Implementation of Pareto smoothed importance sampling (PSIS), a method for stabilizing importance ratios. The version of PSIS implemented here corresponds to the algorithm presented in Vehtari, Gelman and Gabry (2017b). For PSIS diagnostics see the pareto-k-diagnostic page.

psis(log_ratios, ...)

# S3 method for array
psis(log_ratios, ..., r_eff = NULL,
  cores = getOption("mc.cores", 1))

# S3 method for matrix
psis(log_ratios, ..., r_eff = NULL,
  cores = getOption("mc.cores", 1))

# S3 method for default
psis(log_ratios, ..., r_eff = NULL)

# S3 method for psis
weights(object, ..., log = TRUE, normalize = TRUE)



An array, matrix, or vector of importance ratios on the log scale (for PSIS-LOO these are negative log-likelihood values). See the Methods (by class) section below for a detailed description of how to specify the inputs for each method.


Arguments passed on to the various methods.


Vector of relative effective sample size estimates containing one element per observation. The values provided should be the relative effective sample sizes of 1/exp(log_ratios) (i.e., 1/ratios). This is related to the relative efficiency of estimating the normalizing term in self-normalizing importance sampling. If r_eff is not provided then the reported PSIS effective sample sizes and Monte Carlo error estimates will be over-optimistic. See the relative_eff helper function for computing r_eff. If using psis with draws of the log_ratios not obtained from MCMC then the warning message thrown when not specifying r_eff can be disabled by setting r_eff to NA.


The number of cores to use for parallelization. This defaults to the option mc.cores which can be set for an entire R session by options(mc.cores = NUMBER). The old option loo.cores is now deprecated but will be given precedence over mc.cores until loo.cores is removed in a future release. As of version 2.0.0 the default is now 1 core if mc.cores is not set, but we recommend using as many (or close to as many) cores as possible.


For the weights method, an object returned by psis (a list with class "psis").


For the weights method, should the weights be returned on the log scale? Defaults to TRUE.


For the weights method, should the weights be normalized? Defaults to TRUE.


The psis methods return an object of class "psis", which is a named list with the following components:


Vector or matrix of smoothed (and truncated) but unnormalized log weights. To get normalized weights use the weights method provided for objects of class "psis".


A named list containing two vectors:

  • pareto_k: Estimates of the shape parameter \(k\) of the generalized Pareto distribution. See the pareto-k-diagnostic page for details.

  • n_eff: PSIS effective sample size estimates.

Objects of class "psis" also have the following attributes:


Vector of precomputed values of colLogSumExps(log_weights) that are used internally by the weights method to normalize the log weights.


Vector of tail lengths used for fitting the generalized Pareto distribution.


If specified, the user's r_eff argument.


Integer vector of length 2 containing S (posterior sample size) and N (number of observations).

The weights method returns an object with the same dimensions as the log_weights component of the "psis" object. The normalize and log arguments control whether the returned weights are normalized and whether or not to return them on the log scale.

Methods (by class)

  • array: An \(I\) by \(C\) by \(N\) array, where \(I\) is the number of MCMC iterations per chain, \(C\) is the number of chains, and \(N\) is the number of data points.

  • matrix: An \(S\) by \(N\) matrix, where \(S\) is the size of the posterior sample (with all chains merged) and \(N\) is the number of data points.

  • default: A vector of length \(S\) (posterior sample size).


Vehtari, A., Gelman, A., and Gabry, J. (2017a). 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. (published version, arXiv preprint).

Vehtari, A., Gelman, A., and Gabry, J. (2017b). Pareto smoothed importance sampling. arXiv preprint:

See also


log_ratios <- -1 * example_loglik_array() r_eff <- relative_eff(exp(-log_ratios)) psis_result <- psis(log_ratios, r_eff = r_eff) str(psis_result)
#> List of 2 #> $ log_weights: num [1:1000, 1:32] -0.694 -0.941 -0.818 -0.649 -0.816 ... #> $ diagnostics:List of 2 #> ..$ pareto_k: num [1:32] 0.0447 -0.0475 0.0696 -0.052 -0.1073 ... #> ..$ n_eff : num [1:32] 900 888 929 895 858 ... #> - attr(*, "norm_const_log")= num [1:32] 6.22 6.47 6.17 6.47 6.52 ... #> - attr(*, "tail_len")= num [1:32] 99 100 97 100 102 104 99 100 103 99 ... #> - attr(*, "r_eff")= num [1:32] 0.933 0.904 0.968 0.913 0.873 ... #> - attr(*, "dims")= int [1:2] 1000 32 #> - attr(*, "class")= chr [1:2] "psis" "list"
# extract smoothed weights lw <- weights(psis_result) # default args are log=TRUE, normalize=TRUE ulw <- weights(psis_result, normalize=FALSE) # unnormalized log-weights w <- weights(psis_result, log=FALSE) # normalized weights (not log-weights) uw <- weights(psis_result, log=FALSE, normalize = FALSE) # unnormalized weights