petitRADTRANS.retrieval.psis ============================ .. py:module:: petitRADTRANS.retrieval.psis .. autoapi-nested-parse:: Pareto smoothed importance sampling (PSIS) # TODO find a more explicit name for this module in version 4.0.0 This module implements Pareto smoothed importance sampling (PSIS) and PSIS leave-one-out (LOO) cross-validation for Python (Numpy). Included functions ------------------ psisloo Pareto smoothed importance sampling leave-one-out log predictive densities. psislw Pareto smoothed importance sampling. gpdfitnew Estimate the parameters for the Generalized Pareto Distribution (GPD). gpinv Inverse generalised Pareto distribution function. sumlogs Sum of vector where numbers are represented by their logarithms. References ---------- Aki Vehtari, Andrew Gelman and Jonah Gabry (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27(5):1413–1432. doi:10.1007/s11222-016-9696-4. https://arxiv.org/abs/1507.04544 Aki Vehtari, Andrew Gelman and Jonah Gabry (2017). Pareto smoothed importance sampling. https://arxiv.org/abs/arXiv:1507.02646v5 3-Clause BSD License -------------------- Copyright 2017 Aki Vehtari, Tuomas Sivula Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. Neither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. 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Functions --------- .. autoapisummary:: petitRADTRANS.retrieval.psis.psisloo petitRADTRANS.retrieval.psis.psislw petitRADTRANS.retrieval.psis.gpdfitnew petitRADTRANS.retrieval.psis.gpinv petitRADTRANS.retrieval.psis.sumlogs Module Contents --------------- .. py:function:: psisloo(log_lik, **kwargs) PSIS leave-one-out log predictive densities. Computes the log predictive densities given posterior samples of the log likelihood terms :math:`p(y_i|\theta^s)` in input parameter `log_lik`. Returns a sum of the leave-one-out log predictive densities `loo`, individual leave-one-out log predictive density terms `loos` and an estimate of Pareto tail indeces `ks`. The estimates are unreliable if tail index ``k > 0.7`` (see more in the references listed in the module docstring). Additional keyword arguments are passed to the :meth:`psislw()` function (see the corresponding documentation). Parameters ---------- log_lik : ndarray Array of size n x m containing n posterior samples of the log likelihood terms :math:`p(y_i|\theta^s)`. Returns ------- loo : scalar sum of the leave-one-out log predictive densities loos : ndarray individual leave-one-out log predictive density terms ks : ndarray estimated Pareto tail indeces .. py:function:: psislw(lw, r_eff=1.0, overwrite_lw=False) Pareto smoothed importance sampling (PSIS). Parameters ---------- lw : ndarray Array of size n x m containing m sets of n log weights. It is also possible to provide one dimensional array of length n. r_eff : scalar, optional relative MCMC efficiency ``N_eff / N`` overwrite_lw : bool, optional If True, the input array `lw` is smoothed in-place, assuming the array is F-contiguous. By default, a new array is allocated. Returns ------- lw_out : ndarray smoothed log weights kss : ndarray Pareto tail indices .. py:function:: gpdfitnew(x, sort=True, sort_in_place=False, return_quadrature=False) Estimate the paramaters for the Generalized Pareto Distribution (GPD) Returns empirical Bayes estimate for the parameters of the two-parameter generalized Parato distribution given the data. Parameters ---------- x : ndarray One dimensional data array sort : bool or ndarray, optional If known in advance, one can provide an array of indices that would sort the input array `x`. If the input array is already sorted, provide False. If True (default behaviour), the array is sorted internally. sort_in_place : bool, optional If `sort` is True and `sort_in_place` is True, the array is sorted in-place (False by default). return_quadrature : bool, optional If True, quadrature points and weight `ks` and `w` of the marginal posterior distribution of k are also calculated and returned. False by default. Returns ------- k, sigma : float estimated parameter values ks, w : ndarray Quadrature points and weights of the marginal posterior distribution of `k`. Returned only if `return_quadrature` is True. Notes ----- This function returns a negative of Zhang and Stephens's k, because it is more common parameterisation. .. py:function:: gpinv(p, k, sigma) Inverse Generalised Pareto distribution function. .. py:function:: sumlogs(x, axis=None, out=None) Sum of vector where numbers are represented by their logarithms. Calculates ``np.log(np.sum(np.exp(x), axis=axis))`` in such a fashion that it works even when elements have large magnitude.