Keywords: Deforestation, Colonial legacies, Legal origin. Codes JEL / JEL codes : Q12, Q24

This paper investigates whether inherited legacies such as legal origin allow of explaining deforestation in 110 developed and developing countries. The hypothesis is that differences in deforestation between countries can be attributed to their legal systems. Also, since nearly all common law countries are former English colonies, and nearly all civil law countries were colonized by France, Spain or Portugal, legal origin and colonial history are strongly correlated, so that one can not attribute all the variance to the effect of the legal system. What is found overall is that (i) French civil law countries deforest less than English common law ones within the total sample, within the sample of colonized countries, and within the sample of tropical developing countries; (ii) Former French colonies deforest less than former English colonies. These results hold when geography features are controlled for since the process of colonization was not random and depended on initial geographic and climatic conditions.

Introduction

Recent empirical work shows that countries whose legal systems are based on French civil law differ systematically from those whose legal systems are based on English common law. The Legal Origin Theory of development, mainly developed by La Porta, Lopez-Silanes, Vishny and Shleifer (LLSV-several papers La [START_REF] Porta | Legal determinants of external finance[END_REF][START_REF] Porta | Law and finance[END_REF][START_REF] Porta | The economic consequences of legal origins[END_REF]), investigates the role of legal origin on law and regulation on economic performance. The style of a legal system is influenced by political institutions (legal procedures,...), ideology, broader attitudes and philosophy. These latter features rely on the historical background and the historical institutional framework represented by the legal origin which is defined as "the style of social control on economic life" (La [START_REF] Porta | The economic consequences of legal origins[END_REF]. The main assumption is that some national legal systems are sufficiently similar to others to allow a classification of national legal systems into major families of law, though each country has a particular national legal system because each country has experienced its own changes and its own local circumstances. Despite particular adaptations, some fundamental legal features, shared by a group of countries, have persisted.

The Legal Origins Theory relies on three important conceptual ideas. First, by the eighteenth or nineteenth centuries, Continental Europe, particularly France, and Great Britain had developed separate styles of social control of economic activities as well as the underlying institutions supporting these styles. Second, both these styles and their legal institutions were transplanted by the origin countries to most of the world through colonization, rather "than written from scratch." Third, despite country specific changes, these styles have persisted in addressing the social control of economic life.

This way, this theory suggests that all law in a country is influenced by either the English common law or else the French civil law. The first one originates in the laws of England and has been transposed through conquest and colonization to England's colonies, including the United States, Canada, New Zealand, Australia, and many countries in Africa and Asia. The second one originates in the Roman law. Rediscovered in the eleventh century by the Catholic Church, the civil Roman law shaped the law in many European countries. However, the French civil law developed at the beginning of the nineteenth century after the French Revolution with Napoleon Bonaparte. This French law has been exported to many countries such as Spain, Portugal, Belgium, Italy, and the Netherlands through Napoleon's conquests. Moreover, the French civil law has been transposed to many countries in the world through colonization and conquest by France in the nineteenth century in Oceania, Indochina, Africa, and some Caribbean Islands. Besides, the influence of French civil law has also been implemented in Latin South American countries in the nineteenth century after the independence of the Spanish and Portuguese colonies. Finally, French civil law is now present in many South American, African and Asian countries.

The Legal Origins Theory attributes a heavier role of government in the civil law countries compared with the common law ones. This difference is mainly attributed to historical differences between the two legal traditions (see [START_REF] Glaeser | Legal origins[END_REF] for a theoretical presentation, and [START_REF] Klerman | Legal origin?[END_REF] and Roe (2007) for a critical view of the Legal Origin Theory). The French civil law had been developed to promote state control and to prevent judges the opportunity for re-interpreting or changing laws. This marginalization of the judiciary helped extend state control of the economy. In contrast, the English common law had been created to protect private property and private freedom against the Crown. Thereby, a system of decentralized law-making with an independent judiciary where judges could interpret and change laws were implemented as effective checks on the government.

The consequences of these differing legal traditions on current economic performances have been broadly studied. For instance, La [START_REF] Porta | Legal determinants of external finance[END_REF][START_REF] Porta | Law and finance[END_REF] investigates the link between the law and finance. They show that the law in the civil law system has been designed to keep investors poorly motivated and the stock market less developed, contrary to the common law's protecting and motivating investors. This paper highlights the crucial links between the legal environment and finance, i.e., the nexus between legal institutions of the past and current economic ones. Also, in several studies conducted by LLSV jointly with others, they found that such outcomes as government ownership of banks [START_REF] Porta | Government ownership of banks[END_REF], the burden of entry regulations [START_REF] Djankov | The regulation of entry[END_REF], regulation of labor markets [START_REF] Botero | The regulation of labor[END_REF], and government ownership of the media Djankov et al. (2003b) vary across legal families. They argue that civil law is associated with more government ownership and regulation than common law. This has adverse impacts on markets, such as greater corruption, a larger unofficial economy, and higher unemployment. For instance, La [START_REF] Porta | The quality of government[END_REF] studies such determinants of political institutions as government performance (provision of public good, effectiveness, government spending). They highlight some political theories to explain the quality of government and assume that legal origin can be a good proxy for these political theories. They argue that the legal origin of law and regulation can predict "inefficient, interventionist and distortionary policies." They show that common law countries are less interventionist, better public good suppliers, more efficient and democratic than civil law countries. Besides, in other studies, the same authors have investigated the link between legal origin, property rights, and contract enforcement. For instance, Djankov et al. (2003a) have found that common law is less associated with formalism of judicial procedures than French civil law. In another paper, La [START_REF] Botero | The regulation of labor[END_REF] have argued that common law countries enhance their judicial independence which is, in turn, associated with more freedom, better contract enforcement, and greater security of property rights1 . In all, more than 100 papers have used legal origin as an explanatory variable, and LLSV's papers have been cited more than 2500 times. This paper contributes to that literature by investigating the impact of legal origin on deforestation.

In this series of papers, LLSV and others have explained that legal origin is correlated with a wide range of institutions and policies which could be expected to produce higher rates of deforestation such as corruption, mis-defined property rights, or a weak rule of law in French civil law countries as opposed to common law ones. For this reason, the present paper investigates the role of legal origin on deforestation assuming that the legal system matters in explaining deforestation. The main result is that civil law countries deforest significantly less than common law countries. Nevertheless, the implementation of the each legal system is not exogenous, except for previously colonized countries in which the legal system was transplanted by the colonizer. The differential impact of the French civil law remains significant for the previously colonized. Also, because of the close relationship between colonial history and legal origin, it is difficult to attribute the differences in environmental performance to legal origin as opposed to other aspects of colonial policy. In this sense, the differential impact of the the French civil law could be attributed to French colonial legacies. Fortunately, not all previously colonized countries with a French civil law inheritance were colonized by France, which allows of testing a specific French difference. Former French colonies tend to deforest less than former British, Spanish, or other colonies. This difference can be attributed to forest law legacies inherited from the 1827 French Forestry Code. Finally, since colonization strategies were not random and depended on geographical and climatic conditions, geographic features are controlled for. These results remain significant and robust. This paper is organized as follows. Section 2 describes the dataset and variables used in this paper. Section 3 presents the main results and checks for the role of colonial legacies as well as geography. Section 4 provides concluding remarks and points to a interesting possible extension of this study.

Data

The core dataset consists of 110 countries which had to meet one condition: having a French civil law origin, a common law origin, or a German civil law origin. However, two other different samples are used here to investigate more deeply the exogeneous role of legal origin and the influence of colonial legacies. The first one is a core dataset of 70 countries which had to meet two other conditions: (1) they are former colonies, (2) they are not located in Europe. The first criterion ensures that legal origin was exogenous and the second one excludes European countries to reflect the difficulty of classifying the colonial history of most European countries such as Italy or Finland for instance. The second sample is a core dataset of 47 countries which had to meet two other conditions: (1) being located in the tropics and (2) having a forest area above one million hectares in 1990.

Table 9 describes the variables. Table 8 indicates which countries are in the core dataset of 110 countries as well as in the two other sub-samples. Deforestation rates have been calculated from various Forest Resources Assessment (FRA) of the Food and Agriculture Organization (FAO). Although forest area data are available since 1960, only data since 1990 is used here, given the lack of reliability of the data before the 1990-FRA. This way, the dependent variable is the four-year average annual rate of deforestation on the sub-periods 1990-1993, 1994-1997, 1998-2001, and 2002-2005. Legal origin was coded following two classifications. The first coding comes from LLSV's most recent coding [START_REF] Porta | The economic consequences of legal origins[END_REF]. The second is a revaluation of this coding in light of a variety of sources2 . This coding differs from LLSV's coding for five countries which are here classified as "mixed legal origin"3 . This alternative coding represents hybrid legal systems of the common and civil law whereas four countries were classified by LLSV as common law and one, the Philippines, as civil law. This coding could have had a large effect on the analysis, because all four classified common law countries had deforestation rates significantly higher whereas the Philippines had lower rates than average4 . Nevertheless, as discussed further below, the main results remain valid even using LLSV's coding.

The colonial origin variables are coded by the dominant colonial power in the period 1750-2009. Though for most countries the coding was relatively straightforward, it was more complicated where the country was colonized by multiple countries. In this latter case, the assumption is that the last colonialist had had the biggest effect on institutional infrastructure at the time of independence5 . Four categories of formerly colonized countries are created: the groups are the former colonies colonized by (i) France, (ii) England, (iii) Spanish and (iv) other countries (Belgium, the Netherlands, the Ottoman Empire, Portugal, and Spain, Japan or the U.S).

Table 1 shows that legal and colonial origin are highly correlated. First, all common law countries are former British colonies whereas all former French colonies are of French legal origin. Nevertheless, the correlation between legal and colonial origin is not perfect so that some deviations could be exploited. For instance, all former Spanish colonies have French legal origin countries. These heterogeneities help to investigate whether legal traditions or colonial legacies have a greater influence on current deforestation in former colonized countries.

Table 2 presents descriptive statistics of deforestation rates between 1990-2005 along the identity of the former colonizer 6 . As can be seen, countries with different legal systems and colonial histories differ significantly in terms of deforestation performance. First, we see that French civil law countries have deforested less between 1990-2005 than common law ones. Second, the former French colonies have deforested less between 1990 and 2005 than the other former colonies7 . These results seem to highlight that legal and colonial legacies could have influenced some current features which, in turn, influence current deforestation.

Econometric Results

Main results

Table 3 presents the results of regressions using the core dataset of 110 countries as well as the sample of 70 former colonies. In all regressions, the dependent variable is the four-year average annual rate of deforestation on the sub-periods 1990-1993, 1994-1997, 1998-2001, and 2002-2005. In the first two columns, the only independent variables are legal origin dummy variables and the omitted category is that of French legal origin.

As in Table 2, common law countries deforest 0.4% more, on average, than French civil law countries, and this difference is significant at the 1% level (column 1). This result holds in only previously colonized countries in which the implementation of the legal system is more exogenous (column 2)8 . These countries tend to deforest 0.3% more than French civil law countries.

The third column reports a regression with only colonial origin dummies as independent variables on the sample of former colonies (the omitted category is French former colonies). Former British colonies deforest 0.5% more than former French colonies, and this difference is statistically significant at the 1% level. Since all former British colonies are common law ones countries, this result does not allow of concluding about the role of legal origin or colonial legacies on deforestation. In addition, former Spanish and other colonies deforest more than former French colonies. This result is important, because all of these countries had received a version of the French civil law from Spain, Belgium, the Netherlands, the Ottoman Empire, or Portugal. Hence, since these former colonies deforest significantly more than former French colonies, it is obvious that legal origin does not suffice to explain the effects of inherited legacies on deforestation. This fact suggests that other aspects of colonial policy are likely to have been more important than legal origin.

Column 4 presents the results of a regression with both legal and colonial origin on the core dataset of 110 developed and developing countries. The coefficient on the common law dummy remains significant whereas the coefficient on the former British colony variable is now non-significant, although positive. These results seems to suggest that legal origin, not colonial origin matters. Moreover, the coefficient of both former Spanish and other colonies remains significantly positive. In this case, colonial origin seems to matter and some French colonial legacies allow of reducing deforestation compared to Spanish colonial legacies for instance.

Table 4 presents the same regressions and adds the five commonly used explanatory variables of deforestation9 : the lag of forested areas, the log of GDP10 , the rural population density, the population growth and the relative price of timber (see Table 9 for more information). All variables are four-year averages, the sub-periods being 1990-1993, 1994-1997, 1998-2001 and 2002-2005. The core dataset is reduced to 87 countries and the former colonies sample regroups, now, only 61 countries. The previous results concerning the effect of legal and colonial origins remain robust to the introduction of all these variables. All common law countries (previously colonized countries) deforest 0.7% (0.08%) more than French civil law countries between 1990 and 2005.

Table 5 replicates key regressions from Table 4 using the alternative coding. This coding differs from LLSV's coding and uses hybrid legal systems of the common and civil law for five countries. The results remain identical to those in Table 4. Common law countries experience deforestation rates higher than French civil law countries (0.7% higher), and that difference remains significant when colonial origin is controlled for (column 3). Again, the coefficient on the former British colony variable is positive but non-significant, suggesting that legal origin, not colonial origin, mattered.

Finally, Table 6 replicates key regressions from Table 4 using the third sample of tropical countries. The results remains identical to those in Table 4. Common law countries deforest 0.08% more than French civil law countries, and that difference remains significant when colonial origin is controlled for (column 4). Again, the coefficient on the former British colony variable is positive but non-significant, suggesting that legal origin, not colonial origin, mattered in developing countries with a significant tropical forest.

The role of geography

The colonization was not randomly implemented by colonial powers. It was done in accord with imperial aims and relative power which varied over time and between colonial powers [START_REF] Engerman | History lessons: Institutions, factors endowments, and paths of development in the new world[END_REF][START_REF] Joireman | Inherited legal systems and effective rule of law: Africa and the colonial legacy[END_REF]Lange, 2004).

To capture this feature, settler mortality data provided by Daron Acemoglu [START_REF] Acemoglu | The colonial origins of comparative development: An empirical investigation[END_REF] 11 is used. These figures give an indication of the extent to which a particular place was suitable for European settlement. In that study, the authors argue that European colonizers adopted different colonization strategies with correspondingly different institutions, depending on the conditions in the colonies. They explain that the feasibility of European settlement, characterized by the mortality rates of colonizers, determined the colonization strategy. More precisely, the conditions in colonies (the disease environment and the density of population) conditioned the European strategies. Besides they argue that institutions of the past have persisted over time so that current effects of present institutions on economic development could be explained by institutions of the past designed by Europeans in their colonies. They show econometrically that former "extractive colonies" characterized by a high level of mortality of settlers experienced bad institutions which impeded their current ones and so their level of development. Unlike these extractive colonies, the "settler colonies" had good institutions brought by the colonizer which have persisted over time, encouraging economic development in these former colonies.

The fact is that Britain colonized all or nearly all the most favorable countries. A mean comparison test shows that former French colonies had higher settler mortality compared to former British colonies.

Moreover, the geographical position of each country could influence deforestation rates. Also, French civil law countries are farther from the equator than common law countries. To control for that, the geographic position (latitude) of each country is used, to be sure that the results are not driven by the geographic location of the country 12 .

Table 7 reports the regressions concerning the effects of the settlers' strategies and the latitude. The dataset for the first three columns is only for former colonies for which settler mortality data is available. This dataset, covering 54 countries, is slightly different than the sample of former colonies used previously, of 70 countries 13 . The results concerning the superiority of French civil law compared to common law are identical. However, the positive effect of former Spanish colonies becomes non-significant 14 . These results could suggest that the initial superiority of the French colonial legacies are better attributed to colonial strategies.

Columns 4-7 add the control for the distance from the equator. The previous results persist concerning the positive effect of common law origins, and Spanish colonial legacies on deforestation.

Conclusion

Recent empirical work demonstrates that countries whose legal systems are based on French civil law differ systematically from those whose legal systems are based on English civil law. The hypothesis of this paper was that differences in the legal system can explain differences in environmental performance, measured through the loss of forest area between 1990 and 2005.

The main result of this paper is that French civil law countries deforest significantly less 12 The literature of the role of geography on economic development is substantial. See for instance [START_REF] Engerman | Factor endowments, inequality, and paths of development among new world economies[END_REF] and [START_REF] Easterly | Tropics, germs, and crops: how endowments influence economic development[END_REF]. 13 Given that all countries are previously colonized ones, in the third column with both legal and colonial origin, the former British colonies variable dummy has been removed because of its collinearity with the common law dummy variable. 14 The sole Spanish colony removed is Equatorial Guinea, whereas none of the former French colonies have been removed. Thus, the results are not driven by a change in the sample.

than English common law countries. This result is robust in all samples and with the commonly used explanatory variables of deforestation. This result is particularly significant for the sample of previously colonized countries for whom the legal system is mainly an exogenous feature, and of developing countries with a substantial tropical forest. Also, since the choice of colonization strategy was not random, geographical features were controlled for. These results remain significantly robust. Also, former French colonies deforest less than former British colonies, though this result does not persist when legal origin is controlled for. Thus legal origin, not colonial origin, matters for explaining the loss of forest area. Also, former French colonies deforest less than other former colonies in the civil law area, such as formerly Spanish colonies. These results suggest that more than the legal origin on law and regulation, some French colonial legacies have reduced deforestation in the developing countries which had been colonized by France15 .

As a conclusion, it is worth noting that the French civil law seems to matter in reducing deforestation in previously colonized countries, by comparison with English common law. This study highlights the superiority of the French civil law over the English common law in preserving forested areas in developing countries. Moreover, because of the close relationship between colonial history and legal origin, it is difficult to attribute the differences in environmental performance to legal origin as opposed to other aspects of colonial policy. Anyway, one can attribute this effect of the French civil law on deforestation to French colonial or legal legacies such as the forest law legacies inherited from the French Forestry Code of 1827. This result is confirmed by the fact that former French colonies tend to deforest less than former British colonies, reinforcing the hypothesis on the presence of French forest law legacies which is both a colonial origin and a legal origin.

This research can be extended to try to explain more precisely the differential impact of the French civil law. Another way to extend this article is to analyze whether English common law could, in such cases, allow of reducing deforestation since common law is associated to less corruption, better rule of law, and well defined property rights, all associated with low levels of deforestation. In this sense, legal origin could condition the effect of the institutional background on deforestation. La Porta, R., López-de Silanes, F., Pop-Eleches, C., Shleifer, A., 2004. Judicial 275 Note: Columns 1 to 3 (4 to 6) report the results with settlers mortality (latitude) as control variable. OLS robust standard errors in parentheses. In all regressions, regional and year dummies are introduced (omitted region is Asia). The omitted category is French legal origin, and/or Former French colony. *** statistical significance at 1%, ** statistical significance at 5%, * statistical significance at 10%.

Table 1 :

 1 checks and balances. The Journal of Political Economy 112 (2), 445-470. Former colonial status and legal origin

	Lange, M. K., 2004. British colonial legacies and political development. World Development
	32 (6), 905 -922.
	Roe, M. J., 2007. Juries and the political economy of legal origin. Journal of Comparative
	Economics 35 (2), 294 -308.

Table 2 :

 2 Legal origin, colonial origin, and deforestation rate (1990-2005)

		Core dataset (110 countries)
	Legal origin	French	Common German
		0.0022	0.0073 * * * -0.0025 * * *
	Number of countries	63	33	14
	Only previously colonized countries (65 countries)
	Colonial power	French	British	Spanish	Portuguese
		0.0034	0.0091 * * * 0.0051	0.0070 *
	Number of countries	18	25	16	6
	Asterisks indicate results of t-tests. The null hypothesis is that the mean is the
	same as the mean for French legal origin countries/Former French colonies.
	*** statistical significance at 1%, ** statistical significance at 5%, * statistical
	significance at 10%.

Table 3 :

 3 Legal origin, colonial origin, and deforestation (1990-2005) OLS robust standard errors in parentheses. In all regressions, regional and year dummies are introduced (omitted region is Asia). The omitted category is French legal origin, and/or Former French colony. *** statistical significance at 1%, ** statistical significance at 5%, * statistical significance at 10%.

	Dependent variable: Rate of deforestation
		Legal origin	Legal origin	Colonial origin	Both
		All sample Previously Colonized Previously Colonized	All sample
	VARIABLES	(1)	(2)	(3)	(4)
	Common law	0.004 * * *	0.003 * *		0.004 * * *
		(0.001)	(0.001)		(0.001)
	German civil law	0.0005	0.006 * * *		0.001
		(0.001)	(0.001)		(0.001)
	Former British colonies			0.005 * * *	0.002
				(0.002)	(0.001)
	Former Spanish colonies			0.006 * * *	0.007 * * *
				(0.002)	(0.002)
	Former other colonies			0.004 *	0.005 * *
				(0.002)	(0.002)
	Constant	0.002	0.005 *	0.004	0.0008
		(0.002)	(0.003)	(0.003)	(0.002)
	Observations	436	278	278	436
	Countries	110	70	70	110
	Adjusted R2	0.26	0.092	0.108	0.278
	F statistic	33.358	26.002	12.912	26.071
	Note:

Table 4 :

 4 Control variables, legal origin, colonial origin, and deforestation (1990-2005) OLS robust standard errors in parentheses. In all regressions, regional and year dummies are introduced (omitted region is Asia). The omitted category is French legal origin, and/or Former French colony. *** statistical significance at 1%, ** statistical significance at 5%, * statistical significance at 10%.

	Dependent variable: Rate of deforestation
		Legal origin	Legal origin	Colonial origin	Both
		All sample Previously Colonized Previously Colonized	All sample
	VARIABLES	(1)	(2)	(3)	(4)
	Common law	0.007 * * *	0.008 * * *		0.006 * * *
		(0.002)	(0.002)		(0.002)
	German civil law	0.005 * *	0.016 * * *		0.005 * * *
		(0.002)	(0.002)		(0.002)
	Former British colonies			0.008 * * *	0.003
				(0.002)	(0.002)
	Former Spanish colonies			0.004	0.005 *
				(0.003)	(0.003)
	Former other colonies			0.005 *	0.005 * *
				(0.003)	(0.002)
	Lag forest Area	0.0002	-.00003	-.0002	0.00002
		(0.0005)	(0.0008)	(0.0009)	(0.0006)
	Log GDP	-.003 * * *	-.002 * * *	-.002 * * *	-.002 * * *
		(0.0005)	(0.0007)	(0.0007)	(0.0005)
	Rural density	-1.71e-06	-4.54e-06 *	-3.58e-06	-2.48e-06
		(1.86e-06)	(2.37e-06)	(2.27e-06)	(1.96e-06)
	Pop. growth	0.005 * * *	0.006 * * *	0.006 * * *	0.005 * * *
		(0.001)	(0.001)	(0.002)	(0.001)
	Timber	0.003	0.002	0.001	0.002
		(0.002)	(0.003)	(0.003)	(0.002)
	Constant	0.005	0.008	0.006	0.005
		(0.007)	(0.011)	(0.012)	(0.007)
	Observations	348	244	244	348
	Countries	87	61	61	87
	Adjusted R2	0.341	0.24	0.236	0.35
	F statistic	18.217	12.696	8.191	15.557
	Note:

Table 5 :

 5 Alternative coding:Legal origin, colonial origin, and deforestation (1990-2005) OLS robust standard errors in parentheses. In all regressions, regional and year dummies are introduced (omitted region is Asia). The omitted category is French legal origin, and/or Former French colony. *** statistical significance at 1%, ** statistical significance at 5%, * statistical significance at 10%.

	Dependent variable: Rate of deforestation
		Legal origin	Legal origin	Both
		All sample Previously Colonized	All sample
	VARIABLES	(1)	(2)	(3)
	Common law	0.007 * * *	0.008 * * *	0.006 * * *
		(0.001)	(0.002)	(0.002)
	Mixed legal origin	0.018 * * *	0.02 * * *	0.017 * * *
		(0.003)	(0.003)	(0.003)
	German civil law	0.007 * * *	0.018 * * *	0.007 * * *
		(0.002)	(0.002)	(0.002)
	Former British colonies			0.002
				(0.002)
	Former Spanish colonies			0.003
				(0.003)
	Former other colonies			0.003
				(0.002)
	Lag Forest Area	0.0005	0.0004	0.0004
		(0.0005)	(0.0008)	(0.0005)
	Log GDP	-.003 * * *	-.003 * * *	-.003 * * *
		(0.0006)	(0.0007)	(0.0006)
	Rural density	-1.99e-06	-5.43e-06 * * *	-2.49e-06
		(1.54e-06)	(1.81e-06)	(1.67e-06)
	Pop. growth	0.006 * * *	0.007 * * *	0.006 * * *
		(0.001)	(0.002)	(0.001)
	Timber	0.003	0.002	0.003
		(0.002)	(0.003)	(0.002)
	Constant	0.0002	0.003	0.0004
		(0.007)	(0.01)	(0.007)
	Observations	348	244	348
	Countries	87	61	87
	Adjusted R2	0.409	0.326	0.408
	F statistic	18.685	12.879	15.78
	Note:

Table 6 :

 6 Tropical countries: Legal origin, colonial origin, and deforestation (1990-2005) OLS robust standard errors in parentheses. In all regressions, regional and year dummies are introduced (omitted region is Asia). The omitted category is French legal origin, and/or Former French colony. *** statistical significance at 1%, ** statistical significance at 5%, * statistical significance at 10%.

	Dependent variable: Rate of deforestation
		Legal origin Legal origin Colonial origin	Both
	VARIABLES	(1)	(2)	(3)	(4)
	Common law	0.008 * * *	0.008 * * *		0.008 * *
		(0.002)	(0.002)		(0.003)
	German civil law	-.012 * * *			-.004
		(0.004)			(0.004)
	Former British colonies			0.011 * * *	0.003
				(0.002)	(0.003)
	Former Spanish colonies			0.003	0.003
				(0.004)	(0.004)
	Former other colonies			0.014 * * *	0.014 * * *
				(0.003)	(0.003)
	Lag Forest Area	-.002 * * *	-.002 * * *	-.004 * * *	-.004 * * *
		(0.0006)	(0.0006)	(0.0006)	(0.0006)
	Log GDP	-.001 *	-.001	-.001	-.001
		(0.0008)	(0.0008)	(0.0008)	(0.0008)
	Rural density	-5.68e-06 * * * -6.57e-06 * * *	-6.63e-06 * * *	-6.57e-06 * * *
		(1.66e-06)	(2.02e-06)	(1.81e-06)	(1.81e-06)
	Pop. growth	0.003 * *	0.003 * *	0.003 * *	0.003 * *
		(0.001)	(0.001)	(0.001)	(0.001)
	Timber	-.001	-.0006	-.001	-.001
		(0.002)	(0.002)	(0.002)	(0.002)
	Constant	0.032 * * *	0.032 * * *	0.04 * * *	0.041 * * *
		(0.01)	(0.011)	(0.01)	(0.01)
	Observations	176	168	168	176
	Countries	44	42	42	44
	Adjusted R2	0.253	0.164	0.301	0.375
	F statistic	18.568	5.361	7.806	17.549
	Note:

Table 7 :

 7 Geography, legal origin, colonial origin, and deforestation (1990-2005) -2.94e-06 -7.03e-06 * * * -5.06e-06 * * -3.51e-06 *

	Dependent variable: Rate of deforestation
		Legal	Colonial	Both	Legal	Legal	Colonial	Both
		Origin	Origin	Origins	Origin	Origin	Origin	Origins
	VARIABLES	(1)	(2)	(3)	(4)	(5)	(6)	(7)
	Common law	0.01 * * *		0.011 * * *	0.007 * * *	0.008 * * *		0.006 * * *
		(0.002)		(0.002)	(0.002)	(0.002)		(0.002)
	German civil law	0.011 * * *		0.012 * * *	0.006 * * *	0.022 * * *		0.006 * * *
		(0.003)		(0.003)	(0.002)	(0.003)		(0.002)
	Former British colonies		0.011 * * *				0.008 * * *	0.002
			(0.002)				(0.002)	(0.002)
	Former Spanish colonies		0.002	0.003			0.005 *	0.005 * *
			(0.003)	(0.003)			(0.003)	(0.002)
	Former other colonies		0.004	0.005 *			0.004 *	0.005 *
			(0.003)	(0.003)			(0.002)	(0.002)
	Lag Forest Area	0.0003	-1.00e-05	-.0001	0.0002	-.0004	-.0005	0.00006
		(0.0008)	(0.0009)	(0.0009)	(0.0005)	(0.0007)	(0.0008)	(0.0006)
	Log GDP	-.002 * * *	-.002 * *	-.002 * *	-.003 * * *	-.002 * * *	-.002 * * *	-.002 * * *
		(0.0008)	(0.0009)	(0.0009)	(0.0005)	(0.0007)	(0.0007)	(0.0005)
	Rural density	-4.22e-06 -4.24e-06 -4.79e-06 (2.67e-06) (2.71e-06) (2.71e-06)	(1.86e-06)	(2.36e-06)	(2.22e-06)	(1.94e-06)
	Pop. growth	0.007 * * *	0.007 * * *	0.007 * * *	0.005 * * *	0.006 * * *	0.006 * * *	0.005 * * *
		(0.002)	(0.002)	(0.002)	(0.001)	(0.001)	(0.002)	(0.001)
	Timber	0.002	0.001	0.002	0.002	0.002	0.001	0.002
		(0.003)	(0.003)	(0.003)	(0.002)	(0.003)	(0.003)	(0.002)
	Log settler mortality	0.003 * * *	0.003 * * *	0.003 * * *
		(0.0008)	(0.0008)	(0.0009)
	Latitude				-.009	-.020 * * *	-.014 * *	-.008
					(0.006)	(0.007)	(0.007)	(0.006)
	Constant	-.013	-.018	-.013	0.007	0.015	0.011	0.007
		(0.012)	(0.013)	(0.013)	(0.006)	(0.01)	(0.011)	(0.007)
	Observations	216	216	216	348	244	244	348
	Countries	54	54	54	87	61	61	87
	Adjusted R2	0.296	0.294	0.301	0.345	0.258	0.243	0.353
	F statistic	10.875	7.501	9.946	16.31	11.443	7.038	14.

*

The seventeenth-century English revolutionary took control of the judiciary away from the Crown, and marginalised the role of the judiciary to prevent it from doing anything other than apply the existing law. The creation of laws and the administration of justice were then separated contrary to France. In this case, legislatures make laws, and independent judges enforce them, without interference from the legislature or the executive.

Basically, the World Legal Systems Websites (http://www.droitcivil.uottawa.ca/world-legal-systems/engmonde.html) of the University of Ottawa, Canada, is examined.

3 Four of these countries are former colonies: Zimbabwe, Sri Lanka, the Philippines, and South Africa. Also, Thailand is classified as "mixed legal origin" and as a non-colonized

country.4 For instance, the average deforestation rate was 0.03% in the core dataset and 0.05% in the sub-sample of tropical countries. In contrast, the four common law countries had a mean deforestation rate of 0.08% whereas the Philippines had a deforestation rate

of -0.08%. 5 Some countries were colonized by joining colonial powers, such as Cameroon, and were coded according to the colonial power of the more populous part (French in the case

of Cameroon).6 The sample used is of 65 countries formerly colonized by France, Great Britain, Spain or Portugal. The five other former colonies, not presented but in the sample of colonized countries, are: the Democratic Republic of the Congo, Eritrea, Ethiopia, South Korea and Indonesia.

However, the mean of deforestation in former French colonies is not statistically and significantly different from that of the former Spanish colonies.

Non colonized countries (included United States, Canada, Australia and New Zealand) were excluded because they are not former colonies so that their legal origin is not exogenous.

The results do not change with other control variables such as corruption or the rule of law index provided by ICRG or the World Bank (World Governance Indicators). Results available upon request.

The introduction of the squared term of GDP to test the presence of the environmental Kuznets curve does not change the results. Results available upon request.

[START_REF] Acemoglu | The colonial origins of comparative development: An empirical investigation[END_REF] provide data for sixty-two former colonies with the maximum settler mortality for Mali.

Though these results, mainly for former Spanish colonies, are not robust when controlling for "geographic" variables, thus suggesting that the superiority of the French colonial legacies could be attributed to French colonial strategies compared to Spanish ones.

Remerciements

I thank the multi-training program ``International Governance, universalism and relativism rules and institutions: What roles for international institutions?'' for financial support.

 A2). [START_REF] Acemoglu | The colonial origins of comparative development: An empirical investigation[END_REF]


Anne-Marie Jolly
email: anne-marie.jolly@univ-orleans.fr

Christophe Leger
email: christophe.leger@univ-orleans.fr

Guy Lamarque
email: guy.lamarque@univ-orleans.fr

Jorge Bernardino

José Carlos

Design of new engineering curricula: combining enterprises needs and interest of student, the case of smart buildings

Keywords: Smart building, sensors, networks, energetic efficiency. WEE2011, September 27-30, 2011, Lisbon, Portugal

Facing the disaffection of students for electronic engineering studies, we decided to have global brainstorming with students and then with teachers and enterprises to define new engineering courses that could be attractive and useful. The paper describes the problematic addressed, the way we used to build this curriculum and the new course as proposed to the students. We also emphasize all along the text on the problems we had to solve till the attribution of the accreditation.

INTRODUCTION

Since 10 years, electronic engineering, that was in the years 1980 the area attracting the best students, is no more attractive for young people. However the need for such competences [START_REF] Becker | Why don't young people want to become engineers? Rational reasons for disappointing decisions[END_REF] is still very important and engineering universities and enterprise spend together a great energy to find remedies to this crisis of vocation for electronic matters [START_REF]Engineering: issues, challenges and opportunities for development[END_REF].

The development of a new curriculum concerning smart building is a part of this work. It cannot be the only manner to solve the problem, the evolution of secondary schools to prepare the minds of pupils and show the interest of technologic fields (as a fundamental element of the multidisciplinary scope of sustainable development) is another one, and the two approaches are linked.

CREATING AN INTEREST FOR POTENTIAL STUDENTS

In 2008, in our school of engineers, the electronic department was at its lower level concerning the number of students.

The board asked to the head of department to find a solution to this problem, this solution had to be at the same time linked to the learning outcomes of the curriculum and to the communication towards the potential students. The project quickly turned to a curriculum linked both to energy engineering and to building engineering because the teachers were conscious of the strong interest of students concerning both sustainable development and civil engineering

Interest for ecology

From some years, in France there is a very strong interest of students about the environmental part of sustainable development; as an example, our department of civil engineering which is at the same time a department of environment saw the number of students increase in a very important way; in the same time, engineering curricula linked to energetic, as power electronics engineering for instance, become attractive to student if their name becomes energetic engineering. So we first decided that engineering electronic department should be called ecotechnologies engineering department because the techniques taught are applied to problems of energetic efficiency, energy valorization or improvement of everyday life.

Even if some older teachers did not agree, marketing cannot be considered nowadays as a foreign consideration in education; giving two different names to the same course can be a good thing too: the "commercial name" being used for the communication towards students and the more classic one towards enterprise.

Then, during a day where university is open to potential students, we made tombola where it was possible to earn games console. To be a participant to that tombola, it was necessary for the pupil, to order according to its own interest some potential application of electronics. At the end of the tombola, the idea of smart building definitely emerged.

Multidisciplinary and attractive course

In Polytech Orleans, in 2008, there were at the same time an electronic department, a civil engineering department, an energetic department and a material department; this means that the teachers already recruited in Polytech Orléans could possibly teach in all these fields, but the reality was quite different because once arrived they were "enrolled" in only one of these departments. To build a cross-curricular course is not easy because the teachers are not used to work together through departments; but things that could not exist in normal circumstances can be realized when there is a real problem of recruitment and a strong encouragement of stakeholders! Moreover the problem of innovation in Europe is questioned and the responsibility of engineering universities in the lack of technical innovation is criticized in many reports. Innovation supposes the interference of different disciplinary fields to create new ideas. Schools of engineers that integrate those different fields but also management, economy law and marketing should be the "top models" of innovation.

The last strong incitation is the fact that the board, very concerned by the lack of female students, also encouraged this initiative because project like Helena [START_REF]Egalité des chances dans les études et la profession d'ingénieur en Europe[END_REF] show that girls are more attracted towards multidisciplinary curricula.

NEEDS OF ENTERPRISES 3.1 The problem of energetic efficiency

According to the action plan of European Community concerning energetic efficiency, in 2020, European Union must have reduced of 20% (with respect to the level of 1990) the emissions of greenhouse gas, reduce of 20% the electric consumption, increase of 20% the part of renewable energies in the total consumption of energy.

In France buildings use 43% of the total energy and transports use 31% of it.

The buildings constructed before 1975 use more than 450kW/m2, buildings constructed since that date without particular application of norms between 150 and 250 KWh/m2 and buildings constructed in respect of norm RT 2005 between 50 and 100kWh/m2. We see on these figures that it is still too much, so the next steps necessary to satisfy requirement of EU are: Low Consumption Building (50kWh/m2) but even more passive building (<15 kWh/m2) and especially Positive Energy Buildings.

In the house 33% of the energetic use of families is done by the electric devices: for these devices, it is necessary not only to consume less energy when they are used, but also when they are in vigil, 14% of electric use inside the house is the fact of lighting; if there is a good automated management of all these devices, a strong reduction of the consumption can be expected. Concerning electric heating, decrease of the temperature of 1 degree during 24 hours reduces the electricity use of 7%.

A better isolation and a better design of design is not enough to fulfill requirements of EU, it is necessary to make building become "intelligent" thank to electronic devices.

The lack of engineers acting as integrators

The analysis of the needs of enterprises concerning recruitment of engineers in the sector of the technical building management showed that electrical engineers and computer-science or automation engineers only partially meet the needs of professionals. Engineers at consulting firms are able to design an acquisition chain (sensor and conditioner) and the processing (scanning and signal processing) of a quantity related to this technical building management (temperature, degree of humidity, light level...); moreover, they are able to develop multi-sensor architectures that can exchange information using various existing communication media (wired, optical, wireless...). But, their lack of knowledge of the standards relating to building, of the main features of the different materials used in building, and of the thermal aspect of the building does not allow them to offer optimal solutions. At the extreme, the system designed can be totally ineffective when installed in a building, because this ignorance of thermal-energy aspects of the building does not allow them to take into account all the constraints in designing such a system.

Our smart building course aims to train R & D engineers with strong skills in electronics and computer-automation science, able to understand, analyze and take into account all constraints related to building and the environment in which it is located. These field engineers, with a dual skill electronic-automation and computer from one side and energy and thermal of the building on the other part, will be powerful actors in establishing relevant specifications relating to the introduction of intelligent systems in and around the building (e.g.: fire protection, thermal regulation, management of openings, lighting, taking into account different energies, application of labels). Smart building engineers will be able to analyze the problem and find innovative solutions in comfort (technical management of buildings, multimedia...), home support (disabled, elderly), both in new buildings and renovation after an audit of existing buildings, for individual or collective, in manufacturing and services.

The multidisciplinary aspect of their field of expertise will enable these engineers to be more effective in innovation, in fact, this will be quite necessary because these engineers will be the pioneers in their field. To ensure the continuity of this double competence at all levels of the building industry, these engineers will have to train both integrators and installers but also their own employees.

Moreover these new engineers must be able to act on integrators to find global solution for energy management, air management and water management in the buildings. The concept of smart grids leads us to consider not only solution for intelligent building but also to the problem of the new cities because for example the problem of soft travelling impacts with solutions linked to the building itself.

BRAINSTORMING OF TEACHERS FOR INTERDISCIPLINARITY

The first step was to mix teachers and researchers in electronics, renewable energy, materials and mechanics, heat energy, networks, automation, and more generally all those who feel concerned by the concept of smart building.

It is not a trivial demarche because in France teachers are also researchers and choices in superior education impact on both fields.

We decided, after a global explanation of the "building of the future" made by people from enterprise and people already conducting researches in a part of this field, to list, in the first time, the skills necessary for job linked to smart building; we wanted to be able to work from learning outcome to define the content of the course, as our accreditation organism CTI asks us for some years. The teachings necessary should be defined afterwards only. The discussions were conducted in a rather pleasant atmosphere because only people really interested came in the design team.

Discussions to reach an agreement concerning the curriculum: Figure 1. Smart Building according to Electric Power Research Institute

The debate between teachers was about the fields really concerned by Smart Building and in which percentage they should be taught.

The concepts cover large fields (see Figure1) because tomorrow house will be seen as a global entity, moreover the problems attached aren't limited to those of energetic efficiency but they enlarge to care or safety [START_REF] Wang | Intelligent buildings and building automation[END_REF], [START_REF] Sinopoli | Smart buidings: a handbook for the design and operation of buiding technology systems[END_REF].

The team of teachers came to an agreement to define the fields concerned by the course:

-management of the building so as to make it more efficient: new sources of energy, stocking of energy, thermal comfort integrating all the sources of energy, water, quality of air -management of all the channels of information inside the building (voice, data, images)

-caring of people at home -security of buildings and persons at home -link with smart grids

It is important to consider all these points as a part of a more global solution, because the engineer has to propose solutions that can evolve.

Moreover the choice of pedagogy used can be decided for each of the items: apprenticeship based on problem or projects is possible in many courses; at this moment teachers are not in the same degree of knowledge concerning these new way of teaching so, even if these new methods are strongly recommended by CTI for apprenticeship, in a first time, only the presence of projects was considered as mandatory.

The result

The teachers defined first the great disciplinary fields concerned and then in each of them what was the learning outcome expected

The classification of skills leads us to the following items -Technical management of buildings -Design of electric or electronic devices for building -Management of energy inside buildings -Eco-design of buildings But they discussed also about the pedagogy: the necessity of keeping the interest of student, showed us that it was no more possible to teach mathematics for the treatment of signal during the first year of the school and then image processing two years after. The organization of the teaching must be made in units where the full process must be present from the basis of mathematics or physics to the final use of them, each of these units closing by a small application project.

These meetings and discussions were a quite innovative approach for the teachers, they revealed to be completely successful because they listened to each other, many of them feel concerned by environment, and some of them have at the same time a position in the school and in an enterprise.

When the strong points of the course were decided, the group of teachers had to work in two groups to be more effective in the content of the courses: one group was composed with teachers in electronics, computer science, and the other group was composed with teachers in materials, mechanics, sustainable buildings, thermal science.

THE INOVATIVE CURRICULUM

The definition of the name and contents of the units results from the work of the two group that at the end of the process work again together to be sure of the coherence of the curriculum It results in the final course that about 2/3 of the teachings concern the points of view of the IEEE engineer and 1/3 the point of view of the building engineer.

Content

All French engineers have basic knowledge in mathematics, computer science and management of project, but in this particular course, 3 great areas have been considered as the heart of the teaching:

-Management of energetic needs -Management of people and assets

-Habitat and environment

The first item includes energetic optimization, ecodesign, renewable energy, management and optimization of energetic needs, home automation and electric devices for home comfort.

The second one is related to sensors, communication networks, interoperability, multimedia, video surveillance, safety, caring people at home

The third one concerns waste and water treatment, environmental norms about buildings, Electromagnetic compatibility.

Those points are then declined according basic teaching areas: electronics, automation, computer science, mechanics and materials, energetic, heat energy.

As the university is situated close to a school of art and design, common projects between the engineers in smart building and student from the art department will take place to give more interest to the projects of both students. This will also increase acceptability by inhabitants of solutions for home automation too.

In France the studies to become engineer last 5 years after Alevel. Our course correspond to the 3 last years, it has an academic duration of 1800 hours. It addresses to student having a scientific diploma general or specialized, in this way, we respect CTI's recommendations [START_REF]Références et Orientations CTI[END_REF], the French engineering agency for the accreditation of engineering curricula.

The first year of the course gives the students strong skills in mathematics, automation, computer science and thermal. These skills will allow them to understand the more specific courses of the last two years, but will also give them opportunities to reorient their career after some years if they want to.

We will give details of the last two years (see Figure 2 and Figure 3) corresponding to the Masters 'Degree. During the last year of this master degree student will make a trainee in enterprise, it is the reason why the number of hours of lecture, tutorial and practical work is reduced.

1st There are many small projects included in each of the thematic courses, this is necessary, because we want that the students really become implied in each part of the curriculum and not only consumers of the course.

Differences with other curricula dealing with sustainable buildings

In France, from 2 years, the schools of engineers feel concerned by the problematic of energetic efficiency for buildings.

In our school, two approaches started from demarches of the electronic department in front of a necessity to evolve or die! So the two approaches that we realized in Polytech Orléans are electronic oriented:

-designing the new course presented here, completely devoted to smart building, -creation an option of last year in the classical electronic (now called eco-technology) course

While in other schools [START_REF]Master Batiment Intelligent Cergy[END_REF] they started from civil engineering adding teachings in electronics.

The two categories of engineers will have to work together, but engineers whose basic skills are electronic will be able to go deeper in technologic solutions for home automation. Our approach driven by enterprise and attractiveness problematic is coherent with our site strategy; a cluster called S2E2 (Sciences et Systèmes de l'Energie Electrique) is present in Région Centre and develops transversal solutions and experimental platforms from electronics solutions applied to buildings.

CONCLUSIONS

A problem occurred at the time of the accreditation by Commission des Titres d' Ingénieurs. This agency wishes that the title of engineering courses refer to great domains of engineering (American classification of ABET). In our case it is rather impossible because we depend both of two of these fields. After a discussion between the school and the agency, the decision was to keep the name "intelligence of building" for 3 years and to decide what to do on the next accreditation process. The dean of the school thinks that those multidisciplinary courses are necessary to generate innovation and we hope that some of the engineers will decide to create enterprises in these fields.

Concerning the attraction of students, it quickly proved to be a good bet, as soon as the information about the new accreditation was broadcast, many students came with a candidature.

In the fields of engineering such as electronic engineering, attract the student is one thing, keep their interest is another one; the organization of the teachings in units where the mathematics necessary to the study of signal treatment is placed with the applications of those mathematics is a good thing; at these moments we have a strong interest about the use of apprenticeship by problem learning. People from University of Leuven [8] came to explain to the teachers the concepts and interest of this methodology, the final evaluation of this session showed a real interest of the teachers.

Concerning enterprises, we hope that the crisis will let them propose internships to the students. We were very happy to see that those enterprises took time to meet the CTI agency when it comes in the school to make an audit before deciding to accredit ate the course. The pertinent explanations of these enterprises (Wirecom, Legrand) were the best keys to show what were this curriculum concept and the real need for it.

At the same time as we designed this curriculum, the ministry of education decided to change the secondary technical course that closes secondary scholarship and leads to the University: instead of a specific course for example: electronics, power electronics, they decided to create a more transversal teaching including principles of sustainable development. In a certain way, this point of view is very similar to the one that guided the design of our curriculum. At each level, electronics and electrical engineering need to attract good students that would be able to realize postgraduate studies [START_REF][END_REF].

The complete program can be found on our web site [10], the course open in September 2011.

We consider that the methodology used in this curriculum is convenient and necessary in other fields of teaching science and technology, the approaches quoted in [START_REF][END_REF] confirm it.

Figure 3 .

 3 Figure 3. Second year of Master program without the final trainee


Romain Benassi

Julien Bect

Emmanuel Vazquez

Robust Gaussian process-based global optimization using a fully Bayesian expected improvement criterion

We consider the problem of optimizing a real-valued continuous function f , which is supposed to be expensive to evaluate and, consequently, can only be evaluated a limited number of times. This article focuses on the Bayesian approach to this problem, which consists in combining evaluation results and prior information about f in order to efficiently select new evaluation points, as long as the budget for evaluations is not exhausted. The algorithm called efficient global optimization (EGO), proposed by Jones, Schonlau and Welch (J. Global Optim., 13(4): 1998), is one of the most popular Bayesian optimization algorithms. It is based on a sampling criterion called the expected improvement (EI), which assumes a Gaussian process prior about f . In the EGO algorithm, the parameters of the covariance of the Gaussian process are estimated from the evaluation results by maximum likelihood, and these parameters are then plugged in the EI sampling criterion. However, it is well-known that this plug-in strategy can lead to very disappointing results when the evaluation results do not carry enough information about f to estimate the parameters in a satisfactory manner. We advocate a fully Bayesian approach to this problem, and derive an analytical expression for the EI criterion in the case of Student predictive distributions. Numerical experiments show that the fully Bayesian approach makes EI-based optimization more robust while maintaining an average loss similar to that of the EGO algorithm.

Introduction

Let f be a continuous real-valued function defined on some compact space X ⊂ R d . We consider the problem of finding the maximum of f , when f is supposed to be expensive to evaluate because one evaluation takes a long time or a large amount of resources. In this case, the optimization of f must be carried out using a limited number of evaluations. More precisely, given a budget of N evaluations of f , our objective is to choose sequentially N evaluation points X 1 , . . . , X N ∈ X so that ε(X N , f ) = M -M N is small, where X N stands for (X 1 , . . . , X N ), M = max x∈X f (x) and

M N = f (X 1 ) ∨ • • • ∨ f (X N ).
In this article, we adopt a Bayesian approach to this sequential decision problem: the unknown function f is considered as a sample path of a real-valued random process ξ defined on some probability space (Ω, B, P 0 ) with parameter x ∈ X, and a good strategy is a strategy that achieves, or gets close to, the Bayes risk r B := inf X N E 0 (ε(X N , ξ)), where E 0 denotes the expectation with respect to P 0 and the infinimum is taken over the set of all sequential strategies. The reader is referred to the books [START_REF] Törn | Global Optimization[END_REF][START_REF] Pintér | Global optimization. Continuous and Lipschitz optimization: algorithms, implementations and applications[END_REF][START_REF] Zhigljavsky | Stochastic global optimization[END_REF][START_REF] Conn | Introduction to derivative-free optimization[END_REF][START_REF] Tenne | Computational intelligence in optimization: applications and implementations[END_REF] for a broader view on the field of global optimization.

It is well-known [START_REF] Mockus | The application of Bayesian methods for seeking the extremum[END_REF][START_REF] Mockus | Bayesian approach to Global Optimization: Theory and Applications[END_REF][START_REF] Betrò | Bayesian methods in global optimization[END_REF][START_REF] Locatelli | An adaptive stochastic global optimization algorithm for one-dimensional functions[END_REF][START_REF] Auger | Continuous lunches are free plus the design of optimal optimization algorithms[END_REF][START_REF] Ginsbourger | Towards Gaussian process-based optimization with finite time horizon[END_REF][START_REF] Grünewälder | Regret bounds for Gaussian process bandit problems[END_REF] that an optimal Bayesian optimization strategy, i.e. a strategy X ⋆ N such that E 0 (ε(X ⋆ N , ξ)) = r B , can be formally obtained by dynamic programming. Let E n , n = 1, 2, . . ., denote the conditional expectation with respect to the σ-algebra F n generated by the random variables X 1 , ξ(X 1 ), . . . , X n , ξ(X n ). Denote by R N = E N (ε(X N , ξ)) the terminal risk and define by backward induction

R n = min x∈X E n R n+1 | X n+1 = x , n = N -1, . . . , 0. (1)
Then, we have R 0 = r B , and the strategy X ⋆ N defined by

X ⋆ n+1 = argmin x∈X E n R n+1 | X n+1 = x , n = 1, . . . , N -1, (2)
is optimal. Unfortunately, solving (1)-( 2) over an horizon N of more than a few steps is not numerically tractable, for both the space of possible actions and the space of possible outcomes at each step are continuous.

A natural way of dealing with this problem is to consider a suboptimal onestep lookahead strategy; see, e.g., [13, chapter 6]. This leads to choosing each new evaluation point according to

X n+1 = argmin x∈X E n (M -M n+1 | X n+1 = x) = argmax x∈X E n (M n+1 | X n+1 = x) = argmax x∈X ρ n (x) := E n (ξ(X n+1 ) -M n ) + X n+1 = x , (3)
where (z) + = 0 ∨ z. The sampling criterion ρ n , introduced by J. Mockus [START_REF] Mockus | The application of Bayesian methods for seeking the extremum[END_REF] and popularized through the EGO algorithm [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF], is known as the expected improvement (EI). When ξ is a Gaussian process, or in other words, when a Gaussian process prior is chosen for f , it is well-known that the EI can be written in closed form, with the consequence that the maximization of ρ n can be carried out with a moderate computational effort. However, a Gaussian process prior carries a high amount of information about f and it is often difficult to elicit such a prior before any evaluation is made. As a result, the covariance function of ξ is usually assumed to belong to some parametric class of positive definite functions, the value of the parameters assumed to be unknown. In the EGO algorithm, the parameters are estimated from the evaluation results by maximum likelihood, and then plugged in the EI sampling criterion (computed for a Gaussian process with known covariance function). It has been reported [START_REF] Forrester | Global optimization of deceptive functions with sparse sampling[END_REF] that this plug-in strategy can lead to very disappointing results when the evaluation results do not carry enough information about f to estimate the parameters satisfactorily. We advocate a fully Bayesian approach to this problem, following the steps of Locatelli [START_REF] Locatelli | An adaptive stochastic global optimization algorithm for one-dimensional functions[END_REF][START_REF] Locatelli | Bayesian algorithms for one-dimensional global optimization[END_REF] and, more recently, Osborne and co-authors [START_REF] Osborne | Bayesian Gaussian Processes for Sequential Prediction Optimisation and Quadrature[END_REF][START_REF] Osborne | Gaussian processes for global optimization[END_REF][START_REF] Osborne | Towards real-time information processing of sensor network data using computationally efficient multi-output Gaussian processes[END_REF].

The paper is organized as follows. Section 2 recalls the expression of the EI criterion in the case of a Gaussian process prior with known covariance function, and describes the plug-in approach used in the EGO algorithm to handle the parameters of the covariance function when it is only assumed to belong to some parametric class. Section 3 explains how a fully Bayesian approach can be adopted in this problem, in order to take into account the uncertainty on the parameters of the covariance function. Section 4 presents a new closed-form expression of the EI criterion for Student predictive densities, which arises naturally when a conjugate inverse-gamma prior is used for the variance parameter of the Gaussian process prior. Section 5 illustrates with numerical results the benefits of the fully Bayesian approach, focusing more particularly on the tail of the error distribution, i.e., on the occurrence of large errors.

Nota bene. The analytical expression of the expected improvement for Student predictive distributions, presented in Section 4, has in fact already been obtained by Williams, Santner and Notz [START_REF] Williams | Sequential Design of Computer Experiments to Minimize Integrated Response Functions[END_REF] in the special case of an improper Jeffrey prior on the variance. We warmly thank Frank Hutter for pointing out this paper to us during the LION5 conference.

Efficient global optimization

The expected improvement sampling criterion for a Gaussian process

Recall that the distribution of a Gaussian process ξ is uniquely determined by its mean function m(x) := E 0 (ξ(x)), x ∈ X, and its covariance function k(x, y) := E 0 ((ξ(x)m(x))(ξ(y)m(y))), x, y ∈ X. Hereafter, we assume that the mean function is constant on X and write ξ ∼ GP (m, k) to denote that ξ is a Gaussian process with mean function m(x) = m ∈ R and covariance function k. Proposition 1. Let k be a stationary covariance function written as k(x, y) = σ 2 r(xy), x, y ∈ X, where σ 2 > 0 and r(0) = 1 (hence, r is a correlation function). Assume that ξ | m ∼ GP (m, k) and m ∼ U (R), where U (R) denotes the (improper) uniform distribution over R. Then, for all x ∈ X,

ξ(x) | F n ∼ N ξ n (x), s 2 n (x) ,
where

ξ n (x) = m n + r n (x) T R -1 n (ξ n -m n 1 n ) , (4)
with

                         ξ n = (ξ(X 1 ), . . . , ξ(X n )) T , 1 n = (1, . . . , 1) T ∈ R n ,
R n the correlation matrix of ξ n , r n (x) the correlation vector between ξ(x) and ξ n ,

m n = 1 T n R -1 n ξ n 1 T n R -1
n 1n , the weighted least squares estimate of m, and

s 2 n (x) = σ 2 κ 2 n (x) , (5)
with

κ 2 n (x) = 1 -r n (x) T R -1 n r n (x) + (1 -r n (x) T R -1 n 1 n ) 2 1 T n R -1 n 1 n . ( 6
)
Proposition 2. Under the assumptions of Proposition 1, the expected improvement can be written as

ρ n (x) =    s n (x) Φ ′ ξn(x)-Mn sn(x) + ( ξ n (x) -M n ) Φ ξn(x)-Mn sn(x) if s n (x) > 0, ξ n (x) -M n + if s n (x) = 0. ( 7
)
where Φ denotes the Gaussian cumulative distribution function.

Propositions 1 and 2 show that, given a set of evaluation points and a Gaussian prior, the EI sampling criterion can be computed with a moderate amount of resources (computing (4) at q different points in X involves O(qn 2 ) operations).

However, it is rare that a user has enough information about f in order to choose an adequate covariance function k before any evaluation is made. The approach generally taken consists in choosing k in a parametrized class of covariance functions and estimating the parameters of k from the evaluation results.

Classical parametrized covariance functions

There are chiefly three classes of parametrized covariance functions in the literature of Gaussian processes for modeling computer experiments. These are the class of the so-called Gaussian covariances, the class of the exponential covariances, and that of the Matérn covariances. Using Matérn covariances makes it possible to tune the mean square differentiability of ξ, which is not the case with the exponential and Gaussian covariances.

Define

υ ν : R + → R + such that, ∀h ≥ 0, υ ν (h) = 1 2 ν-1 Γ (ν) 2ν 1/2 h ν K ν 2ν 1/2 h , ( 8
)
where Γ is the Gamma function and K ν is the modified Bessel function of the second kind of order ν. The parameter ν > 0 controls regularity at the origin of υ ν .

The anisotropic form of the Matérn covariance on R d may be written as

k θ (x, y) = σ 2 r θ (x, y), with r θ (x, y) = υ ν   d i=1 (x [i] -y [i] ) 2 β 2 i   , x, y ∈ R d , (9)
where the positive scalar σ 2 is a variance parameter (we have

k θ (x, x) = σ 2 ), x [i] , y [i]
denote the i th coordinate of x and y, the positive scalars β i represent scale or range parameters of the covariance, or in other words, characteristic correlation lengths, and finally θ = (ν, β 1 , . . . , β d ) ∈ R d+1 + denotes the parameter vector of the Matérn covariance. Note that an isotropic form of the Matérn covariance is obtained by setting β 1 = . . . = β d = β. Then, the parameter vector of the Matérn covariance is θ = (ν, β) ∈ R 2 + .

The EGO algorithm

The approach taken in the EGO (efficient global optimization) algorithm [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF][START_REF] Schonlau | Computer experiments and global optimization[END_REF][START_REF] Schonlau | Global optimization with nonparametric function fitting[END_REF][START_REF] Schonlau | A data analytic approach to Bayesian global optimization[END_REF] consists in estimating the unknown parameters of the covariance function by maximum likelihood, after each new evaluation. Then, the EI sampling criterion is computed using the current value of the parameters of the covariance. EGO can therefore be viewed as a plug-in approach.

Remark 1 (about maximum likelihood estimation of the parameters of a covariance function of a Gaussian process). Recall that, for ξ ∼ GP(m, k θ ) with k θ (x, y) = σ 2 r θ (x, y), the likelihood of the evaluation results can be written as

ℓ n (ξ n ; m, σ 2 , θ) = 1 (2πσ 2 ) n/2 |R n (θ)| 1/2 e -1 σ 2 (ξ n -m1n) T Rn(θ) -1 (ξ n -m1n) , (10)
where R n (θ) stands for the correlation matrix of ξ n , parametrized by θ. Note that setting to zero the partial derivatives of ℓ n with respect to m and σ 2 yields the following maximum likelihood estimates for m and σ 2 :

m(θ) = 1 T n R n (θ) -1 ξ n 1 T n R n (θ) -1 1 n , (11)
σ 2 (θ) = 1 n ξ n -m1 n T R n (θ) -1 ξ n -m1 n . ( 12
)
Thus the maximum likelihood estimate of θ can be obtained by maximizing the profile likelihood θ → ℓ n (ξ n ; m(θ), σ 2 (θ), θ).

The case of deceptive functions

Deceptive functions is a term coined by D. Jones (see [START_REF] Forrester | Global optimization of deceptive functions with sparse sampling[END_REF][START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF]) to describe functions that appear to be "flat" based on evaluation results. In fact, any function can potentially appear to be flat depending on how it is sampled.

When the available evaluation results do not bring enough information on the objective function f to estimate the parameters of the covariance function with a reasonnable precision, the variance of the error of prediction can be severely under-estimated as depicted in Figure 1. As will be shown in Section 5.1, this can lead to very unsatisfactory behaviors of the EGO algorithm, which tends to waste lots of evalutions in local search around the current maxima (exploitation), very early in the optimization procedure, to the detriment of global search (exploration).

Fully Bayesian one-step lookahead optimization

It has been emphasized in Section 1 that the rationale behind the EI criterion is of a Bayesian decision-theoretic nature. Indeed, maximizing the EI criterion at iteration n is equivalent to minimizing the expected loss E n (max(ξ) -M n+1 ), where the expectation is taken with respect to the value of the next evaluation, which is unknown and therefore modeled as a random variable.

In a fully Bayesian setting, all the unknown parameters of the model have to be given prior distributions. This has already been done for the unknown mean m in Proposition 1. Let π 0 denote the prior distribution of the vector of covariance parameters θ ′ = (σ 2 , θ), and let π n , n = 1, . . . , N , denote the corresponding posterior distributions. According to Bayes' rule, the posterior distribution of ξ(x) is a mixture of Gaussian distributions N ξ n (x; θ ′ ), s 2 n (x; θ ′ ) weighted by π n (dθ ′ ). The expected improvement criterion for this model can thus be written, using the tower property of conditional expectations, as

E n (ξ(x) -M n ) + = E n E n (ξ(x) -M n ) + θ ′ = ρ n (x; θ ′ ) π n (dθ ′ ) . ( 13
)
Note that the plug-in EI criterion of Section 2.3 can be seen as an approximation of the fully Bayesian criterion (13):

ρ n (x; θ ′ ) π n (dθ ′ ) ≈ ρ n (x; θ ′ n ) ,
which is justified only if the posterior distribution is concentrated enough around the MLE estimate θ ′ n . In the general case, we claim that it is safer to use the fully Bayesian criterion [START_REF] Bertsekas | Dynamic programming and optimal control[END_REF], since the corresponding expected loss integrates the uncertainty related to the fact that θ ′ is not exactly known. This claim will be supported by the numerical results of Section 5.

When π 0 is a finitely supported discrete distribution, the posterior distribution π n -and therefore the integral (13)-can be computed exactly using Bayes' rule. For more general prior distribution, the integral can be approximated by stochastic techniques like MCMC sampling or SMC sampling (see [START_REF] Robert | Monte Carlo statistical methods[END_REF][START_REF] Del Moral | Sequential Monte Carlo samplers[END_REF][START_REF] Liu | Monte Carlo strategies in scientific computing[END_REF] and the references therein). An alternative approach using Bayesian quadrature rules [START_REF] O'hagan | Bayes-Hermite quadrature[END_REF] has been proposed in [START_REF] Osborne | Bayesian Gaussian Processes for Sequential Prediction Optimisation and Quadrature[END_REF][START_REF] Osborne | Gaussian processes for global optimization[END_REF][START_REF] Osborne | Towards real-time information processing of sensor network data using computationally efficient multi-output Gaussian processes[END_REF]. In all cases, the EI criterion is approximated by an expression of the form i w i ρ n (x; θ ′ i ), which amounts to saying that π n is approximated by the discrete distribution i w i δ θ ′ i . Remark 2. Although fully Bayesian approaches for Gaussian process models have been proposed in the literature for more than two decades (see [START_REF] O'hagan | Curve Fitting and Optimal Design for Prediction[END_REF][START_REF] Handcock | A Bayesian analysis of Kriging[END_REF] and the references therein), surprisingly little has been written from this perspective in the context of Bayesian global optimization. An early attempt in this direction can be found in [START_REF] Locatelli | An adaptive stochastic global optimization algorithm for one-dimensional functions[END_REF][START_REF] Locatelli | Bayesian algorithms for one-dimensional global optimization[END_REF], where the variance parameter of a Brownian motion is given an inverse-gamma prior and then integrated out as in [START_REF] Bertsekas | Dynamic programming and optimal control[END_REF]. More recently, the fully Bayesian approach has been developed in a more general way by [START_REF] Osborne | Bayesian Gaussian Processes for Sequential Prediction Optimisation and Quadrature[END_REF][START_REF] Osborne | Gaussian processes for global optimization[END_REF][START_REF] Osborne | Towards real-time information processing of sensor network data using computationally efficient multi-output Gaussian processes[END_REF], but the important connection of ( 13) with the usual (Gaussian) EI criterion was not clearly established. Remark 3. Discrete mixtures of Gaussian distributions and the corresponding EI criterion have also been introduced in [START_REF] Ginsbourger | Discrete mixtures of kernels for kriging-based optimization[END_REF] to allow for the use of several parametric classes of covariance functions, in order to provide increased robustness with respect to the choice of a particular class. The approach is not Bayesian, however, since the weights in the mixture are not posterior probabilities.

Let us consider the case of a Gaussian process ξ with unknown mean m and covariance function of the form k(x, y) = σ 2 r(x, y). We assume that m and σ 2 are independent, with m uniformly distributed on R (as in Proposition 1) and σ 2 following an inverse-gamma distribution with shape parameter a 0 and scale parameter b 0 , hereafter denoted by IG (a 0 , b 0 ). We shall prove that, in this setting, the EI criterion still has an explicit analytical expression, which is a generalization of the usual EI criterion given in Proposition 2.

First, recall that the prior chosen for σ 2 is conjugate [START_REF] O'hagan | Some Bayesian numerical analysis[END_REF]:

Proposition 3. The conditional distribution of σ 2 given F n is IG (a n , b n ), with a n = a 0 + n -1 2 , b n = b 0 + 1 2 ξ n -m n 1 n T R -1 n ξ n -m n 1 n .
Using this result and the fact that ξ(x) | σ 2 , ξ n ∼ N 0, σ 2 κ 2 n (x) , it is easy to show that the predictive distribution of ξ(x) is a Student distribution. More precisely: Proposition 4. Let t η denote the Student distribution with η > 0 degrees of freedom. Then, for all x ∈ X,

ξ(x) -ξ n (x) γ n (x) | F n ∼ t ηn ,
with η n = 2a n , and γ 2 n (x) = b n /a n κ 2 n (x). In other words, the predictive distribution at x is a location-scale Student distribution with η n degrees of freedom, location parameter ξ n (x) and scale parameter γ n (x). The following result is the key to our EI criterion for Student predictive distributions:

Lemma 1. Let T ∼ t η with η > 0. Then E (T + u) + = +∞ if η ≤ 1, η+u 2 η-1 F ′ η (u) + u F η (u) otherwise,
where F η is the cumulative distribution function of t η .

Combining Lemma 1 and Proposition 4 finally yields an explicit expression of the EI criterion:

Theorem 1. Under the assumptions of this section, for all x ∈ X,

E n (ξ(x) -M n ) + = γ n (x) η n + u 2 η n -1 F ′ ηn (u) + u F ηn (u) , ( 14
)
with u = ξ n (x) -M n /γ n (x).
It has been assumed, up to this point, that the only unknown parameter in the covariance function is the variance σ 2 . More generally, assume that k(x, y) = σ 2 r(x, y; θ): in this case we proceed by conditioning as in Section 3. Indeed, assume that θ is independent from m, σ 2 with a prior distribution π 0 . Let us denote by ρn (x; θ) = E n (ξ(x) -M n ) + | θ the value of the EI criterion at x provided by Theorem 1 when the value of the unknown parameter is θ. Then

E n (ξ(x) -M n ) + = E n ρn (x; θ) = ρn (x; θ) π n (dθ), (15)
where π n denotes the posterior distribution of θ after n evaluations. As explained in Section 3, the integral (15) boils down to a finite sum that can be computed exactly (using Bayes' rule) when the prior π 0 has a finite support; in the general case, approximation techniques have to be used.

5 Numerical experiments

Optimization of a deceptive function

Experiment. Consider the objective function f :

X = [-1, 1] → R defined by f (x) = x (sin(10x + 1) + 0.1 sin(15x)) , ∀x ∈ X .
We choose an initial set of four evaluation points with abscissas -0.43, -0.11, 0.515 and 0.85, as shown in Figure 1. Our objective is to compare the evaluation points chosen by the plug-in approach (i.e., the EGO algorithm) and those chosen by the fully Bayesian algorithm (FBA) proposed in Section 4.

In both approaches, we consider a Matérn covariance function with a known regularity parameter ν = 2 (see Section 2.2). In the approach of Section 4, we choose an inverse gamma distribution IG(0.2, 12) for σ 2 . Since X has dimension one, there is only one range parameter β. To simplify the implementation of the approach proposed, we shall assume that β has a finite support distribution. More precisely, define a β min and a β max , such that β min < β max , and set, for all i = 0, . . . , I, β i = β min βmax βmin i/I . We assume a uniform prior distribution over the β i s, with β min = 2 × 10 -3 , β max = 2 and I = 100.

The optimization of the two sampling criteria is performed by a Monte Carlo approach. More precisely, we generate once and for all a set of q = 600 candidate points uniformly distributed over X and the search for the maximum of each sampling criterion is carried out at each iteration by determining the value of the sampling criterion over this finite set (the same set of points is used for both criteria).

Results. Figures 2, 3 and4 show that the standard deviation of the error of prediction is severely underestimated when using the EGO algorithm, as a result of the maximum likelihood estimation of the parameters of the covariance from a deceptive set of evaluation points. If the uncertainty about the covariance parameters is taken into account, as explained above, the standard deviation of the error is more satisfactory. Figures 3 and4 show that the maximum is approximated satisfactorily after only four iterations with FBA, whereas EGO needs nine more iterations before making an evaluation in the neighborhood of the maximizer. Indeed, we observe that EGO stays in the neighborhood of a local optimum for a long time, while X remains unexplored. This behavior is not desirable in a context of expensive-to-evaluate functions.

Comparison on sample paths of a Gaussian process

Experiment. In order to assess the performances of EGO and FBA from a statistical point of view, we study the convergence to the maximum using both algorithms on a set of sample paths of a Gaussian process.

We have built several testbeds T k , k = 1, 2, . . ., of functions f k,l , l = 1, . . . , L, corresponding to sample paths of a Gaussian process, with zero-mean and a Matérn covariance function, simulated on a set of q = 600 points in [0, 1] d generated using a Latin hypercube sampling (LHS), with different values for d and for the parameters of the covariance. Here, due to the lack of room, we present only the results obtained for two testbeds in dimension 1 and 4 (the actual parameters are provided in Table 1). Table 1. Parameters used for building the testbeds of Gaussian-process sample-paths.

We shall compare the performance of EGO and FBA based on the approximation error ε(X n , f k,l ), l = 1, . . . , L. For reference, we also provide the results obtained with two other strategies. The first strategy corresponds to using an EI criterion with the same values for the parameters of the covariance function of ξ than those used to generate the sample paths in the testbeds. In principle this strategy ought to perform very well. The second strategy corresponds to space-filling sampling, which is not necessarily a good optimization strategy.

For FBA, we choose the same priors as those described in Section 5.1. More precisely, whatever be the dimension d, we choose an isotropic covariance function (with only one scale parameter) and we set β min = 1/400 and β max = 2 √ d. experiment, as the reference strategy where the true parameters are assumed to be known. Comparing the tails of complementary cumulative distribution function of the error max f -M n makes it clear, however, that using a fully Bayesian approach brings a significant reduction of the occurrence of large errors with respect to the EGO algorithm. In other words, the fully Bayesian approach appears to be statistically more robust than the plug-in approach, while retaining the same average performance. (c) Distribution of errors at iteration 16 Fig. 5. Average results and error distributions for testbed T1, for FBA (solid black line), EGO (dashed black line), the EI with the parameters used to generate sample paths (solid gray line), the space-filling strategy (dashed gray line). More precisely, (a) represents the average approximation error as a function of the number of evaluation points. In (b) and (c), F (x) stands for the cumulative distribution function of the approximation error. We plot 1 -F (x) in logarithmic scale in order to analyze the behavior of the tail of the distribution (big errors with small probabilities of occurrence). Small values for 1 -F (x) mean better results.

Results. Figures 5(a

Fig. 1 .

 1 Fig.1. Example of a deceptive sampling of a function (dashdot line). Evaluation points (black dots) are chosen such that the value of the function is around zero at these points. After having estimated the parameters of the covariance function by maximum likelihood, the prediction is very flat (solid line) and confidence intervals derived from the standard deviation of the error of prediction (gray area) are severely underestimated.

1 -Fig. 2 .

 12 Fig. 2. A comparison of a) EGO and b) FBA at iteration 1. Top: objective function (dashdot line), prediction (solid line), 95% confidence intervals derived from the standard deviation (gray area), sampling points (dots) and position of the next evaluation (vertical dashed line). Bottom: EI criterion.

  Bayesian approach for the parameters

Fig. 3 .Fig. 4 .

 34 Fig. 3. Iteration 3 (see Figure 2 for details)

  Distribution of errors at iteration 34

Fig. 6 .

 6 Fig.6. Average results and distribution of errors for testbed T2. See Figure5for details.