual european consortium for political research general conference[END_REF][START_REF] Tao | Trend towards virtual and hybrid conferences may be an effective climate change mitigation strategy[END_REF][START_REF] Burtscher | The carbon footprint of large astronomy meetings[END_REF]. Shifting to a fully virtual conference format can reduce GHG emissions by up to 99% [START_REF] Klöwer | An analysis of ways to decarbonize conference travel after covid-19[END_REF].

Moreover, the CS community, although quite reactive to the change from in-person to virtual [START_REF] Mubin | The new norm: Computer science conferences respond to covid-19[END_REF], is arguably more sensitive to this change. Conferences are more important to build a career in CS (in terms of publication records, sharing research results, networking, etc.) than in other disciplines, making it important to understand the community view of the shift from in-person to virtual.

This paper is a step towards better understanding these points by presenting a community-guided discussion to understand the environmental effects of post-COVID-19 Computer Science conferencing. We took as a use case the 2020 and 2021 editions of the International European Conference on Parallel and Distributed Computing (Euro-Par). Euro-Par represents a typical CS conference: over more than 25 years, it developed a robust, active community of researchers; gathering annually approximately 200 in-person participants. The 2020 and 2021 editions took place virtually. Our community-guided method consists of (1) estimating the CO2 emissions of the conference, and we compared them to a hypothetical fully in-person conference; (2) collect feedback and comments from the conference's community; and (3) using these comments and feedback to guide our discussions and analyses.

The two key new methodological contributions of this paper are: first, approaching the environmental effects of post-COVID-19 Computer Science conferencing under the particular perception of the CS community; and, second, by approaching the community as an integral part of the study methodology, by tailoring a community-guided discussion on the matter.

We organize the remainder of this paper as follows: Section 2 presents the related work on scientific conferences and their environmental effects. Section 3 presents the methods to estimate the CO2 emissions of a virtual and in-person conference and our method to collect feedback and comments from the community. Section 4 presents an analysis of the emissions estimations applied to Euro-Par 2020 and 2021, as well as our community-guided discussion, and in Section 5 we present our concluding remarks and future perspectives.

Related Work

There is a rapid increase in the number of studies regarding this shift from in-person to virtual/hybrid conferencing. Most studies focus on its environmental impacts, in terms of CO2 equivalent emissions. Prior works [START_REF] Van Ewijk | Emission reduction potentials for academic conference travel[END_REF][START_REF] Jäckle | Reducing the carbon footprint of academic conferences by online participation: the case of the 2020 virtual european consortium for political research general conference[END_REF][START_REF] Burtscher | The carbon footprint of large astronomy meetings[END_REF] took into account the transport and the electricity consumption of devices and servers emissions. Tao et al. [START_REF] Tao | Trend towards virtual and hybrid conferences may be an effective climate change mitigation strategy[END_REF] performed a Life Cycle Assessment (LCA) including food, accommodation, preparation, execution, information and communication technology, and transportation of an in-person, virtual and hybrid conference. Their carbon footprint estimations acknowledged the previous works, showing that participants' transport dominates the emissions of an in-person conference, accounting for more than 80% of the total emissions. These works considered the carbon intensity of electricity generation in the virtual conference emissions by using either a single value [START_REF] Burtscher | The carbon footprint of large astronomy meetings[END_REF] or min-max values, that work as a lower and upper bound of the emissions [START_REF] Jäckle | Reducing the carbon footprint of academic conferences by online participation: the case of the 2020 virtual european consortium for political research general conference[END_REF]. The aforementioned works also consider only one method for estimating the emissions of conferencing. We distinguish from these works by using multiple virtual conference GHG emission models during our evaluation, taking into account the carbon intensity of electricity generation of the conference hosting and participant's country.

Numerous works considered multi-site/hybrid conferences as an in-between alternative for conferencing. The major question in this scenario is to decide on how many hubs to launch and where to place them. Choices are from arbitrary [START_REF] Van Ewijk | Emission reduction potentials for academic conference travel[END_REF][START_REF] Hischier | Environmental impacts of an international conference[END_REF], to driven by optimization methods with criteria such as potential emissions [START_REF] Stroud | Responsible academia: optimizing conference locations to minimize greenhouse gas emissions[END_REF] and geometric distance of participants [START_REF] Astudillo | Estimating the global warming emissions of the lcaxvii conference: connecting flights matter[END_REF]. Tao et al. [START_REF] Tao | Trend towards virtual and hybrid conferences may be an effective climate change mitigation strategy[END_REF] introduced two criteria for selecting hub locations, Maximum Virtual Participation and Maximum Travel Distance. While all these works show significant reductions in energy consumption and GHG emissions, -geographical social effects of conference hub locations are neglected. In contrast, we debate possible social effects of such decisions.

Our work distinguishes from others in the literature in the following factors: (i) we use multiple virtual conference GHG emission models during our evaluation, taking into account the carbon intensity of electricity generation of the conference hosting and participant's country, (ii) to the best of our knowledge, we are the first study to evaluate the GHG emissions of virtual and in-person conferences in Computer Science, which has a greater impact on the scientists' careers as abovementioned, and (iii) we approach the Computer Science community as an integral part of the study methodology, by tailoring a community-guided discussion on the problem.

Methods

We divide our method into two steps. The first step involves using the conference participant's data to estimate the emissions of a conference. This first step gives qualitative and quantitative estimations of the environmental effects of a conference, but their interpretations and possible measures rely on collective knowledge from the conference's community. We address this in the second step, where we build a mechanism to collect feedback from the conference community to tailor a community-guided discussion.

3.1

Step 1: Estimating CO2 emissions of a conference

Estimating in-person travel CO2 emissions

Although many factors add up to the CO2 emissions of an in-person conference, such as hotels, or meals, traveling is arguably the most important factor, accounting for around 85% of the total carbon footprint [START_REF] Tao | Trend towards virtual and hybrid conferences may be an effective climate change mitigation strategy[END_REF]. To estimate the conference participants' travel CO2 emissions, we used an online travel footprint calculator [START_REF] Barret | Estimating, monitoring and minimizing the travel footprint associated with the development of the athena x-ray integral field unit[END_REF]. This calculator estimates the travel footprint associated with round-trip flights. The advantage of this calculator is that it incorporates models obtained through numerous flight emissions data sources. The data sources are:

• ADEME [START_REF]Ademe carbon database[END_REF]: the French Environment and Energy Management Agency Carbon database;

• The French Ministry of Ecology and Inclusive Transition [START_REF] Ecology | Transition, I. Ghg information for transport services[END_REF];

• atmosfair [START_REF] Atmosfair | atmosfair flight emissions calculator[END_REF], a German Carbon offsetting non-profit organization;

• DEFRA [START_REF]government greenhouse gas conversion factors for company reporting[END_REF], UK's Department for Environment, Food and Rural Affairs;

• ICAO [START_REF]Carbon emissions calculator methodology[END_REF], a widely known online calculator;

• KLM [START_REF] Klm | Klm data[END_REF] Carbon compensation service data;

• myclimate [START_REF]The myclimate flight emission calculator[END_REF], a Carbon offsetting non-profit organization.

Among other factors, these sources differ in how they deal with the radiative forcing index (RFI) [START_REF] Jungbluth | Recommendations for calculation of the global warming potential of aviation including the radiative forcing index[END_REF], which is a multiplicative factor for aircraft CO2 emissions that accounts for the increased effects such as burning fuels directly into the upper/lower stratosphere and aviation induced cloudiness introduced by contrails [START_REF] Burkhardt | Global radiative forcing from contrail cirrus[END_REF], impacting atmospheric composition. Setting an appropriate value for the RFI is an ongoing debate in the community. Commonly used RFI values are 1 (no RFI), which is the case for the French Ministry of Ecology and Inclusive Transition, KLM, and ICAO, 1.9 for DEFRA, 2 for myclimate and ADEME, and 3 for atmosfair.

We can have some variability of the estimations by considering multiple estimation methods. For our estimates, we defined two categories of models: (1) the pessimistic models: the ADEME, myclimate, and DEFRA models; and (2), the optimistic models, the French Ministry of Ecology and Inclusive Transition, KLM and ICAO models. We report the corresponding models' CO2 emissions for each category as an average. The used online travel footprint calculator also enables us to evaluate the emissions related to train journeys by setting a threshold distance for flying. When a distance of a certain participant is lower than this threshold, the calculator switches from air to train travel, and CO2 emissions are estimated accordingly.

The calculator requires two inputs, an origin input, and a destination input. The origin input can be a CSV file where each line contains a city and country of a participant, and the destination input is the city and country of the conference. For the origin file, each participant must correspond to a line in the CSV file. The calculator then groups the number of entries per city/country and reports the sum of CO2 emissions of a round trip for all participants from each city/country. The calculator assumes that each participant takes an independent flight and no other participant takes the same flight, acting, therefore, as an upper bound for the number of flights.

Estimating virtual conference CO2 emissions

We made extensions into two distinct methods for estimating the CO2 emissions of a virtual conference and streaming video. Both methods consist of multiplicative models, the overall difference between them is which kind and source of data are considered.

First method: extended IEA video streaming emissions. We developed a method of measuring the CO2 emissions of a virtual conference, by making extensions to a study performed by IEA [START_REF] Iea | The carbon footprint of streaming video: fact-checking the headlines[END_REF], that estimates the consumed energy in kilowatt-hour of streaming video. An assumed hypothesis is that a streaming video interaction corresponds to most of the current virtual interactions in virtual conferences (including Euro-Par 2020 and 2021).

IEA's study provides energy consumption estimations for three different scenarios, considering different viewing device choices, types of network connection, and video resolution. The baseline IEA method for estimating CO2 emissions of streaming video works as follows: for a scenario S, the CO2 emissions of a person p watching a video for one hour can be estimated by the following equation,

CO2 p = (kW h ser (S) + kW h net (S) + kW h dev (S)) • CI (1)
where kW h ser (S), kW h net (S), and kW h dev (S) are the energy intensity of the server, network, and viewing device of a person watching a one-hour video, taking into account the scenario S -such values are provided by IEA's study [START_REF] Iea | The carbon footprint of streaming video: fact-checking the headlines[END_REF] -and CI is a value of carbon intensity of energy generation, that measures how much CO2 is emitted when producing electricity. We extended the above baseline in the following manner: we changed Equation 1 to account for the number of participation hours and the heterogeneity in the carbon intensity.

CO2 p = (CI ser kW h ser (S) + CI net kW h net (S)+ CI p kW h dev (S)) • nbhours p (2)
In this extension (Equation 2), nbhours p is the number of participation hours for participant p and CI ser , CI net = CIser+CI p 2 , and CI p are the carbon intensity of electricity generation for the server, network, and viewing device components. For the former, such data can be easily retrieved in standard videoconferencing software such as Zoom. For the latter, such information can be found on websites such as Electricity Map1 , or the European Environment Agency (EEA) [START_REF] Agency | Greenhouse gas emission intensity of electricity generation in europe[END_REF]. The change in the carbon intensity is to account for differences in the carbon intensity of electricity generation according to the geographical location of each of the components: CI ser and CI p are the carbon intensity of electricity generation where the server and the participant are located, respectively, and CI net is an estimation of the carbon intensity from the server to the participant location.

Finally, total CO2 emissions of a virtual conference taking into account a list of participants P can be estimated using Equation 3.

CO2 total = p∈P CO2 p (3)
Second method: extended Burtscher et al. emissions of virtual conference. We extended Burtscher et al. [START_REF] Burtscher | The carbon footprint of large astronomy meetings[END_REF] method to estimate the carbon emissions of a virtual conference. This method first estimates the energy consumption of the ICT infrastructure in terms of network, end devices, and server, and then converts the energy consumption to respective carbon emissions. In this section, we first present its baseline version, and then we present our extended version.

The baseline method first estimates the overall participant demand dem during the conference,

dem = nbdays • onlinenbhours • att • nbpart ( 4
)
where nbdays is the number of conference days, onlinenbhours is the number of hours that the participants stayed online per day, att is a daily attendance percentage of the total of participants, and nbpart is the total number of participants. After estimating dem, then the method estimates the carbon emissions of network CO2 net , end device CO2 dev and server CO2 ser , and then finally estimates an overall carbon emission of the conference CO2 total (Equation 5).

CO2 net = dem • bdwth • kW h Gb • CI CO2 dev = dem • devpwr • CI CO2 ser = nbdays • nbhours • serpwr • CI CO2 total = CO2 net + CO2 dev + CO2 ser (5)
In the equation above, bdwth is the necessary network bandwidth per participant, kW h Gb is the energy intensity of the network per Gigabit, devpwr is the power of the end viewing device, nbhours is the number of hours per day that the server was online, serpwr is the server power, nbdays is the number of conference days, and CI is a carbon intensity of electricity generation.

We extended the above method to give more accurate estimates of the participation of the conference attendees and the carbon intensity of electricity generation. For a participant p we first estimate the total number of attendance hours nbhours p during the conference, and we search for the carbon intensity of electricity generation of the participant's country CI p . Both values can be retrieved as explained in Section 3.1.2. If we can not find a CI p for a participant's country, we use an aggregate value (i.e., global or per continent average).

We then perform the following estimations for a participant p

CO2 p net = nbhours p • bdwth • kW h Gb • ( CI ser + CI p 2 ) CO2 p dev = nbhours p • devpwr • CI p CO2 p = CO2 p net + CO2 p dev ( 6
)
where CI ser is the carbon intensity of electricity generation where the server is hosted, and bdwth, kW h Gb and devpwr are the same as in Equation 5. Similar to the IEA method (see Section 3.1.2) we also set the carbon intensity of electricity generation as the average of the intensities of the participant's country and the server's hosting country.

Finally, for all participants p in a list of total participants P, we estimate the total CO2 emissions of the conference.

CO2 att = p∈P CO2 p CO2 ser = nbdays • nbhours • serpwr • CI p CO2 total = CO2 att + CO2 ser (7)

Step 2: Building a community-guided discussion of the environmental effects of conferencing

This second step consists of opening a discussion with the community to raise the matter about the ecological impacts of the conference, and collecting comments and feedback from the community. Collecting feedback from the conference's community is essential since the emissions estimations can guide decisions regarding future editions of the conference, which directly impacts the conference community. We collected two types of feedback: (1) verbal comments by the conference participants; and (2) written comments. We collect these written comments by offering the option to the participants to write anonymous comments on a collaborative pad. The pad can be organized into threads, where a participant starts a discussion topic, and others can react and comment on the topic. We foster the community's reaction and potentially reduce any social pressure on the matter by letting participants post comments anonymously.

We afterward used the verbal and written comments to synthesize a discussion about the environmental effects of the conference. We decided on an approach of reacting to the comments directly, with the objective of merging our knowledge with the collective knowledge of the conference's community. The output of this step is a tailored discussion guided by the conference's community feedback.

Results

We had access to Euro-Par 2020 and 2021 participant data collected by their respective videoconferencing software. The data available were the declared participant's city (manually inferred from the participants' affiliation) and country data, and participation in the conference's sessions. For Euro-Par 2020, we had precise information about the number of login hours for each participant and each session, and we used this information to calculate the number of participation hours for each participant. For Euro-Par 2021, we had only the information of whether a certain participant logged in to a certain session. In this case, we calculated the number of participation hours for a participant by summing the number of hours for each of the sessions this participant logged in.

After this pre-processing, we had participant data for 221 and 137 participants for Euro-Par 2020 and 2021, respectively. This discrepancy in attendance data is due to the free registration and attendance of Euro-Par 2020, whereas such registration and attendance were paid for Euro-Par 2021.

All estimated emissions numbers should be taken with some reservation. Energy consumption and emissions significantly depend on the computing/storage/networking hardware and software. The emissions we report are approximations that allow us to compare different modes of conferencing, notably in-person versus virtual. We also only report emission numbers for the usage phase of the computing/storage/networking infrastructure, as opposed to a complete life cycle analysis (LCA), which includes manufacturing, transporting and recycling phases.

For the collaborative part of the methodology, we collected comments and feedback from the participants at Euro-Par 2022, which was a hybrid event hosted in Glasgow, UK. There were roughly three times more people attending the sessions in-person than attending on-line.

Emissions estimations of Euro-Par

Travel emissions estimations

In this section we present the estimated, hypothetical travel emissions for Euro-Par 2020 and 2021. For each of these editions we performed two estimations (i) one estimation that averages estimations of three models: MyClimate [START_REF]The myclimate flight emission calculator[END_REF], ADEME [START_REF]Ademe carbon database[END_REF] and DEFRA [START_REF]government greenhouse gas conversion factors for company reporting[END_REF] (the pessimistic models category, see Section 3.1.1) and (ii) one estimation that averages estimations of three other models: ICAO [START_REF]Carbon emissions calculator methodology[END_REF], KLM [START_REF] Klm | Klm data[END_REF] and the French Ministry for the Ecological and Inclusive Transition [START_REF] Ecology | Transition, I. Ghg information for transport services[END_REF] (the optimistic models category, see Section 3.1.1). We also set a threshold value of 500 km to define if a certain travel distance can be performed by train.

Figures 1 and2 show the cumulative CO2 travel emissions estimations for Euro-Par 2020 and 2021, respectively, using the pessimistic and optimistic model sets. For a certain model set, each point in a figure represents a participant. The CO2 emissions estimations consider the distance of a round trip between the participants' city and Euro-Par's host city (Warsaw for 2020 and Lisbon for 2021), and the participant's travel emissions per kilometer is represented by the marker's color. Participants are sorted in increasing order of their round trip travel distance. Table 1 shows the total travel CO2 emission estimations for Euro-Par 2020 and 2021.

The emissions curves in Figures 1 and2 present two parts. The first represents participants whose travel was considered to be by train -whose emissions are estimated similarly for both model sets, and we can identify this part by the left-most, violet-colored cluster of points in the figures. The second represents participants whose travel was considered to be by plane (the remaining points). Both model sets show the same tendencies of CO2 emissions. The difference lies on how faster the emissions increase in function of flight distance, with the pessimistic model increasing faster. Due to the multiplicative RFI factor of around 2, by looking at Table 1 twice as much total CO2 emissions than the optimistic model set.

On one hand, as expected, the longer the travel distance, the higher is the CO2 emissions. On the other hand, longer trips are more efficient in terms of CO2 emissions per kilometer. Figures 1 and2 show that in both model sets, flights within short distance (round trip distance between 1000 km and 2500 km) result in higher emissions per kilometer when compared to longer flights in the same model set.

For contextualizing these emission numbers in Table 1, we also show some equivalent numbers that correspond to the estimated CO2 emissions [START_REF] Agency | Greenhouse gas equivalencies calculator[END_REF], notably the number of United States homes energy use for one year -to put into perspective the energy consumption -and the number of square meters of United States' forests for one year -to put into perspective how much forest area is required to sequester the emitted CO2. The emissions can reach up to an equivalent of 27 houses' emissions during one year. The forest area needed to sequester the emitted CO2 reaches up to a size equivalent to 143 soccer fields (7140 m2 ).

Virtual emissions estimations

In this section we present the CO2 emissions estimations of the virtual Euro-Par 2020 and 2021, taking into account the extended IEA and Burtscher et al. methods presented in Section 3.1.2. We considered the "Laptop, Wi-Fi, HD" for the former method since it is the closest scenario to a typical virtual conferencing configuration. For the latter method, we used the parameters presented in Table 2. For both methods, we used the carbon intensity of electricity generation data provided by the European Environment Agency (EEA) 2 . If we can not find carbon intensity of electricity generation for a certain participant's country using EEA data, we used the world average of 475 gCO2/kWh [START_REF]Global energy & co2 status report 2019[END_REF].

Table 3 shows the estimated CO2 emissions for the virtual Euro-Par 2020 and 2021. The estimated emission for a virtual conference is three orders of magnitude lower than the in-person equivalent (see Table 1). We can also notice the discrepancies in the emissions when comparing the two models, with the Extended Burtscher et al. model estimating almost twice more emissions as the Extended IEA. This discrepancy between models can be problematic if one wants to evaluate different virtual conferencing configurations and grasp potential benefits, though this discrepancy is less relevant for qualitative studies comparing virtual and in-person conferencing, which is the case in this paper.

Euro-Par community community-guided discussion

We synthesized the comments and feedback obtained from the Euro-Par community in the topics below.

Virtual participation incentives and inclusivity tradeoffs

Numerous comments were on the introduction of the hybrid conference format and incentives to foster virtual participation.

"If we make on-line attendance considerably cheaper than in person this would encourage people to adopt this route" "COVID shone a light on huge opportunities for making academic conferences more accessible than they were. PhD students and researchers who don't have travel budgets from underrepresented institutes and countries were attending events in 2020 and 2021. ... I urge steering committees at conferences like EuroPar to not push for a return to the pre-2020 days: remote participation should be free for non-authors. It would be tragic if remote participation is actively disincentivized by deliberately raising the cost of remote participation to encourage physical attendance, because steering committees decide that physical interactions are essential. Quite the opposite, physically attending delegates should be charged a surcharge to buy carbon credits to make these conferences carbon net-zero. Institutes that can afford to send their people to conferences at thousands of EUROs are very likely able to afford an extra +5% for carbon credits." "... Encouraging people to attend virtually with incentives like cheap admissions costs are a quick and easy way to promote this behavior. It is also a low cost burden on the conference hosts as well" "Some senior members of the community are concerned that virtual attendees will 'cannibalize' (i.e. reduce) in-person delegates. It seems that the Euro-Par'22 experience suggests that this is not the case."

To our interpretation, the hybrid model introduced during the COVID-19 pandemic has brought benefits for certain members of the Euro-Par community, especially for those who do not have enough travel budgets from their institutions, or for those who want to follow the virtual conferencing mode to reduce the emissions from their activities.

However, the high cost to participate in a hybrid conference virtually hinders the realization of these benefits. For instance, a presenting author wanting to participate only at the Euro-Par 2022 main conference (and not the workshops) would pay 370 euros or 310 euros for in-person or virtual participation, respectively3 . We can not distinguish if the discrepancy between in-person and virtual attendance is due to the lack of interest or due to the lack of registration incentives for participating in virtual. We conjecture that this is possibly tied to individual perceptions regarding the purpose of a conference, which we elaborate on in the next section.

A possible measure to increase virtual participation is reducing the cost. Given the magnitude of the conference participation (i.e., a few hundred participants in the Euro-Par use case), increasing virtual participation results in a marginal increase in Information and Communications Technology (ICT) CO2 emissions, especially compared to popular activities such as video streaming. Yet, the energy consumption of ICT increases up to 9% per year [START_REF] Project | Implementing digital sufficiency[END_REF], with a global share of emissions of around 3% [START_REF] Freitag | The real climate and transformative impact of ict: A critique of estimates, trends, and regulations[END_REF]. ICT emissions are in the same order as the aviation industry [START_REF] Data | Climate change and flying: what share of global co2 emissions come from aviation?[END_REF].

We can mitigate the virtual conferencing energy consumption with frugal technologies such as replacing video streaming during presentations with lightweight presentations consisting of local slides (PDF files) synchronized with the presenter's audio. However, we must not hinder the quality of remote participation (e.g.: low-quality streams leading to increased fatigue). Frugal virtual conferencing technologies must achieve the highest level of virtual conferencing experience with the lowest energy consumption and required infrastructure.

A solution supported by the Labo 1point5 initiative in France4 , though arguably coercive, is to impose emissions control in the research laboratories by assigning carbon emission budgets for the activities, which includes conferencing. With such a budget strategy, however, laboratories may seek to blindly optimize their emissions which may deepen the difficulties of performing research activities for specific demographics, going against inclusivity. An example is caretakers: persons in charge of small children or the elderly are likely unable to afford the extra travel time of using low-carbon transportation (trains) and/or will need to bring the children or the elderly with them. In a caricatural scenario, a laboratory seeking to optimize its emissions would always fund three young/late-career men by train to go to three different conferences than a woman with two children to a single conference.

Purpose and effectiveness of virtual Euro-Par

Comments regarding the purpose and effectiveness of the virtual editions were raised. "... maybe many people are keen to experience in-person events again after 3 years of Covid lockdowns?" "Should we normalize these by the 'effectiveness' of in-person vs. virtual? In the past 3 years, I have paid for several virtual conferences which I never ended up attending because I couldn't get away from my normal commitments at home (e.g., teaching, meetings, etc.)."

Although it is commonly agreed on by the Computer Science community that the main purpose of a conference is to share scientific results, one may perceive the main purpose as being the paper published at a prestigious event, or as being the opportunity to meet the community and to do networking, or to have an update on what subject are being studied by the community.

Meeting colleagues and networking is arguably the hardest to achieve with the current virtual technology. Current virtual conferencing technology misses transmitting non-verbal attitudes that say as much as words. We feel psychological subtleties better when interacting in-person, and, therefore, we advance quicker in interchanging research ideas and building professional networks when participating in an in-person conference. Moreover, in-person conferencing activities such as coffee breaks, cocktails, and other side social events foster social interactions more naturally than in virtual. Achieving this purpose by participating in person comes at the cost of increased emissions due to travel.

We can nevertheless employ actions to value these emissions by, for instance, promoting extended stays for long-distance conference participants, with the goal of maximizing the research collaboration opportunities with the local researchers. Such an idea was employed at the 2022 edition of the International Colloquium on Automata, Languages and Programming (ICALP) 5 , one of the main events in the theoretical computer science community. Informally, the idea is to "batch" the research activities with an extended stay in a institution local to the conference, potentially avoiding multiple long-distance trips, and resulting in fewer emissions overall.

Virtual conferencing can be hard to follow for other practical reasons, such as difficulties with significant timezone differences, or lack of dedicated time for conferencing.

Influence of the location: Host city and multi-hub

Comments were raised regarding the geographical location for future Euro-Par editions.

"Since Euro-Par is a European conference and CO2 emissions are dominated by flights, I think they could be reduced considerably by hosting most future conferences somewhere in central Europe. This is a big incentive for a large portion of attendees to come by train instead."

The choice of the host city is an important parameter to mitigate the emissions. A possible approach to reduce emissions from conferencing is to decide on a host city that is more accessible by train to most participants. We can devise, for instance, algorithms that look at the attendant's geographic distribution of previous editions to decide where to host the next edition of a conference. Although this approach could be efficient in terms of CO2 emissions, it also has the potential counterpart of increasing the geographical bias of research, since it is very likely that institutions in the same region -i.e., a centrally-located region with an efficient train network -would be selected every year.

A trade-off solution that is currently being considered by some communities is a multi-hub conference format [START_REF] Van Ewijk | Emission reduction potentials for academic conference travel[END_REF][START_REF] Tao | Trend towards virtual and hybrid conferences may be an effective climate change mitigation strategy[END_REF] taking into account the constraint of avoiding too many moves of people working in far countries. However, most related studies in multi-hub conferencing format transmit geographical biases in the analysis results. Algorithms are created upon biases, and the algorithms institutionalize these biases and all issues that follow, ending up with statements such as "a conference located in the southern hemisphere usually performs much worse in terms of the carbon footprint than the northern hemisphere" [START_REF] Tao | Trend towards virtual and hybrid conferences may be an effective climate change mitigation strategy[END_REF]. Most works propose algorithms and solutions whose results center in the Northern Hemisphere, notably in the US/Europe axis, overlooking the fundamental issue of the international inequality of research, which risks being self-reinforced. We argue that it is the political responsibility of the scientific communities to foster some countries, working towards global inclusivity of research, while also being aware of reducing CO2 emissions.

Even within the US/Europe axis, we risk transmitting geographical biases. For instance, we could evaluate and propose algorithms for optimized/optimal placements of a virtual conference according to the carbon intensity of electricity generation (i.e., energy mix) of the hosting country. However, such contributions would be based upon a more fundamental issue, which in this case is inequality between countries in terms of technology, economy, and access to natural resources (wind, rivers, etc.) to produce low-CO2 electricity. For the above reasons we deliberately decided not to analyze, propose solutions, and provide results regarding the origin country of the participants.

Moreover, it is hard to detach any conference location optimization algorithm, on both single or multi-hub, to indirect or direct rebound effects. For instance, an algorithm that decides the conference location according to the energy mix of the hosting country may lead to an overexploitation of the country's energy, eventually usurping the energy of an essential service. A possible approach to avoid the rebound effect pitfall is, as advocated in [START_REF] Maraninchi | Let us not put all our eggs in one basket[END_REF], introducing the concept of limits as an integral part of the constraints and specifications of the methods. Applying this concept of limits could be, for instance, by creating methods assuming by design that the availability of low-CO2 energy in a particular country will not grow accordingly.

Conclusion

Our work is a step towards better understanding the challenging endeavor to solve the fundamental issue of reducing the greenhouse gas (GHG) emissions of research activities, more specifically in Computer Science conferencing. The COVID-19 pandemic forced us to move towards virtual activities, and this has put into evidence once again the debate of whether to move conferences to virtual, in order to reduce their environmental impacts.

We have taken the opportunity of the changes in our behavior imposed by the COVID-19 pandemic to investigate a concrete case, the effects of the remote conferences on the 2020 and 2021 editions of the European Conference on Parallel and Distributed Computing (Euro-Par), which took place virtually. We made extensions to current methods to estimate virtual conference emissions, and we used them to study the impacts of electricity consumption of terminals and networks of Euro-Par 2020 and 2021. We compared these estimations of the virtual Euro-Par to emissions estimations of traveling as if Euro-Par 2020 and 2021 took place in person. We then collected feedback from the Euro-Par's community to build a Euro-Par's community-guided discussion on the matter.

This work reinforces the qualitative result that virtual conferences are several orders of magnitude less pollutant than in-person conferences. The comments and reactions of the Euro-Par community were positive on the matter, indicating a positive attitude from the community towards reducing Euro-Par environmental impact.

In summary, the introduction of virtual participation increased the inclusivity of the scientific results. We suggest deploying registration incentives for virtual registration to keep this inclusivity post-COVID-19. We observe that this increase in inclusivity comes at a marginal cost of increasing the demand for information and communications technology (ICT). Still, developing and deploying frugal virtual conferencing technologies that achieve the highest level of virtual conferencing experience with the lowest energy consumption and the required infrastructure is essential to keep the cost of virtual conferencing marginal.

We also observed that the current virtual conferencing experience could be more or less valuable according to the purpose of the participant. Social and networking interactions are currently more effective by participating in person, with the drawback of increased emissions. We should reinvent the current way we work and collaborate by, for instance, establishing a ratio of benefit (purpose) versus environmental effects of participating in a conference. We establish this ratio by knowing the purposes we want to achieve when participating in a conference and what environmental effects result in achieving such objectives.

We finally emphasized the complex objective of minimizing the environmental impacts of a conference, while not passing forward more fundamental social issues, especially the global inequality in research. This objective is especially hard when considering optimizing the conference's host location or the conference's location in a multi-hub scenario. We claim that future research on conference host and multi-hub locations must be aware of these social issues when proposing new methods.

Data and code availability

All data generated in this study have been provided in the Supplementary Information and deposited in GitHub [https://github.com/danilo-carastan-santos/europar-travel-emissions]. Python 3.9.12 was used to analyse data.

Figure 1 :

 1 Figure 1: Per participant cumulative travel emissions estimations for Euro-Par 2020

Table 1 :

 1 we observe that the pessimistic model set estimated around Travel CO2 emissions estimations for Euro-Par 2020 and 2021.

	Estimated cumulative travel emissions (KgCO2e)	0 25000 50000 75000 100000 125000 150000 175000	0	Round trip travel distance (km) 5000 10000 15000 20000 25000 30000 35000 40000 Euro-Par 2021: Estimated travel CO2 emissions CO2 per km 0.04 0.08 0.12 0.16 0.20 Model Set Pessimistic Optimistic
		Figure 2: Per participant cumulative travel emissions estimations for Euro-Par 2021
	Year Model Set	Estimated Emissions (kgCO2eq)	Equivalent Emissions (U.S. homes' energy use for one year [3])	Equivalent to Carbon Sequestered by (m 2 of U.S. forests in one year [3])
	2021 Pessimistic	1.73 • 10 5	21.8	8.30 • 10 5
	2021 Optimistic	8.46 • 10 4	10.7	4.05 • 10 5
	2020 Pessimistic	2.14 • 10 5	27	1.02 • 10 6
	2020 Optimistic	1.06 • 10 5	13.4	5.10 • 10 5

Table 2 :

 2 Values used for the parameters in the extended Burtscher et al. method.

			Parameter Value
			bdwth		4.21 Gb/h
			kW h Gb	0.0075 kWh/Gb
			devpwr	0.03 kW
			serpwr		0.3 kW
		Table 3: Virtual Euro-Par CO2 emissions estimations
	Year	Method	Emissions (kgCO2eq) Laptop Network Server Total	Equivalent to Carbon Sequestered by (m 2 of U.S. forests in one year [3]
	2021 Extended IEA	9.26	6.48	1.74	17.50	84.98
	2020 Extended IEA	9.07	9.19	3.591 21.86	105.21
	2021 Extended Burtscher et al.	12.63	11.27	3.73	27.64	133.54
	2020 Extended Burtscher et al.	12.37	15.99	12.16 40.53	194.24

https://app.electricitymaps.com/map

https://www.eea.europa.eu/api/SITE/data-and-maps/daviz/co2-emission-intensity-9/download.csv

https://2022.euro-par.org/attending/registration/

https://labos1point5.org/

https://icalp2022.irif.fr/?page_id=50#support

Acknowledgment

This work was supported by the research chair on Edge Intelligence of the Multi-disciplinary Institute on Artificial Intelligence MIAI at Grenoble Alpes (ANR-19-P3IA-0003), ENERGUMEN (ANR-18-CE25-0008), a Polish National Science Center grant Opus (UMO-2017/25/B/ST6/00116) and the EuroHPC EU Regale project (g.a. 956560).

Author contributions

The authors are listed in alphabetic order. Denis launched the idea of this study. All the authors participated to the discussions and elaboration of this work. Krzysztof and Leonel provided the access to the data. Danilo and Krzysztof contributed to the data processing, analysis, and results interpretation with the help of Denis. Danilo was the main writer, and Denis was the main reviewer of Sections 4 and 5. Finally, all authors reviewed the final manuscript.