Pablo Josue Rojas

Carlos Jaime

Barrios Hernan- Dez
email: cbarrios@uis.edu.co

Luiz Angelo Steffenel
email: angelo.steffenel@univ-reims.fr

Low Energy Consumption on Post-Moore Platforms for HPC Research

Keywords: Edge Computing, Embedded Systems, Internet of Things, Manycore and Heterogeneous Computing, Low Cost Computing

The increase in computational capacities has helped in the exploration, production and research process, this has allowed the use of applications that were infeasible years ago. This increase brings us a new Era (known as Post-Moore Era) and a wide range of promising devices, devices such as Single Board Computers (SBC) or Personal Computers (PC) that achieve performance that a decade ago was only found on a Server. This work presents high performance computing devices with low monetary cost and low energy cost that meet the needs for the development of research in Artificial Intelligent (AI) applications, in-situ data analysis and simulations that can be implemented on a large scale, these devices are compared in different tests, presenting advantages such as its performance per watt consumed, smart form, among others.

Introduction

Two of the largest projects in Latin America have been the Fênix [START_REF]TOP500 The List. Fênix -SYS-1029GQ-TRT[END_REF] supercomputer and the Santos Dumont Hybrid [START_REF]TOP500 The List[END_REF] supercomputer, both of which were manufactured by Atos but Fênix was commissioned by Petróleo Brasileiro S.A. to be used in the oil and gas industry, while the Santos Dumont was built for the Laboratório Nacional de Computação Científica for the academic segment and meets in the post 476 of the TOP500 list. Fênix is among the top three on the continent and at number 195 on the TOP500 list, provide a theoretical capacity of 4,297.42 𝑇𝐹𝑙𝑜𝑝/𝑠, its Linpack result is 1,836 𝑇𝐹𝑙𝑜𝑝/𝑠 with a consumption of approximately 287 𝑘𝑊 . These capabilities have contributed to geophysical data processing using complex algorithms to generate images that are essential in oil exploration and production processes. With these computational capabilities will be possible to provide higher resolution images, reducing operational and geological risks directly impacting the profitability of projects. The Fênix project demonstrate the great benefit that is obtained when investing in techno-logical tools, but developing countries cannot always invest in these options, therefore, it is necessary to look for alternatives that can offer us decent computing capabilities that, in turn, give us a good cost/benefit ratio.

The evolutionary way that hardware has traveled until today has been ups and downs, as well as long, but exponentially beneficial. Moore's law [START_REF] Waldrop | The chips are down for Moore's law[END_REF] was the guide for planning, development and investigation for decades. Numerous advances have arisen under your guide. This trigger an improvement in the performance thousands of millions of times. The problem is that nothing is forever and leads us to find ourselves at the gates of the end of one of the main boosters of this advance. Thus, in 2016, the roadmap for semiconductor technology no longer focused on "Moore's law", it ends is expected for the period 2025-2030 [START_REF] Matsuoka | From FLOPS to BYTES: Disruptive change in High-Performance Computing towards the Post-Moore Era[END_REF]. This new path describes how application needs drive the advancement of technology. This leads to an "explosion" [START_REF] Matsuoka | Cambrian explosion of computing and big data in the Post-Moore era[END_REF] of notorious computing capabilities today, algorithms that were not viable (Artificial Intelligence, Autonomous and deep learning), are at the top of research and are the ones that generate the technological evolution of countries. At the same time, there is an abundance of hardware with high computing capacity, low power consumption and an affordable price.

This paper focuses on the energy efficiency of low cost devices, there are different techniques and tools such as commands, monitors, hardware, and benchmark suites that can help with these issues. Thus, the first topic to be exposed will be the context in which computing is currently, followed by referent devices, techniques and tools, to finally present the results obtained from the tests carried out.

Post-Moore Era

Any attempt to overcome the limitations presented by Moore's law can be considered as Post-Moore [START_REF] Matsuoka | From FLOPS to BYTES: Disruptive change in High-Performance Computing towards the Post-Moore Era[END_REF]. Moore's law says that around every two years the number of transistors in a microprocessor doubles, in addition to this, the improvement was characterized by an increase in the clock frequency, a decrease in lithography [START_REF] Matsuoka | From FLOPS to BYTES: Disruptive change in High-Performance Computing towards the Post-Moore Era[END_REF], but limitations such as size the lithography or quality of silicon have caused Moore's law to begin to falter. This end has generated new research paths such as multicores or heterogeneous computing. Furthermore, these emerging architectures do not focus on a single chip but on the combination of multiple chips [START_REF] Matsuoka | From FLOPS to BYTES: Disruptive change in High-Performance Computing towards the Post-Moore Era[END_REF], this allows these architectures to adapt to the needs of each application, generating a sudden appearance and rapid diversification of the hardware. In order to achieve this integration, strategies such as Heterogeneous System Architecture (HSA), languages such as CUDA, OpenCL or OpenACC, unification of CPU, GPU, FPGA, NPU, non-volatile 3D memory, reconfigurable communication grids, inductive wireless couplings, among others in a SoC [START_REF] Barker | On the feasibility of optical circuit switching for high performance computing systems[END_REF][START_REF] Take | 3D noc with inductive-couplings for building-block SiPs[END_REF][START_REF] Kagami | Efficient 3-D bus architectures for inductive-coupling ThruChip Interfaces[END_REF][START_REF] Inadomi | Analyzing and mitigating the impact of manufacturing variability in power-constrained supercomputing[END_REF], must be taken into account.

As discussed above, the investment in computing resources is helpful for research and development of countries, but developing countries do not have as one of its priori-ties, creating shortages in resources. To reduce research costs can make or buy low cost devices such as the Single Board Computer (SBC) or computer equipment heterogeneous desktop (PC) that can meet the needs of the applications used in research. Added to this, the applications must be developed or transcribed with the objective of being implemented in these architectures as a test base, these applications must maintain a portability and scalability that allows them fluidity and flexibility of deployment on architectures with greater capabilities. To verify the performance of different architectures, two reference devices are chosen that classify in the characteristics proposed by the Post-Moore Era, these devices will be treated in the next section.

Reference Devices

In order to compare performance, various devices are chosen to perform the experiments and compare the results. After several searches, there are several candidates, among which stand out: Raspberry Pi, Orange Pi, Asus Tinker Board, Odroid and Jetson. Of all these proposals, the Jetson family stands out above all. The Jetsons are SBCs developed by Nvidia, in this group there are cards such as TK1, TX2 or Xavier, even so, their costs are somewhat high or their production was suspended, but not everything is lost, therefore, from this family we chose to younger brother Jetson Nano.

The Nano features provide us with an ARM CPU, a Nvidia GPU. The other architecture chosen is somewhat more traditional, the features it provides us with are a Ryzen CPU, GPU Nvidia. The components that the experiments are going to focus on are the CPU and GPU, so it is good to give them a little review.

3.1

The CPUs ARM Cortex-A57 is primarily composed of a finder, decoder, and instruction dispatcher, integer executor, load / storage unit (L1), L2 memory system, floating point unit, advanced SIMD, generic CPU interrupt control interface, generic timer, debug and trace.

AMD Ryzen 5 3600 is manufactured in a Zen 2 microarchitecture, this design is given around small 8-core chiplets separated into 2 groups of 4, this allows the cores to be organized in a "central complex" or CCX, which contains the 4 cores and an L3 cache set. Regardless of the number of chiplets, it paired with a central I/O via Infinity Fabric, this I/O acts as a central hub for all off-chip communications, as it houses all PCIe lanes, memory channels, and Infinity Fabric links to other chiplets or CPUs. This separation greatly benefits its scalability and manufacturing capabilities, plus it makes it easier to build processors with lots of cores.

The GPUs

The selected GPUs are from successive architectures, Maxwell and Pascal.

Maxwell focused primarily on energy efficiency, his SM (Streaming Multiprocessor) was restructured, partitioned and renamed SMM. The structure of the warp scheduler, along with the FP64 CUDA core and texture unit was inherited from Kepler, but the design of most execution units was partitioned so that each warp scheduler in an SMM handles a 32-core FP32 CUDA package. This enables better resource management than Kepler, saving more energy when the workload is not optimal for sharing resources. Maxwell would be succeeded by the Pascal microarchitecture.

Pascal architecture improvements are based on five technological advancements, 1) a 16nm fabrication process that increases performance and improves energy efficiency, 2) increased double precision performance for HPC workloads (in Deep Learning offers more than 12 times of neural network training and a 7-fold increase in Deep Learning inference performance compared to previous generation GPU architectures), 3) it is the first architecture to integrate revolutionary NVIDIA NVLink™ bidirectional interconnect High Speed (this technology is designed to scale applications across multiple GPUs, delivering faster 5X acceleration of interconnect band), 4) using an innovative approach to memory design, CoWoS® (Chip-on-Wafer -on-Substrate) with HBM2 gives you a 3X boost in memory bandwidth performance over NVIDIA Maxwell ™ architecture, 5) the new in 16-bit medium-precision floatingpoint instructions and the new 8-bit integer instructions allow AI algorithms to provide real-time responsiveness for Deep Learning inference. With these descriptions, it gets an idea of the specifications that these devices offer.

To measure performance, techniques and tools must be able to be implemented in the reference devices, this topic will be covered in the next chapter.

Techniques and Tools

As it was shown in the introduction, the computational capacities are immense, which requires large investments, for this fact, it cannot be directly compared to the devices chosen with these titans, based on the Post-Moore approach, more affordable partici-pants can be chosen. In order to simplify the choice, these devices should have at least one CPU and one GPU, it should also be possible to measure their power consumption. In this way, its capabilities can be verified by means of benchmarks.

There are many benchmark tools, from the reliable Linpack (HPL) [START_REF]HPL-A Portable Implementation of the High-Performance Linpack Benchmark for Distributed-Memory Computers[END_REF] to Phoronix Test Suite (PTS) [START_REF]Phoronix Test Suite[END_REF], in terms of energy consumption, hardware or other measurement variants can be used to obtain this data in a controlled way.

Benchmarks

Stress-ng will test a computer system in various selectable ways. Execute a wide range of CPU specific stress tests that exercise floating point, integer, bit manipulation and control flow. It was designed to show thermal problems, errors in the operating system, this can result in device failure, because of this, care must be taken in the execution of the tests.

HPL (High-Performance Linpack) solves a random dense linear system in double precision (64 bits) arithmetic on distributed-memory computers. HPL provides a testing and timing program to quantify the accuracy of the obtained solution as well as the time it took to compute it. The generic implementations of MPI, the BLAS and VSIPL are available for a large variety of systems.

PTS (Phoronix Test Suite

) is a testing and evaluation platform. The software is designed to effectively carry out qualitative and quantitative benchmarks in a clean, reproducible and user-friendly way, it will take care of the entire testing process, from dependency management to test download/installation, execution and aggregation of results. PTS has access to more than 100 test suites through [START_REF]Open benchmarking[END_REF]. If there is a test that is not currently covered in PTS, they can be added through its extensible architecture.

Measure Power Consumption

As for measuring the power consumption, the aim is to generate the least possible load on the chosen devices, therefore, a measurement tool external to the device is chosen, which will measure the consumption second by second. For this task we choose a smart outlet of the VTA brand.

VTA Smart Outlet

Wi-Fi

GHz

Operating Voltage

VAC 60Hz Maximum Current 10 A Maximum Power 1250 W

The smart outlet counts a mobile APP, which carries a monthly history consumption, but video of second-by-second consumption can be taken when testing is performed. The devices are thought to have the lowest possible cost and decent capacities, therefore, SBCs are the best option that can be found, followed by Personal Computers (PC), although their cost is not comparable to SBCs, they have higher power.

An SBC is a computer (CPU, RAM, GPU, etc.) on a single circuit or board. Its applications range from industrial environments to home IoT systems. Due to high component integration and small footprint, these devices feature a higher reliability, better power handling, less weight and an SBC can mass-produce to reduce its costs.

On the other hand, the PC has been heterogeneous for quite some time. In addition, over time, the cost of calculations per second has been decreasing, improving access to new hardware with greater power, allowing the development and implementation of countless applications. To measure the performance of each reference device, several tests must be carried out, the process of carrying out these tests is presented in the next section.

Experiments

The tests perform two branches of experiments to measure the behavior of the chosen devices, the first branch focuses on the use of the CPU, the second on the use of the GPU. For CPU testing, Stress-ng and HPL were nominated. Stress-ng offers us a variety of tests such as operations with floating numbers, integers, random numbers, matrices, among others. The following configuration is used for this test:

stress-ng --cpu N --cpu-load P --cpu-method method --metrics-brief --timeout T

Where, --cpu N starts N jobs by stressing the CPU.

--cpu-load P loads the CPU with a percentage P of load for stress jobs on the CPU. Accuracy depends on overall processor load and planner responsiveness, so the actual load may be different from the desired load. Also, the number of bogo operations may not scale linearly with load, as some systems employ CPU frequency scaling, and therefore heavier loads result in higher CPU frequency and higher bogo operations.

--metrics-brief enables metrics and only shows non-zero metrics.

--timeout T stops the stress test after T seconds. You can also specify units of time in seconds, minutes, hours, days, or years with the suffix s, m, h, d, or y.

--cpu-method method specifies a CPU stress method. By default, all stress methods are exercised sequentially, however, only one method can be specified to use if necessary. Some methods were selected to measure the behavior of the CPU:

• cfloat are 1000 iterations of a combination of complex floating-point operations.

• correlate make a correlation of random doubles: 16384 × 1024.

• union performs integer arithmetic on a combination of bit fields in a C union. This shows how well (the compiler and the CPU) can perform loading and storing of integer bit fields. • hyperbolic calculates sinh(θ) × cosh(θ) + sinh(2θ) + cosh(3θ) for hyperbolic sine and cosine functions on float, double and long double, where θ = 0 a 2π in 1500 steps.

• prime finds all prime numbers in the range 1 to 1,000,000 using a slightly optimized brute force search. • matrixprod is a matrix product of two 128 × 128 double floating matrices.

Testing in hardware shows that this provides a good combination of memory, cache, and floating-point operations and is probably the best method of measuring CPU performance.

Bogo ops are the number of iterations of the stress test during the race. This is the metric for how much overall "work" has been accomplished in stress operations.

In addition to the Stress-ng tests, HPL tests are performed. HPL is software that solves a dense, random linear system with double-precision (64-bit) arithmetic on computers with distributed memory. For the HPL test, it is configured as presented in the Table 3: In Test 1, the HPL.dat was configured to use ± 50% of the RAM capacity, Test 2 only uses 75% of the total capacity, it was configured in this way to take advantage the capabilities that each device offered.

In the next branch we have the GPU usage tests, in the GPUs tests are carried out with different tests that use OpenGL, CUDA, among others. The tests consist of: building a terrain in a random way, simulating a colloid in a liquid medium and the CUDA nbody test, using the PTS as a benchmark tool for this test. The tests are performed at 1080p for the Simulation-Visualization cases and with the power limiters disabled for all the tests. In order to better explain the tests, the Diagram of General Test Workflow is presented (Figure 1), this diagram presents six steps and is described below:

Set up the different test requirements

As explained in this same section, each test is different and you can modify values such as the duration time or the percentage of the test load, in addition, it can modify the OS configuration files to increase the Hertz or the models of consumption of the devices, for this reason it is necessary to configure the test.

Configure the energy consumption monitor

The consumption monitor is always active and presenting data, therefore, after configuring the test we must find a way to capture the data at the time of the test, if possible we should look for monitors that keep a record of their measurements, if it's not possible, we can capture on video or by hand the behavior of the monitor during the test.

Start energy monitoring and launch test

Once the test is configured and it is ready to capture the data from the monitor, we proceed to run the test, we must be very aware of the beginning and end of the test since at this time is when the probability of failure is highest.

Store and label test results

This step is as important as the first step, this is due to the labeling, almost always when it is done wrong, it results in confusion when comparing the results of each device and the test should be repeated, if possible, the label should bear the name of the device, the test carried out, % of workload, its duration or the consumption model.

Repeat the test to average the results or start a new test

This step is a bifurcation, in certain cases it is necessary to have greater certainty in the results, it is good to repeat the test several times and vary its configuration, this allows us to see patterns of behavior in the devices, the test can also be changed, these Modifications can show problems like the thermal bottle. collars, memory saturation, poor performance when loading or storing data types, etc.

Group the results and generate the graphs of the tests

Based on the labels, the data is processed to generate the graphs. For this document, the number of operations (Ops) per second performed was taken together with the number of watts to generate the Ops/W.

With this description, the following section shows the results obtained in each of the tests carried out for the different configurations.

Results

In the previous section, the tests carried out for the analysis of the two reference devices were explained, it should be clarified that the ARM Cortex-A57 is presented with the A57 label, the AMD Ryzen 3600 uses the 3600 label. As for GPUs, the tag used for the GPU of the Jetson Nano is Nano and the GP107 tag is used for the GTX 1050ti. Each test is performed five times per core (5 x 1 core, 5 x 2 core, etc.), the results presented are the average of the values obtained in each test. The first tests to be carried out are those carried out on the CPU. The first test performed is Cfloat, the unit of measurement will be Operations per watt (Ops / W), as indicated above, Cfloat is 1000 iterations of a combination of complex floating-point operations. The completion of these 1000 iterations is counted as an Operation (Ops), the following graph (Figure 2) shows the results of this test: As we can see in Figure 2, the 3600 has superior performance and better scalability in this test than the A57, it should be noted that the A57 suffers from a bottleneck when using all its processors for the task, which makes scalability growth flattened out. The second test run is Correlation, an Ops consists of performing a random double correlation (16384 × 1024). We see the results in Figure 3. In this test, the A57 stands out notably in its performance, but it is still affected by the bottleneck, even so, it almost triples the performance of the 3600. The third test is union; the results are shown in Figure 4. The A57 shows an excellent handling (Compiler-CPU) of the loading and storage of bit fields, which is the strength of this test, almost quadrupling the performance of the 3600. The fourth test carried out is Hyperbolic, it calculates hyperbolic sine and cosine functions with 1500 steps in each Ops, Figure 5 shows these results. The 3600 offers excellent performance for this test, showing great performance for hyperbolic operations with float, double and long double, almost five times more than the A57. The fifth test is a calculation of the primes between 1 and 1,000,000, this calculation is carried out by means of brute force, once it is finished, it compares the results and if the results coincide, it is marked as an Ops, the results are seen in Figure 6. In this test, the A57 gives us superior performance per watt used, making it a good candidate for brute force tasks. The penultimate test performed is a multiplication of two 128x128 matrices, each element in the matrix is in a double precision floating point format, once the task is completed it is marked as an Ops, the results can be seen in Figure 7. The Results shown by the 3600 overwhelm the A57, with this preliminary matrix multiplication test we move on to the next and last test, the last test is HPL. HPL is one of the best-known benchmarks to test the performance of a CPU, we see the results in Figure 8. The test was performed with two different loads and was done conservatively so as not to have the losses presented in the previous tests, as can be seen, the CPU 3600 achieves a performance of almost one Gflop per Watt consumed, after completing the task the 3600 has consumed almost 8 Watts while the A57 has consumed about 7 Watts. The CPU 3600 provides us with a greater computational force than the A57 but it should be noted that the AMD Ryzen 3600 works at a frequency of 3.6 Ghz, while the ARM Cortex-A57 works at a frequency of 1.4 Ghz among other differences.

The previous tests were carried out on the CPU, the next round of tests was carried out on the GPUs of the chosen devices, the GPUs are the GPU of the Jetson Nano (Nano) and the GTX 1050ti (GP107), Figure 9, 10 and 11 present the results. Terrain (Figure 9) is a simulation of a field that is generated randomly, this generation is measured by frames per second (FPS), during this test the ratio of FPS per Watt consumed during the task is shown, the Jetson Nano is far superior to that of the GTX 1050Ti (GP107). The Colloid test has the same configuration as Terrain and presents its result in the same unit of measurement, Figure 10. shows the results. Colloid like Terrain shows us an excellent FPS/W ratio, almost double the FPS per Watt consumed, in addition these two simulations were performed at 1080p. As the last test, a deployment of the Phoronix Test Suite tool is performed, this tool provides us with several benchmarks, Mini-Nbody is selected as the test benchmark, the result is shown in Figure 11. The Nbody per Watt consumed ratio is notably better in the Jetson Nano, it should be noted that each of the tests was performed in the maximum consumption mode for the Jetson Nano, as for the GP107, the configuration given by the manufacturer Asus in its ROG Strix model is maintained. Once these tests are performed according to Figure 1. Several conclusions are reached, in the next segment these and other conclusions that this work has generated are presented.

Conclusions

As discussed at the beginning, the objective of this work is to propose low-cost computational options that take advantage of the characteristics of the Post-Moore Era devices to increase the benefit / cost ratio. With the abundance of devices this era has brought, two different devices are selected to perform the same tests. SBCs are devices that stand out for their low cost and PCs for their wide use in different fields. Both offer computational capabilities that less than a decade ago were only found in servers or supercomputers, the investment in these devices is in most cases less than 5 ~ 10% of what it would be in equipment for HPC.

The results presented in the previous section show that SBCs like the Jetson Nano are a great option to be used as the computational force of an investigation. These devices provide researchers with computational strength on par with more traditional options. Due to its shape, we can take advantage of its energy efficiency to be implemented in different tasks and environments.

Both SBCs and PCs are excellent options from an economic point of view to encourage investment in research and development. The current PC offers us considerable computational strength at affordable prices, while SBCs take advantage of their low energy consumption, both options can be implemented in multiple tasks, achieving great performance.

In addition to this, the specifications of these devices make them essential for the development of applications designed to be scalable, portable, simple and efficient. Since, when developing an application for these devices, tests can be carried out with different amounts of data or their deployment on different platforms, improving the fluidity of the applications when implemented on servers or supercomputers.

Further Work

The most intense tests were performed on the CPU. For future work, a series of easyto-implement, high computational effort tests will be developed on the GPU. In the first instance, the tests are written in CUDA, then be transcribed into other languages. Once this stage is complete, an implementation methodology will be proposed to measure the impact on performance when testing is performed non-natively (using methods such as containers).

Figure 1 .

 1 Figure 1. Diagram of General Test Workflow.

Figure 2 .

 2 Figure 2. Cfloat test on CPU.

Figure 3 .

 3 Figure 3. CPU Correlation Test.

Figure 4 .

 4 Figure 4. CPU Union test.

Figure 5 .

 5 Figure 5. CPU Hyperbolic Test.

Figure 6 .

 6 Figure 6. CPU Prime Test.

Figure 7 .

 7 Figure 7. CPU Matrix Multiplication Test.

Figure 8 .

 8 Figure 8. HPL for CPU.

Figure 9 .

 9 Figure 9. Colloid Simulation Tests.

Figure 10 .

 10 Figure 10. Terrain Simulation Test.

Figure 11 .

 11 Figure 11. N-body test.

Table 1 .

 1 Specifications of the Reference Devices.

		Device 1	Device 2
	CPU	ARM Cortex-A57 Quad-Core 64-bit @ 1.43 GHz	AMD Ryzen 5 3600 Hexa-Core 64-bit @ 3.6GHz
	RAM	4GB LPDDR4 @ 1600 MHz	16GB DDR4 @ 3200 MHz
	GPU	Nvidia Maxwell 128-Core @ 921 MHz	Nvidia Pascal 768-Core @ 1350~1800 Mhz (GTX 1050Ti)

Table 2 .

 2 Specifications of the Measure Power Consumption.

Table 3 .

 3 Configuring the HPL parameters.

				Parameter
		Device	N	NB P Q
	Test 1	Jetson Nano 11584 64 1 4
		PC	23168 64 1 4
	Test 2	Jetson Nano 17376 96 1 4
		PC	34752 96 1 4


Camille Sonneville
email: camille.sonneville@insa-lyon.fr

Dominique Planson
email: dominique.planson@insa-lyon.fr

Viet Luong

Phung

Pascal Bevilacqua

Besar Asllani

Interest of using a micro-meter spatial resolution to study SiC semiconductor devices by Optical Beam Induced Current (OBIC)

Keywords: OBIC characterization, Silicon Carbide High Voltage Diodes, Wide Band Gap semiconductor, TCAD

In this paper we present a new test bench called micro-OBIC used to characterized wide band gap semi-conductor. Micro-OBIC allows to measure an Optical Beam Induced Current (OBIC) signal with a microscopic spatial resolution. We used micro-OBIC to characterize peripheral protection such as MESA, JTE or JTE rings in high voltage SiC device.

Introduction

High voltage devices, up to 10 kV, are mandatory for high power converters as they will be used in HVDC energy transmission network in the future [START_REF] Kimoto | Fundamentals of Silicon Carbide Technology[END_REF]. For the sake of power losses, bipolar devices are interesting due to the conductivity modulation of the thick and low doped layer. Efficient peripheral protections are requested to avoid electric field crowding close to the edges of the device. Thus, non-destructive characterization techniques are useful to study these features.

UV Laser is used to generate electron-hole pairs into Wide Band-Gap (WBG) semiconductors (SiC, GaN or Ga2O3). In the space charge region (SCR), the electric field drives the collected carriers and a current, named Optical Beam induced Current (OBIC), can be measured. This induced current is then directly related to the electrical field in the device. OBIC is a non-destructive technique, which has been previously successfully used to characterize High Voltage (HV) SiC devices [START_REF] Stengl | High-voltage planar junctions investigated by the OBIC method[END_REF][START_REF] Hamad | OBIC technique applied to wide bandgap semiconductors from 100 K up to 450 K[END_REF][START_REF] Raynaud | Optical beam induced current measurements: principles and applications to SiC device characterization[END_REF][START_REF] Planson | Experimental and simulation results of optical beam induced current technique applied to wide bandgap semiconductors[END_REF]. In order to fully benefit of the advantages provided by WBG semiconductors materials and to avoid premature breakdown of the high voltage devices, it is mandatory to have efficient peripheral protections such as a combination of MESA with JTE or MESA with JTE and JTE rings as described in [START_REF] Nguyen | Edge termination design improvements for 10 kV 4H-SiC bipolar diodes[END_REF]. The OBIC characterization can help Technology Computer-Aided Design (TCAD) and the device process to optimize the efficiency of the peripheral protections by analyzing the electric field distribution in the structure and especially at the junction periphery. In this paper, we will present an in-house testbench called micro-OBIC which will allow us to characterize HV PiN diodes with a micro-meter spatial resolution. Theoretically, a small spot size should allow the accurate determination of the OBIC signal at specific location such as MESA, JTE and JTE rings.

Experimental and set-up

Description of the micro-OBIC test bench

We developed in Ampere laboratory a custom OBIC test-bench called micro-OBIC. A laser emitting UV is used to generate electron-hole pairs into WBG semiconductors. The laser is a Nd:YLF pulsed laser from spectra physics emitting a gaussian beam at 349 nm with a mean power around 4.10 -3 Watt. The original beam exiting the laser has a waist diameter around 170 𝜇m. In order to reduce the laser beam spot size, we developed a testbench represented in figure 1. The laser beam power is first controlled by a set of neutral density filter, before passing through a X10 beam expander. Finally, the laser spot is focused by a long-distance X15 Thorlabs microscope objective. A beam expander allows to illuminate a wider part of the microscope objective aperture, in order to reduce diffraction and then optimize the final laser spot size. To estimate the final diameter of the gaussian beam, we use the following equation [START_REF] Hauer | Spot Size Engineering in Microscope-Based Laser Spectroscopy[END_REF]:

𝜔 # = 𝜆𝑓 # 10𝜔𝜋
Where 𝜔 # is the final waist diameter, 𝜆 is the laser wavelength (349 nm), f' is the objective equivalent focal length (13 mm) and 𝜔 is the original laser beam waist diameter (170 µm). Thanks to Eq. 1, we estimated 𝜔 # and thus the spot diameter around 1 µm. The (x,y) spatial resolution of this new OBIC test-bench is estimated around 1 µm. The z focus was manually realized by optimizing the OBIC signal in the p + region (top anode level). Moreover, the focus depth is greater than the mesa height so there is no need to readjust the focus on the JTE level.

The HV device under test (DUT) is fixed on a X, Y, Z motorized stage. The motorized stage, controlled by a homemade Labview program, allows us to move the DUT under the laser beam focus point. Induced current is then measured with a Keithley 237 High-Voltage Source-Measure Unit (SMU). The DUT can be reverse biased by applying voltage thanks to the SMU (source measure unit). The testbench allows us to realize a spatial mapping of OBIC signal, either on a (X,Y) plane or only on X (or Y) lines. For X or Y lines, the step between two displacements of the laser beam is 0.25 µm. For (X, Y) plane, the step displacement of the laser beam in both directions is 3 µm. Fig. 1 represents schematically the micro-OBIC test-bench.

Diode description

Experimental micro-OBIC scans were performed on high voltage PiN diode (10 kV-class). 10KVclass PiN diode means that these diodes were designed by TCAD in collaboration with CALY Technologies to sustain up to 10kV when reverse biased. However, due to technological issues during ion implantation of the JTE, the best sustainable breakdown voltage is around 9kV. In the frame of this paper, we used "test" diodes, i.e without polyimide passivation. These "test" diodes exhibit a lower breakdown voltage.

The high voltage PiN diodes were fabricated on a 4H-SiC wafer using a 110 µm thick epilayer with a doping concentration of 7.10 14 cm -3 . These diodes are protected by a combination of a MESA and a JTE of 400 µm length (p doping concentration of 7.10 16 cm -3 , thickness 0.7 𝜇m) and have whether (diode 2) or not 8 JTE rings (diode 1), as shown in Fig. 2a) and b).

The diodes fabrication was led by CALY Technologies on a 4 inches wafer and realized in the cleanroom of the IBS company. Square shape diodes with a bend radius of 600 µm have been implemented with different edge terminations. An elementary cell view of diode mask is shown in There is still a thin SiO2 layers as primary passivation, which is transparent to UV-light. The breakdown voltage has been measured in a vacuum chamber at ISL [START_REF] Vergne | Brosselard Precise automated semiconductor characterization under ambient control (PASHAC) -An advanced test bench for highvoltage semiconductor[END_REF] for both diodes (diode 1: Vbr = 5743 V, diode 2: Vbr = 5578 V).

Simulation set-up

The diode structure design was concurrently simulated thanks to Synopsys Sentaurus TCAD package [START_REF]Sentaurus TCAD[END_REF]. The simulator supports all the WBG materials currently used for commercial devices. It does also support custom physical parameters that may be used to feed the available models (Arora, SRH). Based on previous study on the absorption coefficients of UV light in 4H-SiC [START_REF] Hasegawa | Simulation-Based Study About the Lifetime and Incident Light Properties Dependence of the Optically Triggered 4H-SiC Thyristors Operation[END_REF], the OBIC and micro-OBIC setup was simulated. The schematic 2D cross-section of the simulated device is shown in Fig. 4. Except the 8 guard rings of the diode 1, both structures share the same features. The n epilayer is doped at 7.10 14 cm -3 . Its thickness which should be above 100 µm has been cut down to 10 µm since it will be sufficient for reverse voltage up to 200 V. A JTE layer is formed on the top of the n epilayer. It is fully covered by a SiO2 layer. The anode sits on the top of a heavily doped p+ region. The distance between the JTE inner edge and the anode contact one is named L2. In order to limit the computing time, only the interesting part of the structure is modelled. The guard rings are properly spaced and their depth is equal to the JTE thickness. The laser beam (349 nm), located above the structure is defined by a gaussian profile. Its width is adjustable at will. In any case, the beam moves at a constant step of 0.5 µm. The origin axis is set at the left edge of the structure. The optical generation inside the structure is computed thanks to raytracing. The current obtained across the reverse-biased structure is the simulated OBIC current.

Results and discussion

Micro-OBIC mapping versus optical image

2D (X,Y) mapping of the micro-OBIC signal were realized on diode 2 at 0V bias. This same diode was observed with an optical microscope. Fig. 5a) and b). compares the optical image and the micro-OBIC signal of upper left rounded corner the diode. The studied area is represented by a square in Fig. 3.

There is no OBIC signal when the laser beam scans the metallic anode, indeed the anode reflects all the UV light. When the laser beam hits the p + /MESA region, the micro-OBIC signal reaches a peak (around 120 nA), then it decreases to a lower value to stay almost constant (around 30 nA) in the JTE region. Once it approaches the outer edge of the JTE, the micro-OBIC signal decreases slowly to zero. The optical image shows clearly a first "black ring", around 24 µm thick, along the anode metallization. Micro-Raman analysis was performed on this first "black ring" and shows that this black layer is not SiC. This layer can be either a mask or a secondary passivation (polyimide) residues. Further chemical and structural investigations such as SEM/EDX (scanning electron microscopy with energy dispersive X ray spectroscopy) or TOF-SIMS (time of flight second ion mass spectroscopy) are needed to identify this "black layer". It seems that this layer absorbs the visible light and may absorb the UV-light too. If it is the case, an important part of the diode p + will be optically hidden and then OBIC signal would not be observable in this hidden area. A second thinner "black ring" can be observed at 50 µm from the anode and is probably due to the mesa edge. Fig. 6a) and b) represent a (X,Y, IOBIC(nA)) representation in false colors of the OBIC signal with a 0V bias (Fig. 6.a)) and at 100V reverse bias (Fig. 6b)). The 100 V reverse bias induced an increase of the OBIC maximum from 120 nA to 200 nA and an increase of the OBIC signal in the JTE region from 30 nA to 130 nA. Moreover, the microscopic spatial resolution of the micro-OBIC testbench highlights very small features in the p + /MESA region. At 100V reverse bias, some defaults in the JTE region are visible. These electrical defaults could be inhomogeneity in the JTE doping concentration. Estimation of the local SiC doping concentration by complementary technics such as micro-Raman spectroscopy could be interesting to realize.

Comparison between experimental OBIC and experimental micro-OBIC

X-line scans were performed with a former OBIC testbench at various reverse voltage on diodes 1 and 2. This former OBIC testbench has been already described in [START_REF] Planson | Phung Near breakdown voltage optical beam induced current (OBIC) on 4H-SiC bipolar diode[END_REF] and related measurements published in [START_REF] Planson | Experimental and simulation results of optical beam induced current technique applied to wide bandgap semiconductors[END_REF]. X-line scans were performed on diodes 1 and 2 at 0V with the new micro-OBIC testbench. Fig. 7a) shows the comparison of the normalized OBIC (in black) and the normalized micro-OBIC (in red) signal for the diode 1. Fig. 7b) shows the comparison between the OBIC (in black) and the micro-OBIC (in red) signal for the same diode 2. The approximative position of the anode and peripheral protections (mesa, JTE, rings) are represented in Fig. 7. The OBIC signals were normalized to have 1 a.u (arbitrary unit) in the JTE region. Fig. 7 a) and b) show an important increase of the maximum peak intensity with this new micro-OBIC set-up compared to the OBIC set-up. This observation can be explained by the increase of the spatial resolution.

In order to estimate experimentally the spatial resolution of the micro-OBIC test bench, we used the experimental procedure described in [START_REF] Stengl | High-voltage planar junctions investigated by the OBIC method[END_REF]. For that, we calculated the OBIC and micro-OBIC derivatives in the anode/p + boundary (i.e when an OBIC signal starts appearing), as shown in Fig. 8. As explained in ref [START_REF] Stengl | High-voltage planar junctions investigated by the OBIC method[END_REF], the fullwidth at half maximum (FWHM) of the derivative can well approximate the spatial resolution of the OBIC device. With this method, we estimated the spatial resolution of the former OBIC testbench around 60 µm. The spatial resolution of the new micro-OBIC testbench is then 1µm as estimated theoretically.

Comparison between experimental and simulated micro-OBIC study of the p + /MESA region

Interestingly the use of well-focused laser beam shows smaller features in the p + /MESA region, which were invisible with the previous test bench because of its low spatial resolution. Fig. 9a) is a zoomin of the Fig. 7a) in this p + /MESA region. In Fig. 7a) when the micro-OBIC signal appears (relative displacement around 0.1mm), one can observe a sharp current peak with at least two main contributions. To better understand these contributions, we performed some micro-OBIC simulations. Fig. 9b) shows the OBIC current profile obtained from TCAD simulations for a beam width of 1 µm. In order to be the closest possible to the fabricated diode structure, L2 was set to 7 µm, which is exactly the part of p + optically visible in Fig. 5a). The simulated beam exposes several features that would merit to be described. First, the left part, from 0 to 22 µm on the x axis, shows no current as expected because the whole beam is reflected. The 7 µm wide constant signal (L2, from 22 µm to 29 µm) comes from the p + region exposed to the beam where the SCR stays constant. The profile shows 2 peaks located respectively at 30 and 31 µm. These peaks are precisely the edges of the MESA. An electric field crowding is responsible for such a profile. As the laser moves towards the outer area of the device, the simulated OBIC current remains constant as it directly mirrors the constant SCR that spreads along the JTE [START_REF] Planson | Experimental and simulation results of optical beam induced current technique applied to wide bandgap semiconductors[END_REF]. Fig. 9a) and Fig. 9b) correspond respectively to the experimental and the simulated micro-OBIC signal of diode 1 in the p+/MESA region and have exactly the same scale. Thanks to the simulation results, we can reasonably assume that the MESA edges are responsible for some of the contributions to the OBIC signal experimentally measured.

It should be noted that the simulated OBIC signal does not match completely yet with the experimental one. Firstly, we do not experimentally observe the constant OBIC signal in the p+ area. It may be related to the presence of the "black" ring observed in Fig. 5a). Secondly the amplitudes of the OBIC peaks don't match either as the device area is not strictly the same and some other technological uncertainties on samples that must be confirmed.

Fig. 1

 1 Fig. 1 Schematic representation of the micro-OBIC testbench

Fig. 3

 3 Fig. 3 Elementary cell view of diode mask layout (1cm*1cm)

Fig. 2a )

 2a Fig. 2a) (left) Structure of the 8 JTE rings HV PiN diodes (diode 1), Fig. 2b) (right) Structure of the JTE without ring HV PiN diodes (diode 2). Schema are not to scale.

Fig. 4

 4 Fig. 4 2D Diode schematic cross-section (a) Mesa JTE diode + 8 rings (TCAD diode 1) (b) Mesa JTE diode (TCAD diode 2).

Fig. 5a )

 5a Fig. 5a) (left) Optical image of the diode (top view), microscope objective x50. Fig. 5b) (right) micro-OBIC 2D-scan of the SiC HV PiN diode at 0V bias.

Fig. 6a )

 6a Fig. 6a) (left) Micro-OBIC 2D-scan of the SiC HV PiN diode biased at 0V. Fig.6b) (right) Micro-OBIC 2Dscan of the SiC HV PiN diode reverse biased at 100V.

Fig. 7a )

 7a Fig. 7a) (left) OBIC (black dashed) and micro-OBIC (red) X-line scan performed on the 8 JTE rings diode at 0V. Fig. 7b) (right) OBIC (black dashed) and micro-OBIC (red) X-line scan performed on the no JTE ring diode at 0V.

Fig. 8a )

 8a Fig. 8a) (left) Derivative of the diode 1 OBIC signal (black), Fig. 8b) (right) derivative of the diode 1 micro-OBIC signal (red)

Acknowledgements

The authors want to acknowledge SuperGrid Institute for providing the HV devices and CALY technologies and IBS for the device realization.

Comparison between experimental and simulated micro-OBIC study of the JTE outer edge

 7a) and Fig. 7b) showed that the diode 1 and the diode 2 present some differences in their experimental micro-OBIC signal at the JTE outer edge for 0V bias. Indeed, the experimental micro-OBIC signal of diode 1 (Fig. 7a)) is null for a relative displacement around 0.65 mm (i.e around 550 µm from the anode edge). Meanwhile, the experimental micro-OBIC signal of diode 2 (Fig. 7b)) is null for a smaller relative displacement around 0.58 mm (i.e around 480 µm from the anode edge). As the OBIC signal only exists in presence of a space charge region (SCR), these experimental results mean that there is a further lateral extension of the SCR in diode 1 than in diode 2. This further lateral extension is around 80 µm, which is close to the total length occupied by the 8 guard rings.

In order to understand, if these experimental observations can be explained by the presence of the 8 guard rings, we performed some micro-OBIC simulations in the JTE outer edge region. Fig. 10 shows the simulated OBIC in this precise area for diode 1 (with rings) and diode 2 (without rings), the latter having ripples and a slight bigger extension of around 3 µm. Those slight ripples are attributed to a partial depletion of the first guard ring. However, this slight extension of 3 µm is far smaller than the one observed experimentally (around 80 µm). Our experimental measurements (see Fig. 7a) and Fig. 7b) seem to show that the 8 guard rings have been merged together during fabrication and act more like a longer JTE of 80 µm.

Summary

We designed a new custom test-bench called micro-OBIC. This test-bench has a well improved spatial resolution compared to the previous OBIC test-bench. We estimated theoretically and experimentally the spatial resolution of the micro-OBIC to be around 1 µm. A better resolution is interesting to characterize peripheral protections, such as JTE rings or MESA, whose lateral extension is of the order of magnitude of the micron. Our Micro-OBIC experiment and simulations showed that micro-OBIC can be used to characterize the MESA and JTE regions. For example, our micro-OBIC and optical study highlighted some defaults (i.e. "black" rings, JTE inhomogeneities, ...).

Even if JTE rings are not directly observable in these experiments, we observed a further lateral extension of the space charge region (SCR), interpreted as a "merge" of the 8 guard rings to create a longer JTE. It would be then interesting to complete this study by measurements at higher reverse bias.

In order to better understand the defaults, highlighted by our micro-OBIC results, it would be interesting to study these same diodes by other complementary characterization methods such as micro-Raman spectroscopy, SEM/EDX and TOF-SIMS.


Jérôme A Dupuis

Estimating the size of small populations from incomplete lists via graphical models

Keywords: Bayesian model averaging, Capture-recapture, Graphical models, Hyper-Dirichlet distribution, Incomplete lists, Population size estimation, Small populations. 1

We consider the problem of estimating the size N of a closed population from q incomplete lists. Estimation of N is based on capture-recapture type models. We use graphical models to deal with possible dependencies between lists. The current parametrization involves clique probabilities which have no simple concrete meaning and are delicate to manipulate in a Bayesian context insofar as hyper-Dirichlet distributions are used as priors. Our parametrization involves marginal and conditional capture probabilities. We develop our approach with q = 3. We show that there is a one-to-one and onto correspondence between both parametrizations and that placing hyper-Dirichlet distributions on the clique parameter boils down to place independent beta distributions on the capture parameters. When N is small, the non informative Bayesian analysis encounters difficulties. The posterior distribution of N may not exist for a particular graphical model: we give a necessary and sufficient condition of existence for each. Moreover, it is highly desirable that the priors on capture are compatible across the different models. Now, due to the small size of N , fulfilling this requirement demands a particular attention. We conclude by extending our approach to q = 4 lists.

Introduction

Estimating the size N of a closed population is an important issue in several scientific fields: such as medecine, ecology, computer science (eg Pollock, 1990). As far as human populations are concerned, estimating N is typically based on q ≥ 2 incomplete lists and the resulting data are thus capturerecapture type data: an individual which appears on a given list being, in a way, 'captured' by this list (eg Hook and Regal, 1995;[START_REF] Chao | The applications of capture-recapture models to epidemiological data[END_REF].

Owing to this analogy, the satistical analysis uses capture-recapture type models. The estimation of N can be based on two lists; but, in such a case, it is not possible to take into account in the model a possible dependence between the two lists, and then there is a risk of overestimating (or underestimating) N if such a dependence exists. In fact, at least three lists are necessary to model dependences between lists (eg [START_REF] Chao | Capture-recapture for human populations[END_REF], what underlines the fact that the value q = 3 is an important particular case on which we will focus in this paper.

Two approaches have been proposed for modelling some possible dependences between the lists: the one of [START_REF] King | On the Bayesian estimation of population size[END_REF] which uses loglinear models, and the one of [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF] which uses decomposable graphical models. The paper takes place in the latter. Graphical models are a particularly attractive tool to formulate in a rigorous way all the conditional (or marginal) independence assumptions between the different random variables. Moreover, this tool allows to visualize such assumptions what makes it very popular among the researchers; eg [START_REF] Dupuis | Bayesian estimation of movement and survival probabilities from capture-recapture data[END_REF], [START_REF] Hojsgaard | Capture-recapture methods in epidemiology: methods and limitations[END_REF]. The graph of a graphical model m is deter-mined by its (maximal) cliques, and [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF] parametrize any model by the corresponding cliques probabilities. This way of proceeding is quite natural. However, most of researchers are used to work with capture parameters which are quantities having a very concrete meaning (contrary to clique probabilities). Hence the interest to examine in which extend it is possible to reparametrize the approach of [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF] via capture probabilities. We show that it is effectively the case and that both parametrizations produce the same bayesian inference on N . Accordingly, the theoretical results established in this paper will apply without any modification to the Madigan & York's approach. As in [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF], inference on N is based on a Bayesian model averaging which includes all the possible decomposable graphical models (for fixed q). The paper focuses on populations of which the size N is small; see for example [START_REF] Wang | Bayesian population estimation for small capture-recapture data using noninformative priors[END_REF] for motivations. In human populations, estimating the number of people affected by a rare disease typically enters in this framework. When N is small, it may occur that one (or more) count associated with a particular capture-recapture history (different from the one of an individual never captured) is null or very low. In such circumstancies, statistical difficulties may occur when one wishes to perform a non informative Bayesian analysis of the data.

-A first difficulty, not mentioned in [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF], is related to the existence of the posterior distribution of N when an improper prior is placed on N (Jeffreys or uniform). Sufficient and necessary conditions of existence are thus stated for each graphical model: our results extend the result of [START_REF] Wang | Bayesian population estimation for small capture-recapture data using noninformative priors[END_REF] which concerned only the simplest model (namely the independant model). A analoguous result of existence is also stated for the Bayesian averaging model procedure.

-A second difficulty is related to the priors put on the capture parameters.

When inference involves several candidate models, it is strongly desirable that the priors are compatible across the different models (eg [START_REF] Dawid | Hyper Markov laws in the statistical analysis of decomposable models[END_REF]. Here, the prior distribution on any capture parameter of any sub-model is derived from the prior distribution put on the parameter of the saturated model (afterwards denoted by θ Sat ). This strategy is the one adopted by [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF] to derive the prior distributions on the clique probabilities. Standard non informative priors on θ Sat are the uniform distribution on the 2 q -simplex and the Jeffreys prior. Now, we note that [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF] advice to use the uniform prior. But this prior induces informative priors on the marginal capture probabilities since they follow a beta (2 q-1 , 2 q-1 ) which cannot be considered as non informative when q ≥ 3 (especially when N is small).

-In fact these two difficulties are closely linked, since the prior adopted for θ Sat plays a part in the condition of existence of the posterior distribution of N (see Section 7). In this paper, we propose a distribution on θ Sat which induces non informative priors on marginal and conditional capture probabilities, and ensures -whatever the graphical model considered -the existence of the posterior mean of N when the Jeffreys prior is adopted for N .

In conclusion, we briefly indicate how to extend our approach to the particular case q = 4, and how to take into account a possible individual heterogeneity at the capture level (see Section 6).

Data description

For any individual i of the population of interest, we denote by x i the vector (x ir ; r = 1, 2, 3) where x ir = 1 if individual i appears on list r and zero otherwise; x i is called the history of individual i. There are 8 possible histories, namely: (0 0 0), (0 0 1), (0 1 0), (0 1 1), (1 0 0), (1 0 1), (1 1 0), (1 1 1).

The set of all these histories is denoted by H, and the set H minus the history 000 is denoted by H * . We denote by n h the number of individuals whose history is h. Note that the count n 000 is not observable and that d = h∈H * n h represents the number of individuals appearing in at least one list. Data is denoted by y; thus, one has y = {n h ; h ∈ H * }.

Assumptions, models and parameters

We assume that the N random vectors X 1 , . . . , X i , . . . , X N are independent and identically distributed. Therefore, the probabilistic assumptions concerning the components of the random vector X i = (X ir ; r = 1, 2, 3) do not depend on i; and, for convenience, we will afterwards omit index i in X ir .

The assumptions on X 1 , X 2 , and X 3 are formulated via graphical models. As [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF], we consider eight models: the saturated model and seven sub-models obtained by removing one or several arrows in the graph of the saturated model.

-The saturated model is denoted by [123]. It is characterized by the fact that no independence assumption concerning X 1 , X 2 , and X 3 is made.

-A model which assumes a conditional independence assumption between two nodes of the graph is said of type I. The model which assumes that Table 1: Characteristics of each type of graphical model (q = 3 lists) [12,13]. The two other models of type I are denoted by [23,21] and [31,32] (with obvious notation).

model name assumption factorization 1 3 2 satured no p(x 1 , x 2 , x 3 ) 1 3 2 type I X 2 ⊥ X 3 |X 1 p(x 1 )p(x 2 |x 1 )p(x 3 |x 1 ) 1 3 2 type II X 1 ⊥ (X 2 , X 3 ) p(x 1 )p(x 2 , x 3 ) 1 3 2 independent ⊥ (X 1 , X 2 , X 3 ) p(x 1 )p(x 2 )p(x 3 ) X 2 ⊥ X 3 |X 1 is denoted by
-A model which assumes a marginal independence assumption between one node and the two others is said of type II. The model which assumes that X 1 ⊥ (X 2 , X 3 ) is denoted by [1,23]; The two other models of type II are denoted by [2, 13] and [3, 21] (with obvious notation).

-The independant model denoted by [1,2,3]. It assumes that the three random variables X 1 , X 2 , and X 3 are independant.

The graphs of the above models appears in Table 1. In column 1, X 1 , X 2 , X 3 are, for convenience, respectively represented by 1 , 2 , 3 . Note that all these graphs are decomposable. Let p(x 1 , x 2 , x 3 ) be denote the probability

that X 1 = x 1 , X 2 = x 2 , X 3 = x 3 where x 1 , x 2 , x 3 belongs to {0, 1}.
For each sub-model we can deduce from its graph a specific factorization of p(x 1 , x 2 , x 3 )

given in Table 1. Each factorization induces a natural parametrization of the sub-model in terms of marginal and conditional capture-recapture probabilities. We thus introduce the following notation. Marginal capture probabilities are denoted by θ r which represents the probability that an individual appears on list r ∈ {1, 2, 3}. Conditional capture probabilities are denoted as follows: θ s|r represents the probability that an individual appears on list s given that he appears on list r = s, and θ s|r represents the probability that he appears on list s given that he does not appear on list r = s. We thus define twelve conditional probabilities and three marginal probabilities.With this notation, the independent model includes three parameters, namely θ 1 , θ 2 , θ 3 . A Model of type I includes five parameters. For example, the parameters of model [12,13] are θ 1 , θ 2|1 , θ 2|1 , θ 3|1 and θ 3|1 . As far as models of type II are concerned, two parametrizations are possible (both are natural). For example, if one considers the model [1,23], one can decompose p(x 1 , x 2 , x 3 ) as p(x 1 )p(x 2 )p(x 3 |x 2 ) or as p(x 1 )p(x 3 )p(x 2 |x 3 ). In the first case, the resulting parametrization includes parameters θ 1 , θ 2 , θ 3|2 , θ 3|2 , and parameters θ 1 , θ 3 , θ 2|3 and θ 2|3 in the second case.

The above notation concern the parameters of sub-models. The saturated model is parametrized by the θ h 's where θ h denotes the probability that an individual has h as history; thus, one has: θ Sat = (θ 111 , . . . , θ 000 ). We stress that this additional notation indexes θ by a capture-recapture history, contrary to the one adopted in sub-models which indexes θ by the lists.

Priors

4.1 Priors on the size N of the population.

We will use the greek letter π to designate any prior or posterior density, continuous as well as discrete. Moreover, for a graphical model m, the set of all the capture probabilities (marginal and conditional) is denoted by θ m .

First, we assume that, for all models m, N and θ m are a priori independent, thus one has: π(N, θ m ) = π(N )π(θ m ). In the absence of any prior information on N , one usually adopts either the Jeffreys prior π(N ) = 1/N , or the uniform prior π(N ) = 1 (eg [START_REF] Basu | Bayesian capture-recapture methods methods for error detection and estimation of population size: heterogeneity and dependence[END_REF][START_REF] Dupuis | A Bayesian approach to the multistate Jolly-Seber capture-recapture model[END_REF]. Note that both are improper and the existence of the posterior distribution of N is thus not guaranteed (hence the study made in Section 6).

Prior distributions on the capture probabilities

For each sub-model m, we assume that the elementary parameters present in sub-model m are a priori independent; we mean by elementary parameter any capture probability (marginal as well as conditional) present in model m. For example, if m = [12, 13], we assume that θ 1 , θ 2|1 , θ 2|1 , θ 3|1 and θ 3|1 are a priori independant. As in the paper of [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF], we adopt for θ Sat a Dirichlet distribution with parameters (a 000 , . . . , a 111 ), afterwards denoted by D(a 000 , . . . , a 111 ); thus, one has:

π(θ Sat ) ∝ h∈H θ a h -1 h
where the a h are all stricly positive. In a non informative set-up, a standard choice is the uniform distribution. The main alternatives to the uniform prior are the Jeffreys prior and the Perks prior which respectively corresponds to a D(1/2, . . . , 1/2) and to a D(1/2 3 , . . . , 1/2 3 ).

For j, k in {0, 1}, it is convenient to introduce the following notations:

θ jk+ = 1 l=0 θ jkl and θ j++ = 1 k=0 θ jk+ .
Recall that θ jkl denotes the probability that an individual has jkl as history.

Notations θ j+l , θ +kl , θ +k+ and θ ++l are defined similarly.

As far as a Bayesian model averaging procedure is implemented for estimating N , it is strongly desirable that the priors are -as much as possiblecompatible across the different models. In particular, it is natural to require that the prior distribution put on any fixed elementary parameter is the same from model to model. A simple way to fulfill this requirement is to derive its density from the one put on θ Sat , considering that all elementary parameters express in function of the θ h 's; for example, one has:

θ 3 = j,k∈{0,1} θ jk1 , θ 2|1 = θ 110 + θ 111 θ 1++ and θ 2|1 = θ 010 + θ 011 θ 0++ .
The following Proposition allows to derive the prior distribution on any marginal and conditional capture probabilities from the one placed on θ Sat .

Proposition 1. For any j, k ∈ {0, 1}, we have:

θ j++ ∼ beta(a j++ , a -a j++ )
where a = h∈H a h and θ jk+ /θ j++ ∼ beta(a jk+ , a j++ -a jk+ ) with obvious notations for a jk+ and a j++ .

Proof. The first part of the Proposition is an immediate consequence of the agregation property of the Dirichlet distribution. See Appendix A for the second part.

Similar results of course hold for θ +k+ , θ ++l , and for all the possible ratios similar as the one appearing in Proposition 1. Fixing the hyperparameters a h thus induces a prior distribution for all the marginal and conditional capture probabilities. If we put a uniform distribution on θ Sat , the conditional capture probabilities follow a Beta (2, 2) and the marginal probabilities follow a Beta (4, 4). These beta distributions cannot be considered as non informative. As far as marginal probabilities are concerned, the Jeffreys prior suffers from the same drawback than the uniform prior (but to a lesser extent). On the contrary, if we put a D(1/4, . . . , 1/4) on θ Sat , all the marginal and conditional capture probabilities follow non informative distributions, namely a uniform distribution for the former, and a Jeffreys distribution for the latter.

The Perks prior also induces non informative priors on capture probabilities; but the Dirichlet D(1/4, . . . , 1/4) will turn out to be preferable to ensure the existence of the posteriors (see Section 7).

The links with the Madigan & York's paper

Proposition 2 below clarifies the links between the parametrisation of Madigan and York (1997) which uses the notion of clique probability, and ours which involves marginal and conditional capture probabilities.

Proposition 2. For each fixed sub-model m, there is a one-to-one and onto correspondence between the clique probabilities parametrisation and the capture probabilities parametrisation.

Proof. The proof appears in Appendix B.

Proposition 3 below clarifies the links exiting between the prior adopted by [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF] Proof. The proof appears in Appendix C.

In this Appendix we recall, for each model m, the density of the hyper-Dirichlet distribution adopted by [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF]. In Proposition The assumption of independence between the X i 's (see Section 3) implies that:

(N 001 , . . . , N 111 )|N, θ m ∼ Multinomial(N ; θ 001 , . . . , θ 111 ), from which we deduce that:

L(θ m , N ; y) = N ! (N -d)! h∈H * n h ! 1 - h∈H * θ h N -d h ∈H * θ n h h . ( 5
+ d 2 + d 3 ) -d + a 1++ + a +1+ + a ++1 > 2.
Proof. It appears in Appendix E.

-If the existence of the posterior distribution of N is of interest, replace 2 by 1 in the right member of inequalities. For the posterior variance of N , replace 2 by 3.

-As far the model [1,23] is concerned, the above result shows that the obtained condition does not depend on the factorization adopted for p(x 1 , x 2 , x 3 ) since each of the terms n 011 , a 000 and a 011 remain unchanged when one permutes the index k (related to list 2) and l (related to list 3).

-The conditions for models [23,21], [31,32], [2,31], and [3,12] are obtained by an appropriate permutation of the indices j, k, l in a jkl and n jkl .

We now briefly comment Proposition 4.

-Except for the saturated model, we observe that the left term which appears in the inequalities decomposes in three parts: the first one (namely λ)

is related to the prior put on N , the second one is related to the data, and the last one is related to the prior put on θ Sat . In small populations, it may occur that, for a given sub-model, the second term may be null and thus the choice of the prior on θ Sat becomes crucial to meet the corresponding condition of existence (it is the case for the two data sets considered in Section 7). For example, if one adopts the Perks prior for θ Sat and the Jeffreys prior for N , the posterior mean of N does not exist under the model [12,13] when n 011 is null (in this configuration, only the posterior distribution of N exists). If, always in this configuration, we replace the Perks prior by a D(1/4, . . . , 1/4) thus the posterior mean of N will exist (recall that this Dirichlet ensures that the priors on the capture proabilites are all non informative, see section 4.2).

-For the saturated model, comments are reported in Section 6.2.

Condition of existence of E[N |y].

In Proof. See Appendix F.

We note that data play no part in the condition given by Proposition 4;

the reason is that the condition λ + a -a 000 > 2, which appears in Proposition 2, is the stongest one (see Appendix F). Consequently, the result of Proposition 5 is very general and applies to any type of data (relatively to the sizes of the n h 's). That means that basing inference on a Bayesian model averaging procedure is possible only if appropriate priors are choose for N and θ Sat . For example, if one adopts the Jeffreys prior for N , the Perks's prior cannot be used; on the opposite, the averaged-model posterior mean of N will exist in all the other cases (that is with the uniform and Jeffreys priors, as well as a D(1/4, . . . , 1/4)).

7 Generalizations and extensions.

• The parametrization in terms of marginal and conditional capture probabilities can be extended to q ≥ 4 lists. In practice, the number of lists rarely exceeds 4, and we will thus limit ourself (for brievety) to the case q = 4. Table 6 provides, for each type of graphical model, the corresponding conditional and marginal independence assumptions. All these models are decomposable, except the third (from top). For each model (except the third) we provide the factorization of p(x 1 , x 2 , x 3 , x 4 ) derived from its graph. The third model has no factorization and will be typically removed from a Bayesian averaging model. As for the case q = 3, the definition of the parameters is, for each decomposable sub-model, derived from its factorization (details are omitted).

Note that the second and sixth models from top, involve conditional capture probabilities where the conditioning is on two lists.

Concerning the averaged-model posterior mean of N , it is easy to check that it exists if and only if λ + a -a 0000 > 2 where the convention concerning λ is unchanged (see Section 6). The proof proceeds as for q = 3 and is omitted for concision. Concerning the prior placed on θ Sat we advocate to use a D(1/2 q-1 , . . . , 1/2 q-1 ) with q = 4, for two reasons. First, this choice ensures the existence of E[N |y] when π(N ) = 1/N . Second, it ensures that all the marginal and conditional capture probabilities follow non informative prior distributions. Indeed, it is easy to check that: the marginal capture probalities follow a uniform distribution, the conditional probabilities capture follow a Jeffreys distribution when the conditioning is on one list, and a Beta (1/4, 1/4) when the conditioning is on two lists. The use of the uniform distribution will suffer, in a way more accute than for q = 3, from the drawback mentioned in Section 4.2 since the prior on a marginal capture probability will now follows a Beta(8, 8) (which is far from being able to be considered as non informative). A similar remark applies to the Jeffreys prior (though to a lesser extent).

• The approach developped in this paper assumes that the capture probabilities do not depend on individual characteristics. To take into account some discrete individual characteristics (for example, the sex) two approaches have been considered. Hook and Regal (1985) propose to conduct separate analyses (one for men and another one for women). [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF] will include the individual variable (in our example, the sex) in the graph, at the same level as the lists.

• In public health, it is important to know if a given disease falls within the field of rare diseases, or not. For a population of interest P, a disease is qualified of rare, if its prevalence (equal to the ratio of the number N of people belonging to P and affected by this disease over the size of P) is smaller than a threshold fixed by the competent authorities. Because the size of P is known in practice, this issue can be addressed by the test:

H 0 : N ≤ N 0 vs H 1 : N > N 0
where N 0 is the product of the above threshold by the size of P. We will typically conclude for H 0 , if Pr( N ≤ N 0 |y) is enough high (typically 0.95);

see [START_REF] Robert | The Bayesian Choice: From Decision-Theoretic Fundations to Computational Implementation[END_REF]. and θ 0++ + θ 1++ = 1, the clique parameter (θ C 1 , θ C 2 ) involves in fact five unconstrained parameters, for example: θ 1++ , θ 11+ , θ 01+ , θ 1+1 , θ 0+1 (all the other parameters being redundant). Recall that the capture probabilities parametrisation involves five parameters, namely:

θ 1 , θ 2|1 , θ 2|1 , θ 3|1 , θ 3|1 .
Both parametrizations are linked as follows:

θ 1++ = θ 1 , θ 11+ = θ 1 θ 2|1 , θ 01+ = (1-θ 1 )θ 2|1 , θ 1+1 = θ 1 θ 3|1 , θ 0+1 = (1-θ 1 )θ 3|1 .
Considering the above equalities, it is immediate to check that there is a one-to-one and onto correspondence between both parametrisations.

2. The model [1,23].

The graph of [1,23] includes two maximal cliques {C 1 , C 2 } where C 1 = {X 1 } and C 2 = {X 2 , X 3 }. We have thus two clique probabilities:

θ C 1 = (θ 1++ , θ 0++ ) and θ C 2 = (θ +00 , θ +01 , θ +10 , θ +11 )
where

θ 1++ + θ 0++ = 1 and θ +00 + θ +01 + θ +10 + θ +11 = 1.
Due to these constraints, we have in fact four unconstrained parameters, for example: θ 1++ , θ +01 , θ +10 , θ +11 . As far as the capture probabilities parametrisation is concerned, we have the choice between both parametrisations. Assume that we adopt the following parametrisation θ 1 , θ 2 , θ 3|2 and θ 3|2 . The clique probabilities parametrisation and the capture probabilities parametrisation are linked as follows:

θ 1++ = θ 1 , θ +01 = (1 -θ 2 )θ 3|2 , θ +10 = θ 2 (1 -θ 3|2 ), θ +11 = θ 2 θ 3|2 .
It is clear that there is a one-to-one and onto transformation between both parametrisations.

and are compatible with the prior placed on θ sat since θ Sat ∼ D(a 000 , a 001 , a 010 , a 011 , a 100 , a 101 , a 110 , a 111 ).

Recall now that the clique probabilities parametrisation involves in fact only five unconstrained parameters, for example: θ 1++ , θ 11+ , θ 01+ , θ 1+1 and θ 0+1 ;

see Appendix B. Linked to this parametrisation, we introduce the transformation:

Φ : θ 1 , θ 2|1 , θ 2|1 , θ 3|1 , θ 3|1 -→ θ 1 , θ 1 θ 2|1 , (1 -θ 1 )θ 2|1 , θ 1 θ 3|1 , (1 -θ 1 )θ 3|1 .
Starting from the density of (θ

C 1 , θ C 2 ), we deduce that π(θ m ) is proportional to J Φ T 2 T 3
T 1 where the Jacobian J Φ is equal to θ 2 1 (1 -θ 1 ) 2 and where the term

T 1 is equal to θ a 1++ -1 1 (1 -θ 1 ) a 0++ -1 , T 2 is equal to (1 -θ 1 )(1 -θ 2|1 ) a 00+ -1 (1 -θ 1 )θ 2|1 ) a 01+ -1 θ 1 (1 -θ 2|1 ) a 10+ -1 θ 1 θ 2|1 a 11+ -1
and T 3 is equal to

(1 -θ 1 )(1 -θ 3|1 ) a 0+0 -1 (1 -θ 1 )θ 3|1 ) a 0+1 -1 θ 1 (1 -θ 3|1 ) a 1+0 -1 θ 1 θ 3|1 a 1+1 -1 .
Gathering the terms θ 1 , θ 2|1 , θ 2|1 , θ 3|1 , and θ 3|1 , it is easy to check that π(θ m )

is proportionnal to D 1 D 2|1 D 2|1 D 3|1 D 3|1 where D 1 = θ a 1++ -1 1 (1 -θ 1 ) a 0++ -1 , D 2|1 = θ a 11+ -1 2|1 (1 -θ 2|1 ) a 10+ -1 , D 2|1 = θ a 01+ -1 2|1 (1 -θ 2|1 ) a 00+ -1 and D 3|1 = θ a 1+1 -1 3|1 (1 -θ 3|1 ) a 1+0 -1 , D 3|1 = θ a 0+1 -1 3|1 (1 -θ 3|1 ) a 0+0 -1 ,
from which we immediately deduce that θ 1 , θ 2|1 , θ 2|1 , θ 3|1 , θ 3|1 follow independently beta distributions. Moreover, it is immediate to check (using Proposition 1) that they are all compatible.

The converse. It is now assumed that θ 1 , θ 2|1 , θ 2|1 , θ 3|1 , θ 3|1 follow independently compatible beta distributions. Compatibility and Proposition 1 imply:

θ 1 ∼ beta(a 1++ , a 0++ ), θ 2|1 ∼ beta(a 11+ , a 10+ ), θ 2|1 ∼ Beta(a 01+ , a 00+ ), θ 3|1 ∼ beta(a 1+1 , a 1+0
) and θ 3|1 ∼ beta(a 0+1 , a 0+0 ). Re-finding the density of (θ C 1 , θ C 2 ) -as it appears in (1) -from the one of θ 1 , θ 2|1 , θ 2|1 , θ 3|1 , and θ 3|1 proceeds similarly as the direct sense of the proof; therefore, details are omitted.

1. The model [1,23].

The hyper-Dirichlet put on the clique parameter (θ C 1 , θ C 2 ) where

θ C 1 = (θ 1++ , θ 0++ ) and θ C 2 = (θ +00 , θ +01 , θ +10 , θ +11 ) is such that θ C 1 ⊥ θ C 2 where θ C 1 follows a Beta(a 1++ , a 0++
) and θ C 2 follows a D(a +00 , a +01 , a +10 , a +11 ). Consequently, one has:

π(θ C 1 , θ C 2 ) ∝ θ a 1++ -1 1++ (1 -θ 1++ ) a 0++ -1 k,l∈{0,1} θ a +kl -1 +kl . (2)
We have to prove that (2) is equivalent to θ 1 , θ 2 , θ 3|2 and θ 3|2 follows independently compatible beta distributions. If the other parametrisation is of concern, one has to prove the same equivalence with θ 1 , θ 3 , θ 2|3 , θ 3|2 .

From now, we work with the former parametrisation (but similar developments hold with the latter). Recall that the clique probabilities parametrisation involves in fact only four unconstrained parameters: θ 1++ , θ +01 , θ +10 , θ +11 and that there is a one-to-one and onto transformation between both parametrisations (see Appendix B). Considering that θ C 1 ⊥ θ C 2 and that θ C 1 ∼ Beta(a 1++ , a 0++ ) it is clear that to prove Proposition 2, we have to prove that θ C 2 ∼ D(a +00 , a +01 , a +10 , a +11 ) if and only if θ 2 , θ 3|2 and θ 3|2 follows independently compatible beta distributions.

Assume that θ C 2 ∼ Dirichlet(a +00 , a +01 , a +10 , a +11 ) and consider the transformtion:

Ψ : θ 2 , θ 3|2 , θ 3|2 -→ (1 -θ 2 )θ 3|2 , θ 2 (1 -θ 3|2 ), θ 2 θ 3|2 .
Starting from the density of θ C 2 , we deduce that π(θ 2 , θ 3|2 , θ 3|2 ) is proportional to J Ψ P where the product P is equal to

(1 -θ 2 )(1 -θ 3|2 ) a +00 -1 (1 -θ 2 )θ 3|2 a +01 -1 θ 2 (1 -θ 3|2 ) a +10 -1 θ 2 θ 3|2 a +11 -1
and where the Jacobian J Ψ is equal to θ 2 (1 -θ 2 ). By gathering the terms θ 2 ,θ 3|2 and θ 3|2 we deduce that π(θ 2 , θ 3|2 , θ 3|2 is proportional to:

θ a +1+ -1 2 (1 -θ 2 ) a +0+ -1 θ a +11 -1 3|2 (1 -θ 3|2 ) a +10 -1 θ a +01 -1 3|2 (1 -θ 3|2 ) a +00 -1
what means that θ 2 , θ 3|2 and θ 3|2 follows independently compatible beta distributions. The converse follows similar lines and is omitted for brievety.

3. The independent model.

The proof is trival since the hyper-Dirichlet distribution put on the clique parameter (C 1 , C 2 , C 3 ) is the product of 3 independant beta distributions.

L(θ m , N ; y) ∝ N ! (N -d)! θ d 1 1 (1 -θ 1 ) N -d 1 E 2|1 E 2|1 E 3|1 E 3|1
where

E 2|1 = θ n 11+ 2|1 (1 -θ 2|1 ) n 10+ E 2|1 = θ n 01+
2|1 (1 -θ 2|1 ) N -d+n 001 and

E 3|1 = θ n 1+1 3|1 (1 -θ 3|1 ) n 1+0 , E 3|1 = θ n 0+1 3|1 (1 -θ 3|1 ) N -d+n 010
Morever, one has:

I m (N ) = N ! (N -d)! h∈H * n h ! B 1 B 2|1 B 2|1 B 3|1 B 3|1
where • Under the model m = [1, 23], one has:

B 1 = B(d 1 + a 1++ , N -d 1 + a 0++ ),
L(θ m , N ; y) ∝ N ! (N -d)! θ d 1 1 (1 -θ 1 ) N -d 1 θ d 2 2 (1 -θ 2 ) N -d 2 E 3|2 E 3|2 ,
where

E 3|2 = θ n +11 3|2 (1 -θ 3|2 ) n +10 E 3|2 = θ n +01
3|2 (1 -θ 3|2 ) N -d+n 100 and when one adopts the factorization: p(x 1 , x 2 , x 3 ) = p(x 1 )p(x 2 )p(x 3 |x 2 ).

Morever, one has:

I m (N ) = N ! (N -d)! h∈H * n h ! B 1 B 2 B 3|2 B 3|2

  for the clique parameter of model m and the prior we put on the capture parameter present in model m. Proposition 3. Sub-model m being fixed, if one adopts the hyper-Dirichlet distribution of Madigan and York as prior for the clique parameter of submodel m, thus the marginal and the conditional capture probabilites playing a part in sub-model m follow independently beta distributions which are all compatible; moreover the converse holds.

  3, we mean by clique parameter of model m the set of the clique probabilities present in model m (a similar definition is adopted for the term capture parameter). Moreover, we mean by compatible beta distributions that the prior distributions placed on the marginal and conditional capture probabilities are derived from the Dirichlet distribution placed on θ Sat , as[START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF] did for the clique probabilities.6 Conditions of existence of posteriors.The graphical model being fixed, we provide a necessary and sufficient condition of existence of different posteriors: namely, the posterior distribution of N , as well as the posterior mean and variance of N . We also give a necessary and sufficient condition of existence of the averaged-model posterior mean ofN , that is of E[N |y].In this paper, we focus on the posterior mean which is the quantity the most often retained for estimating N ; see eg George &Robert (1992),[START_REF] King | On the Bayesian estimation of population size[END_REF]; as well as[START_REF] Wang | Bayesian population estimation for small capture-recapture data using noninformative priors[END_REF] for small populations. Note that[START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF] use the absolute quadratic loss which yields an estimate of N different from the posterior mean.

6. 1

 1 Conditions of existence of E[N |y, m]. Model m and data y being given, we have: π(N |y, m) = p(y|N, m)π(N ) N ≥d p(y|N, m)π(N ) (5.1) with p(y|N, m) = Pr(Y = y|N, m) = Θm L(θ m , N ; y)π(θ m ) d (θ m ), (5.2) where L(θ m , N ; y) denotes the likelihood of (θ m , N ) under model m. Note that the distribution of N |y, m will exist if and only if the integral appearing in (5.2) is finite (for all N ≥ d) and the series of general term p(y|N, m)π(N ) converges. For obtaining L(θ m , N ; y) we have to compute Pr(Y = y|θ m , N ).

  a Bayesian model averaging procedure, N is typically estimated by the averaged-model posterior mean of N , that is by E(N |y); see eg Hoeting et al. (1999). Now, one has: E(N |y) = m p(m|y) E(N |y, m) (5.1) where p(m|y) represents the posterior probability of model m. Proposition 5. E(N |y) exists if and only if λ + a -a 000 > 2 where λ = 0 when π(N ) = 1 and λ = 1 when π(N ) = 1/N .

  Now, computing Pr (N ≤ N 0 |y) is straightforward since Pr(N ≤ N 0 |y) = m p(m|y) Pr(N ≤ N 0 |m, y) and Pr(N ≤ N 0 |m, y) can be easily computed by the Gibbs sampling implementing for obtaining E(N |m, y) because Pr(N ≤ N 0 |m, y) = E[1I (N ≤N 0 ) |m, y].

B

  2|1 = B(n 11+ +a 11+ , n 10+ +a 10+ ), B 2|1 = B(n 01+ +a 01+ , N -d+n 001 +a 00+ )andB 3|1 = B(n 1+1 +a 1+1 , n 1+0 +a 1+0 ), B 3|1 = B(n 0+1 +a 0+1 , N -d+n 010 +a 0+0 ).

  E[N |y, m] will thus exist if and only if the series of general

	term N I m (N )π(N ) converges. In Proposition 4 below, we provide a necessary
	and sufficient condition so that this series converges; in our statement, λ = 1
	is associated with the Jeffreys prior π(N ) = 1/N , and λ = 0 with the uniform
	prior π(N ) = 1.
	Proposition 4. The posterior mean of N exits:

.3) The expression of L(θ m , N ; y) in function of the capture parameters present in model m, is now easily obtained by taking into account the factorization given in Table 1 (see Appendix D). For obtaining p(y|N, m) one has to integrate L(θ m , N ; y) over θ m . This integral, afterwards denoted by I m (N ), can be write down in a closed form (for each model m); see Appendix D. As stressed in this Appendix, I m (N ) exists for all model m, for all N ≥ d and for all data set y, because the a h 's are strictly positive.

-under the saturated model, if and only if, λ + a -a 000 > 2, -under

[12, 13]

, if and only if, λ + n 011 + (a -a 000 ) + a 011 > 2, -under [1, 23], if and only if, λ + d 1 -n 100 + (a + a 1++ -a +00 ) > 2, -under [1, 2, 3], if and only if, λ + (d 1

4 ⊥ (1, 2, 3) p(x 1 , x 2 , x 3 )p(x 4 ) 1 2

From the agregation property we deduce that (θ 1-j,+,+ , θ jk+ , θ j,1-k,+ ) ∼ D(a 1-j,+,+ , a jk+ , a j,1-k,+ ) for all j, k ∈ {0, 1}. The second part of the Proposition uses the following property: if

for all j = 1, ..., k: see [START_REF] Kotz | Continuous Mutivariate Distributions[END_REF]. From this property, we deduce that if (α 0 , α 1 , α 2 ) ∼ D(a 0 , a 1 , a 2 ) then α 1 /(α 1 +α 2 ) ∼ beta (a 1 , a 2 ). Consequently, one has θ jk+ /θ j++ ∼ beta(a jk+ , a j++ -a jk+ ) since θ jk+ + θ j,1-k,+ = θ j++ and a j,1-k,+ = a j++ -a jk+ .

Appendix B

We limit ourself to one model by type. The proof for the other models of same type being similar, it has been omitted for concision.

1. The model [12,13].

The graph of model m = [12, 13] includes two maximal cliques, we denote by C 1 and C 2 , where [START_REF] Madigan | Bayesian methods for estimation of the size of a closed of population[END_REF], we denote the corresponding clique probabilities as follows:

Due to the following constraints:

3. The independant model.

The graph of the independent model includes three cliques {C 1 , C 2 , C 3 } where

We have thus three clique probabilities:

where

The clique probabilities parametrisation thus involves only three parameters;

for example: θ 1++ , θ +1+ and θ ++1 . Since, the capture probabilities parametrisation involves the parameters θ 1 , θ 2 , θ 3 it is clear that there is a one-to-one and onto transformation between both parametrisations.

4. The saturated model.

The graph of the saturated model includes one (maximal) clique. One has only one clique probability which coincides with θ Sat defined in Section 2.

Appendix C

For concision, we limit ourself to one model by type.

1. The model [12,13].

The direct sens. The hyper-Dirichlet put on the clique parameter (θ

Its density is:

See eg [START_REF] Dawid | Hyper Markov laws in the statistical analysis of decomposable models[END_REF] for details. Margins satisfy:

In this Appendix we provide the expressions of the likelihood and of I m (N ) for each model m.

• Under the saturated model m = [123], the likelihood is the one given by (5.2) and one has:

• Under the model m = [12, 13], the likelihood is proportional to:

where u and v denote reals which do not depend on N . To obtain this equivalent, we start from the well known equivalent:

from which we deduce that:

Now, it is easy to check that:

Since, one has:

The result follows from

• We first consider the model m = [12,13]. We have to examine the series of general term I m π(N ) where

It is straightforward to see that one has actually to examine the convergence of the series of general term w N = N d-t T 1 T 2 T 3 where

Using now the above equivalent of Γ(N + v)/Γ(N + u), one finds that:

By observing that d -d 1 -n 01+ -n 0+1 = -n 011 and that -a + a 0++ -a 01+a 0+1 = -a + a 000 -a 011 it comes that:

w N ∼ N -(t+n 011 +a-a 000 +a 011 ) .

The posterior distribution of N thus exits if and only if: t + n 011 + a -a 000 + a 011 > 1.

• We now consider the saturated model [123]. One has actually to examine the convergence of the series of general term:

we deduce that

The posterior distribution of N thus exits if and only if: t + a -a 000 > 1.

Appendix F

The averaged posterior mean of N exists if and only if the posterior mean of N exists under each model m. Now, the condition which ensures the existence of the posterior distribution of N under the saturated model is the strongest. Indeed, it is clear that, on one hand, (d 1 + d 2 + d 3 ) -d ≥ 0, d 1 -n 100 ≥ 0 (idem for d 2 -n 010 , and for d 3 -n 001 ) and that, on the other hand, a 1++ + a +1+ + a ++1 , (a + a 1++ -a 100 ), and (a -a 000 ) + a 011 are all strictly greater than a -a 000 .