Abstract
We introduce the DNNLikelihood, a novel framework to easily encode, through deep neural networks (DNN), the full experimental information contained in complicated likelihood functions (LFs). We show how to efficiently parametrise the LF, treated as a multivariate function of parameters of interest and nuisance parameters with high dimensionality, as an interpolating function in the form of a DNN predictor. We do not use any Gaussian approximation or dimensionality reduction, such as marginalisation or profiling over nuisance parameters, so that the full experimental information is retained. The procedure applies to both binned and unbinned LFs, and allows for an efficient distribution to multiple software platforms, e.g. through the framework-independent ONNX model format. The distributed DNNLikelihood can be used for different use cases, such as re-sampling through Markov Chain Monte Carlo techniques, possibly with custom priors, combination with other LFs, when the correlations among parameters are known, and re-interpretation within different statistical approaches, i.e. Bayesian vs frequentist. We discuss the accuracy of our proposal and its relations with other approximation techniques and likelihood distribution frameworks. As an example, we apply our procedure to a pseudo-experiment corresponding to a realistic LHC search for new physics already considered in the literature.
Highlights
We propose to present the full likelihood functions (LFs), as used by experimental collaborations to produce the results of their analyses, in the form of a suitably trained Deep Neural Network (DNN), which is able to reproduce the original LF as a function of physical and nuisance parameters with the accuracy required to allow for the four aforementioned tasks
A dedicated Python package allowing to sample LFs and to build, optimize, train, and store the corresponding DNNLikelihoods is in preparation
We have argued above that the mean absolute error (MAE) or Minimum loss test (MSE) on log L(xi ) are the most suitable loss functions to train our deep neural networks (DNN) for interpolating the LF on the sample xi
Summary
Experimental analyses usually deliver only a small fraction of the full information contained in the LF, typically in the form of confidence intervals obtained by profiling the LF on the nuisance parameters (frequentist approach), or in terms of probability intervals obtained by marginalising over nuisance parameters (Bayesian approach), depending on the statistical method used in the analysis This way of presenting results is very practical, since it can be encoded graphically into simple plots and/or simple tables of expectation values and correlation matrices among observables, effectively making use of the Gaussian approximation, or refinements of it aimed at taking into account asymmetric intervals or one-sided constraints. We propose to present the full LF, as used by experimental collaborations to produce the results of their analyses, in the form of a suitably trained Deep Neural Network (DNN), which is able to reproduce the original LF as a function of physical and nuisance parameters with the accuracy required to allow for the four aforementioned tasks
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