Complex simulators are used to express stochastic generative models of data across a wide segment of the scientific community, with applications as diverse as hazard analysis in seismology (Heinecke et al., 2014), supernova shock waves in astrophysics (Endeve et al., 2012), market movements in economics (Raberto et al., 2001), and blood flow in biology (Perdikaris et al., 2016). In these generative models, complex simulators are composed from low-level mechanistic components. These models are typically non-differentiable and lead to intractable likelihoods, which renders many traditional statistical inference algorithms irrelevant and motivates a new class of so-called likelihood-free inference algorithms (Hartig et al., 2011).
There are two broad strategies for this type of likelihood-free inference problem. In the first, one uses a simulator indirectly to train a surrogate model endowed with a likelihood that can be used in traditional inference algorithms, for example approaches based on conditional density estimationUria et al. (2016); Papamakarios et al. (2017); Rezende and Mohamed (2015); Kingma et al. (2016) and density ratio estimation (Cranmer et al., 2015; Dutta et al., 2016). Alternatively, approximate Bayesian computation (ABC) (Wilkinson, ; Sunnåker et al., 2013) refers to a large class of approaches for sampling from the posterior distribution of these likelihood-free models, where the original simulator is used directly as part of the inference engine. While variational inference (Blei et al., 2017) algorithms are often used when the posterior is intractable, they are not directly applicable when the likelihood of the data generating process is unknown.
The class of inference strategies that directly use a simulator avoids the necessity of approximating the generative model. Moreover, using a domain-specific simulator offers a natural pathway for inference algorithms to provide interpretable posterior samples. In this work, we take this approach, extend previous work in universal probabilistic programming (Gordon et al., 2014) and inference compilation (Le et al., 2017) to large-scale complex simulators, and demonstrate the ability to execute existing simulator codes under the control of general-purpose inference engines. This is achieved by creating a cross-platform probabilistic execution protocol (Figure 1) through which an inference engine can control simulators in a language-agnostic way. We implement a range of general-purpose inference engines from the Markov chain Monte Carlo (MCMC) (Brooks et al., 2011) and importance sampling (Doucet and Johansen, 2009) families. The execution framework we develop currently has bindings in C++ and Python, which are languages of choice for many large-scale projects in science and industry, and it can be used by any other language pending the implementation of a lightweight front end.
We demonstrate the technique in a particle physics setting, introducing probabilistic programming as a novel tool to determine the properties of particles at the Large Hadron Collider (LHC) (Aad et al., 2012; Chatrchyan et al., 2012) at CERN. This is achieved by coupling our framework with SHERPA111Simulation of High-Energy Reactions of Particles. https://sherpa.hepforge.org/ (Gleisberg et al., 2009), a state-of-the-art Monte Carlo event generator of high-energy reactions of particles, which is commonly used with Geant4222Geometry and Tracking. https://geant4.web.cern.ch/ (Allison et al., 2016), a toolkit for the simulation of the passage of the resulting particles through detectors. In particular, we perform inference in the case of (tau) lepton particle decay in a realistic detector, controlling the simulation within the standard SHERPA software with minimal modification and extracting posterior distributions in agreement with ground truths. To our knowledge this is the first time that universal probabilistic programming has been applied in this domain and in this scale, controlling a codebase of nearly one million lines of code. Our approach is readily scalable to more complex events and full detector simulators, paving the way to its use in the discovery of new fundamental physics.
2 Particle Physics and Probabilistic Inference
Our work is primarily motivated by applications in high-energy physics (HEP), which studies elementary particles and their interactions using energetic events created in particle accelerators such as the LHC at CERN. In this setting, the observed data are the result of interactions of particles generated in a collision event and observed through particle detectors. From these observations, we would like to infer the properties of the particles and interactions that generated them. Collisions happen millions of times per second, creating cascading particle decays in complex detectors instrumented with millions of electronics channels. These experiments then seek to filter the vast volume of (petabyte-scale) resulting data to make discoveries that shape our understanding of fundamental physics.
The complexity of the underlying physics and of the detectors have, until now, prevented the community from employing inference techniques. However, they have developed sophisticated simulator packages such as SHERPA (Gleisberg et al., 2009), Geant4 (Allison et al., 2016), Pythia8 (Sjöstrand et al., 2006), Herwig++ (Bähr et al., 2008), and MadGraph5(Alwall et al., 2014) to model physical processes and the interactions of particles with detectors. This is interesting from a probabilistic programming point of view, because HEP simulators are essentially very accurate probabilistic algorithms implementing the Standard Model and the passage of particles through matter (i.e., particle detectors). These simulators are coded in languages with unbounded recursion, and performing inference in such a setting requires using inference techniques developed for universal probabilistic programming that cannot be handled via more traditional inference approaches that apply to, for example, finite probabilistic graphical models (Koller and Friedman, 2009). Thus we focus on creating an infrastructure for the interpretation of existing simulator packages as probabilistic programs, which lays the groundwork for running inference in scientifically-accurate models using general-purpose probabilistic inference algorithms.
The Lepton Decay. The specific HEP setting we focus on in depth in this paper is the decay of a lepton particle inside an LHC-like detector. This is a real use case in particle physics currently under active study by LHC physicists (Aad et al., 2016) and it is also of interest due to its importance to establishing the properties of the recently discovered Higgs boson (Aad et al., 2012; Chatrchyan et al., 2012) through its decay to particles. Once produced, the
decays to further particles observed within the detector according to certain decay channels. The probabilities of these decays or “branching ratios” are shown in Figure2, which have been measured by other experiments and provide prior estimations for inference.
3 Related Work
3.1 Probabilistic Programming
Our work belongs in the family of sampling-based approximate inference techniques, which have been conventionally based on importance sampling (Agapiou et al., 2015) and Markov chain Monte Carlo (MCMC) methods (Brooks et al., 2011) such as the Metropolis–Hastings (MH) algorithm (Gilks et al., 1995). These are computationally inefficient on large-scale models, due to the difficulty in choosing correct proposal distributions and handling increasing model dimensionality. Recent developments in inference algorithms, such as variational methods (Wingate and Weber, 2013)
, extensions that combine deep learning(Kingma and Welling, 2013; Rezende and Mohamed, 2015), MCMC samplers based on physical dynamics such as the No-U-Turn Sampler (NUTS) (Hoffman and Gelman, 2014) and Stochastic Gradient Langevin Dynamics (Welling and Teh, 2011), and methods that use deep neural networks to amortize the cost of inference (Gershman and Goodman, 2014) such as inference compilation (IC) (Le et al., 2017), have been targeting fast and scalable inference.
Probabilistic programming languages (PPLs) attempt to decouple inference algorithms from model building, by creating a simple, yet expressive, syntax that allows one to take advantage of these powerful inference algorithms on any probabilistic generative model expressed as a regular computer program. Universal PPLs allow the expression of unrestricted probability models in a Turing-complete fashion (Wingate et al., 2011; Goodman et al., 2012; Wood et al., 2014), and there is a recent trend in combining these with variational inference and deep learning, leading to tools such as Pyro (Eli et al., 2017), ProbTorch (Siddharth et al., 2017), and Edward (Tran et al., 2017). This is in contrast to languages such as Stan (Gelman et al., 2015) that target the more restricted model class of probabilistic graphical models (Koller and Friedman, 2009).
3.2 Data Analysis in High-Energy Physics
Inference for an individual collision event in HEP is often referred to as reconstruction (Lampl et al., 2008)
. Reconstruction algorithms can be seen as a form of structured prediction: from the raw event data they produce a list of candidate particles together with their types and point-estimates for their momenta. The variance of these estimators is characterized by comparison to the ground truth values of the latent variables from simulated events. Bayesian inference on the latent state of an individual collision is rare in HEP given the complexity of the latent structure of the generative model. Until now, inference for the latent structure of an individual event has only been possible by accepting a drastic simplification of the high-fidelity simulators(Kondo, 1988; Abazov et al., 2004; Artoisenet and Mattelaer, 2008; Gao et al., 2010; Alwall et al., 2011; Bolognesi et al., 2012; Avery et al., 2013; Andersen et al., 2013; Campbell et al., 2013; Artoisenet et al., 2013; Gainer et al., 2013; Schouten et al., 2015; Martini and Uwer, 2015; Gritsan et al., 2016; Martini and Uwer, 2017; Soper and Spannowsky, 2011)
. In contrast, inference for the fundamental parameters is based on hierarchical models and probed at the population level. Recently, machine learning techniques have been employed to learn surrogates for the implicit densities defined by the simulators as a strategy for likelihood-free inference(Brehmer et al., 2018).
Currently HEP simulators are run in forward mode to produce substantial datasets that often exceed the size of datasets from actual collisions within the experiments. These are then reduced to considerably lower dimensional datasets of a handful of variables using physics domain knowledge, which can then be directly compared to collision data. Machine learning and statistical approaches for classification of particle types or regression of particle properties can be trained on these large pre-generated datasets produced by the high-fidelity simulators developed over many decades (Asquith et al., 2018; Kasieczka, 2018). The field is increasingly employing deep learning techniques allowing these algorithms to process high-dimensional, low-level data (Baldi et al., 2014; de Oliveira et al., 2016; Aurisano et al., 2016; Racah et al., 2016; Hooberman et al., 2017). However, these approaches do not estimate the posterior of the full latent state nor provide the level of interpretability our probabilistic inference framework enables by directly tying inference results to the latent process encoded by the simulator.
4 Probabilistic Inference in Large-Scale Simulators
In this section we describe the main components of our probabilistic inference framework, which consists of (1) pyprob
, a PyTorch-based(Paszke et al., 2017) PPL and associated inference engines in Python, (2) PPX, a probabilistic programming execution protocol that defines a cross-platform interface for connecting models and inference engines implemented in different programming languages and executed in separate processes, (3) pyprob_cpp, a lighweight C++ front end that allows the execution of models written in C++ under the control of pyprob.
4.1 Designing a PPL for Existing Large-Scale Simulators
A shortcoming of the current state-of-the-art in PPLs is that they are not designed to directly support existing codebases, severely limiting their applicability to a very large body of existing probabilistic models implemented as domain-specific simulators in many fields across academia and industry. A PPL, by definition, is a programming language with additional constructs for sampling
random values from probability distributions andconditioning
values of random variables via observations(Gordon et al., 2014). Domain-specific simulators in HEP and other fields are commonly probabilistic in nature, thus satisfying the behavior random sampling, albeit generally from simplistic distributions such as the continuous uniform. By automatically “reinterpreting” these existing codebases with a proper rewiring of the (pseudo-)random number generator and introducing a construct for conditioning, we can execute existing simulators under the control of general-purpose inference engines designed for probabilistic programming. This enables the application of Bayesian inference techniques in these simulators, essentially treating the existing simulator as a joint prior distribution of latent and observed variables of a model, and obtaining posterior distributions over latent variables conditioned on realizations of observed variables.
To realize our framework, we implement pyprob,333https://github.com/probprog/pyprob a universal PPL specifically designed to control models written not only in Python but also in other languages. Because the main inference technique we use in this PPL is based on deep neural networks, we base our PPL on PyTorch (Paszke et al., 2017), whose automatic differentiation (AD) (Baydin et al., 2018) feature with support for dynamic computation graphs has been crucial in our implementation. Our PPL currently has two families of inference engines: (1) MCMC of the lightweight Metropolis–Hastings (LMH) (Wingate et al., 2011) and random-walk Metropolis–Hastings (RMH) (Le, 2015) varieties, and (2) sequential importance sampling (IS) (Arulampalam et al., 2002; Doucet and Johansen, 2009) with its regular (i.e., sampling from the prior) and inference compilation (IC) (Le et al., 2017) varieties. The IC technique, where a deep neural network is trained in an amortized inference setting to guide (control) a probabilistic program conditioning on observed inputs, forms our main inference method for performing efficient inference in large-scale simulators. The LMH and RMH engines we implement are specialized for sampling in the space of execution traces of probabilistic programs, and provide way of sampling from the true posterior—at a high computational cost.
A probabilistic program can be expressed as a sequence of random samples , where , , and are respectively the value, address, and instance (counter) of a sample, the execution of which describes a joint probability distribution between latent (unobserved) random variables and observed random variables given by
denotes the prior probability distribution of a random variable with addressconditional on all preceding values , and is the likelihood density given the sample values preceding observation . A PPL is a regular programming language equipped with sample and observe statements (Gordon et al., 2014) for sampling random variables with given prior probability distributions and conditioning random variables upon particular observed values.
Once a model is expressed as a probabilistic program, we are interested in performing inference in order to get posterior distributions of latent variables conditioned on observed variables . In the sequential IS scheme, a weighted set of samples is used to construct an empirical approximation of the posterior distribution , where is the Dirac delta function. The importance weights for a probabilistic program are expressed as
where is known as the proposal distribution and may be identical to the prior (as in regular IS). In the IC technique, we are training a deep neural network to receive the observed values and return a set of adapted proposals such that their joint is close to the true posterior
. This is achieved by using a Kullback–Leibler divergence training objective
where represents the neural network weights. The neural network weights are optimized to minimize this objective by continually drawing training pairs from the probabilistic program (i.e., the generative model, or the simulator). To simplify the task of training, only a subset of all addresses are handled by the neural network, and the remaining addresses are left to use the prior as proposal during inference. The IC controlling of an address is exposed as a boolean flag called control, which can be applied to individual sample statements or delimited regions of the codebase. Expressed in simple terms, taking a desired outcome from the probabilistic program as its input, the neural network learns to control the random number draws of latents during the execution in such a way that makes the desired outcome likely.
The neural network architecture in IC is based on a stacked LSTM (Hochreiter and Schmidhuber, 1997) recurrent core that gets executed for as many time steps as the probabilistic trace length. The input to this LSTM in each time step is a concatenation of embeddings of the observation , the previously sampled value , the current distribution type , and the current address .
is a neural network specific to the domain (such as a 3D convolutional neural network for volumetric inputs),are feed-forward modules,
are one-hot vectors denoting a prior distribution type from the set of supported distributions,
are learned address embeddings optimized via backpropagation for eachpair encountered in the program execution. The addressing scheme (Wingate et al., 2011) is the main link between semantic locations in the probabilistic program and the inputs to the neural network. The addressing scheme in Python is based on an analysis of Python bytecode of the location where the PPL sample or observe statement is called, and in the PPX protocol (Section 4.2) the addresses are produced and supplied by the side hosting and executing the model.
The joint proposal distribution of the neural network is factorized into proposals in each time step , whose type depends on the type of the prior . In the experiments presented in this paper (Section 5
) the system uses categorical and continuous uniform distributions in the prior, for which we use, respectively, categorical and mixture of Kumaraswamy(Kumaraswamy, 1980; Mitnik and Baek, 2013) distributions as proposals parameterized by the neural network.
A common challenge for inference in real-world scientific models, such as those in HEP, is the presence of large dynamic ranges of prior probabilities for various outcomes. For instance, some particle decays are much more probable than others (Figure 2), and the prior distribution for a particle momentum can be steeply falling. Therefore some cases may be much more likely to be seen by the neural network during training relative to others. For this reason, the proposal parameters and the quality of the inference would vary significantly according to the frequency of the observations in the prior. To address this issue, we apply a technique called “prior inflation” for automatically adjusting the measure of the prior distribution during training to generate more instances of these unlikely outcomes. This applies only to the training data generation for the IC neural network, and the unmodified original model is used during inference, ensuring that the importance weights (Eq. 2) and therefore the empirical posterior are correct under the unmodified real model.
4.2 A Cross-Platform Probabilistic Execution Protocol
Besides its use with pyprob, the PPX protocol defines a very flexible way of coupling any PPL system and model so that they can be (1) implemented in different programming languages and (2) executed in separate processes and on separate machines across networks. Thus PPX is similar in spirit to, and indeed inspired by, the Open Neural Network Exchange (ONNX)777https://onnx.ai/ project for interoperability between machine learning frameworks. Note that, more than a serialization format, PPX enables runtime execution of probabilistic models under the control of inference engines in separate processes. We are releasing this language-agnostic protocol as a separately maintained project, together with the rest of our work in Python and C++.
4.3 Controlling SHERPA and the Standard Model
In this paper our target simulator is SHERPA (Gleisberg et al., 2009), a Monte Carlo event generator of high-energy reactions of particles, which is a state-of-the-art simulator of the Standard Model of particle physics developed as an international effort within the HEP community. SHERPA, like many other large-scale scientific projects, is implemented in C++, and therefore we implement a C++ front end for PPX, called pyprob_cpp.888https://github.com/probprog/pyprob_cpp
We couple SHERPA to pyprob_cpp by a system-wide rerouting of the calls to the random number generator, which is made easy by the existence of a third-party random number generator interface (External_RNG) already present in SHERPA. Through this setup, we can repurpose, with little effort, any stochastic simulation written in SHERPA as a probabilistic generative model in which we can perform inference using probabilistic programming techniques.
Differing from the conventions in the probabilistic programming community, random number draws in C++ simulators are commonly performed at a lower level than the actual prior distribution that is being simulated. This applies to SHERPA where the only samples are from the standard uniform distribution , which subsequently get used for different purposes using transformations or rejection sampling. In our experiments (Section 5) we work with all uniform samples except for a problem-specific single address that we know to be responsible for sampling from a categorical distribution for choosing the lepton decay channel. The modification of this address to use the proper categorical prior allows an effortless application of the prior inflation technique (Section 4.1) to generate training data equally representing each channel.
Rejection sampling (Gilks and Wild, 1992) sections in the simulator pose a problem for our approach, as they define execution traces that are a priori unbounded; and since the inference network has to backpropagate through every sampled value, this makes the training significantly slower. Rejection sampling is key to the application of Monte Carlo methods for evaluating matrix elements (Krauss, 2002) and other stages of event generation in particle physics; thus an efficient treatment of this construction is primal. We address this problem by implementing a novel trace evaluation scheme where during training we only consider the last (thus accepted) instance of any address that fall within a rejection sampling loop. During inference, we use this same proposal distribution in each loop execution. In other words, this corresponds to training the inference network with the state that concludes the loop (i.e., satisfies the acceptance criterion), effectively selecting proposal distributions such that the rejection loop is concluded in as few iterations as possible. This scheme works by annotating the sample statements within long-running rejection sampling loops with a boolean flag called replace, which, when set true, enables the behavior described for the given sample address.
An important decay of the Higgs boson is to leptons, whose subsequent decay products interact in the detector. This constitutes a rich and realistic case to simulate, and directly connects to an important line of current research in particle physics. During simulation, SHERPA stochastically selects a set of particles to which the initial lepton will decay—a “decay channel”—and samples the momenta of these particles according to a joint density obtained from underlying physical theory. These particles then interact in the detector leading to observations in the raw sensor data. While Geant4 is typically used to model the interactions in a detector, for our initial studies we implement a fast, approximate detector simulation for a calorimeter with longitudinal and transverse segmentation (with resolution 203535). The fast detector simulation deposits most of the energy for electrons and into the first layers and charged hadrons (e.g., ) deeper into the calorimeter with larger fluctuations. Given raw 3D calorimeter observations, we would like to infer primarily the decay channel that the lepton followed and the initial momenta , , and . Using our framework, we compute posterior distributions for the decay channel, initial momenta, and other latent quantities in the model conditioning on various simulated observations with known ground truth. The discrete variable for decay channel has a known prior distribution (Figure 2) given by the branching ratio of the into possible decay channels (Patrignani et al., 2016).
In Figure 3 we show inference results obtained from the IC and RMH MCMC engines, for a single observation sampled from the model joint prior by running the simulation. The IC proposals are generated by an inference network trained with 1.6 million execution traces, and the IC engine controls (i.e., makes proposals different from the prior for) 47 addresses,999The controlled addresses are those that fall within sections of the codebase deemed fundamental in the solution of the decay problem, based on domain knowledge. 17 of these in replacement (rejection sampling) mode, out of a total of 24,429 addresses. The RMH engine, by its very nature, controls all addresses encountered in the simulation. During IC network training, 440 trace types are encountered (Table 1), which represent the reoccurrence of the same sequence of addresses with different actual sample values. Traces reach lengths up to 7,514 and 1,190 respectively when looking at all and controlled-only samples (Figure 4).
As can be seen in Figure 3 (a), the network proposes values in agreement with the ground truth values. Figure 3 (b) shows the posterior after importance sampling guided by these proposals: this shows the correct posterior otver particle decays was identified and also that related decays are shown as possible alternatives with correct uncertainty, in agreement with RMH samples from the correct posterior in Figure 3 (c). Furthermore, correlations between the final state particle momenta are well reproduced. In order to make the RMH results for the test observation tractable, we start the chain from the ground truth trace (i.e., eliminating the need for burn in), which would not be possible for inference on real experimental data. This shows good agreement with the decay channel and event composition posteriors obtained from the IC engine that has access to observation only (i.e., did not start from ground truth), and that would be used for fast inference with real experimental data.
The ability to connect posterior samples to the simulator code is a key advantage of our method in scientific applications. This connection enables inference results to be interpretable in the context of the physically-motivated latent process encoded by the simulator. Note that the posterior distributions presented in Figure 3 do not show all of the posterior information this technique encodes. Probabilistic programming gives us posteriors over the full space of execution traces covering the entire latent structure of the simulator, which we show at different levels of detail in Figure 5. Table 2 provides several examples of how the actual addresses
look like within C++, allowing us to pinpoint all individual nodes in the codebase where the model behaves probabilistically. For instance, this approach gives us the ability to inspect aspects such as the chain of particle decays and interactions within the detector that led to particular posterior predictions, mirroring the standard task of event reconstruction(Mankel, 2004). This capability is not present in inference techniques that do not have access to the simulator, such as those solely based on neural networks.
Our work is the first step in subsuming the vast existing body of scientific simulators, which are essentially accurate generative models with decades of development behind them in many instances, into a universal probabilistic programming framework. The ability to scale probabilistic programming to large-scale simulators is of fundamental importance to the field of probabilistic programming and the wider modeling community. It is a hard problem that requires innovations in many areas such as model–inference engine interface, handling of priors with long tails and rejection sampling routines, addressing schemes, and IC network architecture, which make it difficult to cover in depth in a single paper. A main limitation of the introduced technique, currently, is the need for domain expert decisions in marking regions of codebase as controlled (only needed for the IC engine, and not needed for MCMC), which can potentially be automated in future work.
Our advancement allows one to perform model-based machine learning with interpretability, meaning that we understand the exact processes behind how the predictions are produced and the uncertainty in each prediction. With this novel framework providing a clearly defined interface between existing scientific simulators and probabilistic machine learning techniques, we expect to influence both communities to perform research at the intersection of science and machine learning.
Baydin and Wood are supported under DARPA PPAML through the U.S. AFRL under Cooperative Agreement FA8750-14-2-0006, Sub Award number 61160290-111668. Wood is supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1 and DARPA D3M, under Cooperative Agreement FA8750-17-2-0093. Cranmer, Louppe, and Heinrich are supported through NSF ACI-1450310, PHY-1505463, PHY- 1205376, and the Moore-Sloan Data Science Environment at NYU. Gram-Hansen is supported by the UK EPSRC CDT in Autonomous Inteligent Machines and Systems. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the U.S.Department of Energy under Contract No. DE-AC02-05CH11231. This work was partially supported by the NERSC Big Data Center; we acknowledge Intel for their funding support.
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Appendix A Appendix
|Freq.||Length||Addresses (showing controlled only)|
|0.106||72||A1, A2, A3, A5, A6, A32, A33, A31|
|0.105||41||A1, A2, A3, A5, A6, A499, A31|
|0.078||1,780||A1, A2, A3, A5, A6, A7, A8, A9, A10, A31|
|0.053||188||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A26, A31|
|0.053||100||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A99, A100, A101, A102, A31|
|0.039||56||A1, A2, A3, A5, A6, A499, A17, A18, A26, A31|
|0.039||592||A1, A2, A3, A5, A6, A499, A17, A18, A99, A100, A101, A102, A31|
|0.038||162||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A500, A99, A100, A101, A102, A31|
|0.030||240||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A20, A21, A41, A42, A26, A99, A100, A101, A102, A31|
|0.029||836||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A20, A21, A41, A42, A99, A100, A101, A102, A26, A31|
|0.027||643||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A507, A99, A100, A101, A102, A31|
|0.023||135||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A20, A21, A41, A42, A44, A45, A26, A99, A100, A101, A102, A31|
|0.023||485||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A20, A21, A41, A42, A44, A45, A99, A100, A101, A102, A26, A31|
|0.019||316||A1, A2, A3, A5, A6, A32, A33, A17, A500, A99, A100, A101, A102, A31|
|0.014||68||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A20, A21, A26, A99, A100, A101, A102, A31|
|0.013||422||A1, A2, A3, A5, A6, A32, A33, A17, A500, A20, A1496, A99, A100, A101, A102, A31|
|0.013||298||A1, A2, A3, A5, A6, A32, A33, A17, A18, A20, A21, A26, A31|
|0.013||283||A1, A2, A3, A5, A6, A32, A33, A17, A18, A20, A21, A26, A99, A100, A101, A102, A31|
|0.013||608||A1, A2, A3, A5, A6, A32, A33, A17, A18, A20, A21, A99, A100, A101, A102, A31|
|0.013||424||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A20, A21, A99, A100, A101, A102, A31|
|0.013||50||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A20, A21, A26, A31|
|0.013||204||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A20, A21, A99, A100, A101, A102, A26, A31|
|0.013||252||A1, A2, A3, A5, A6, A32, A33, A17, A18, A20, A21, A99, A100, A101, A102, A26, A31|
|0.010||234||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A20, A21, A41, A42, A99, A100, A101, A102, A31|
|0.010||58||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A18, A20, A21, A41, A42, A26, A31|
|0.010||502||A1, A2, A3, A5, A6, A499, A17, A18, A20, A1496, A99, A100, A101, A102, A31|
|0.009||216||A1, A2, A3, A5, A6, A499, A17, A500, A20, A21, A99, A100, A101, A102, A31|
|0.009||1,053||A1, A2, A3, A5, A6, A499, A17, A18, A20, A1496, A26, A99, A100, A101, A102, A31|
|0.009||800||A1, A2, A3, A5, A6, A499, A17, A500, A20, A21, A99, A100, A101, A102, A26, A31|
|0.007||92||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A500, A26, A31|
|0.007||32||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A507, A26, A31|
|0.007||78||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A507, A99, A100, A101, A102, A510, A511, A898, A31|
|0.006||120||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A507, A20, A508, A99, A100, A101, A102, A31|
|0.006||118||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A507, A99, A100, A101, A102, A510, A511, A882, A883, A884, A885, A31|
|0.005||553||A1, A2, A3, A5, A6, A7, A8, A9, A10, A17, A500, A20, A21, A41, A42, A99, A100, A101, A102, A26, A31|
|Address ID||Full address|
|A1||[forward(xt:: xarray_container<xt:: uvector<double, std:: allocator<double> >, (xt:: layout_type)1, xt:: svector<unsigned long, 4ul, std:: allocator<unsigned long>, true>, xt:: xtensor_expression_tag>)+0x5f; SherpaGenerator:: Generate()+0x36; SHERPA:: Sherpa:: GenerateOneEvent(bool)+0x2fa; SHERPA:: Event_Handler:: GenerateEvent(SHERPA:: eventtype:: code)+0x44d; SHERPA:: Event_Handler:: GenerateHadronDecayEvent(SHERPA:: eventtype:: code&)+0x45f; ATOOLS:: Random:: Get(bool, bool)+0x1d5; probprog_RNG:: Get(bool, bool)+0xf9]_Uniform_1|
|A2||[forward(xt:: xarray_container<xt:: uvector<double, std:: allocator<double> >, (xt:: layout_type)1, xt:: svector<unsigned long, 4ul, std:: allocator<unsigned long>, true>, xt:: xtensor_expression_tag>)+0x5f; SherpaGenerator:: Generate()+0x36; SHERPA:: Sherpa:: GenerateOneEvent(bool)+0x2fa; SHERPA:: Event_Handler:: GenerateEvent(SHERPA:: eventtype:: code)+0x44d; SHERPA:: Event_Handler:: GenerateHadronDecayEvent(SHERPA:: eventtype:: code&)+0x477; ATOOLS:: Random:: Get(bool, bool)+0x1d5; probprog_RNG:: Get(bool, bool)+0xf9]_Uniform_1|
|A3||[forward(xt:: xarray_container<xt:: uvector<double, std:: allocator<double> >, (xt:: layout_type)1, xt:: svector<unsigned long, 4ul, std:: allocator<unsigned long>, true>, xt:: xtensor_expression_tag>)+0x5f; SherpaGenerator:: Generate()+0x36; SHERPA:: Sherpa:: GenerateOneEvent(bool)+0x2fa; SHERPA:: Event_Handler:: GenerateEvent(SHERPA:: eventtype:: code)+0x44d; SHERPA:: Event_Handler:: GenerateHadronDecayEvent(SHERPA:: eventtype:: code&)+0x48f; ATOOLS:: Random:: Get(bool, bool)+0x1d5; probprog_RNG:: Get(bool, bool)+0xf9]_Uniform_1|
|A4||[forward(xt:: xarray_container<xt:: uvector<double, std:: allocator<double> >, (xt:: layout_type)1, xt:: svector<unsigned long, 4ul, std:: allocator<unsigned long>, true>, xt:: xtensor_expression_tag>)+0x5f; SherpaGenerator:: Generate()+0x36; SHERPA:: Sherpa:: GenerateOneEvent(bool)+0x2fa; SHERPA:: Event_Handler:: GenerateEvent(SHERPA:: eventtype:: code)+0x44d; SHERPA:: Event_Handler:: GenerateHadronDecayEvent(SHERPA:: eventtype:: code&)+0x8f4; ATOOLS:: Particle:: SetTime()+0xd; ATOOLS:: Flavour:: GenerateLifeTime() const+0x35; ATOOLS:: Random:: Get()+0x18b; probprog_RNG:: Get()+0xde]_Uniform_1|
|A5||[forward(xt:: xarray_container<xt:: uvector<double, std:: allocator<double> >, (xt:: layout_type)1, xt:: svector<unsigned long, 4ul, std:: allocator<unsigned long>, true>, xt:: xtensor_expression_tag>)+0x5f; SherpaGenerator:: Generate()+0x36; SHERPA:: Sherpa:: GenerateOneEvent(bool)+0x2fa; SHERPA:: Event_Handler:: GenerateEvent(SHERPA:: eventtype:: code)+0x44d; SHERPA:: Event_Handler:: GenerateHadronDecayEvent(SHERPA:: eventtype:: code&)+0x982; SHERPA:: Event_Handler:: IterateEventPhases(SHERPA:: eventtype:: code&, double&)+0x1d2; SHERPA:: Hadron_Decays:: Treat(ATOOLS:: Blob_List*, double&)+0x975; SHERPA:: Decay_Handler_Base:: TreatInitialBlob(ATOOLS:: Blob*, METOOLS:: Amplitude2_Tensor*, std:: vector<ATOOLS:: Particle*, std:: allocator<ATOOLS:: Particle*> > const&)+0x1ab1; SHERPA:: Hadron_Decay_Handler:: CreateDecayBlob(ATOOLS:: Particle*)+0x4cd; PHASIC:: Decay_Table:: Select() const+0x76e; ATOOLS:: Random:: Get(bool, bool)+0x1d5; probprog_RNG:: Get(bool, bool)+0xf9]_Uniform_1|
|A6||[forward(xt:: xarray_container<xt:: uvector<double, std:: allocator<double> >, (xt:: layout_type)1, xt:: svector<unsigned long, 4ul, std:: allocator<unsigned long>, true>, xt:: xtensor_expression_tag>)+0x5f; SherpaGenerator:: Generate()+0x36; SHERPA:: Sherpa:: GenerateOneEvent(bool)+0x2fa; SHERPA:: Event_Handler:: GenerateEvent(SHERPA:: eventtype:: code)+0x44d; SHERPA:: Event_Handler:: GenerateHadronDecayEvent(SHERPA:: eventtype:: code&)+0x982; SHERPA:: Event_Handler:: IterateEventPhases(SHERPA:: eventtype:: code&, double&)+0x1d2; SHERPA:: Hadron_Decays:: Treat(ATOOLS:: Blob_List*, double&)+0x975; SHERPA:: Decay_Handler_Base:: TreatInitialBlob(ATOOLS:: Blob*, METOOLS:: Amplitude2_Tensor*, std:: vector<ATOOLS:: Particle*, std:: allocator<ATOOLS:: Particle*> > const&)+0x1ab1; SHERPA:: Hadron_Decay_Handler:: CreateDecayBlob(ATOOLS:: Particle*)+0x4cd; PHASIC:: Decay_Table:: Select() const+0x9d7; ATOOLS:: Random:: GetCategorical(std:: vector<double, std:: allocator<double> > const&, bool, bool)+0x1a5; probprog_RNG:: GetCategorical(std:: vector<double, std:: allocator<double> > const&, bool, bool)+0x111]_Categorical(length_categories:38)_1|