Truncated importance sampling (TIS)

Implementation of truncated (self-normalized) importance sampling (TIS), truncated at S^(1/2) as recommended by Ionides (2008).

tis(log_ratios, ...)

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

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

# S3 method for default
tis(log_ratios, ..., r_eff = 1)

Arguments

log_ratios

An array, matrix, or vector of importance ratios on the log scale (for Importance sampling 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.

r_eff

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 (T)IS effective sample sizes and Monte Carlo error estimates can be over-optimistic. If the posterior draws are (near) independent then r_eff=1 can be used. r_eff has to be a scalar (same value is used for all observations) or a vector with length equal to the number of observations. The default value is 1. See the relative_eff() helper function for computing r_eff.

cores

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.

• Note for Windows 10 users: it is strongly recommended to avoid using the .Rprofile file to set mc.cores (using the cores argument or setting mc.cores interactively or in a script is fine).

Value

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

log_weights

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

diagnostics

A named list containing one vector:

• pareto_k: Not used in tis, all set to 0.

• n_eff: Effective sample size estimates.

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

norm_const_log

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

r_eff

If specified, the user's r_eff argument.

tail_len

Not used for tis.

dims

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

method

Method used for importance sampling, here tis.

Methods (by class)

• tis(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.

• tis(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.

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

References

Ionides, Edward L. (2008). Truncated importance sampling. Journal of Computational and Graphical Statistics 17(2): 295--311.

See also

• psis() for approximate LOO-CV using PSIS.

• loo() for approximate LOO-CV.

• pareto-k-diagnostic for PSIS diagnostics.

Examples

log_ratios <- -1 * example_loglik_array()
r_eff <- relative_eff(exp(-log_ratios))
tis_result <- tis(log_ratios, r_eff = r_eff)
str(tis_result)
#> List of 2
#>  $ log_weights: num [1:1000, 1:32] 2.37 2.12 2.24 2.41 2.25 ...
#>  $ diagnostics:List of 3
#>   ..$ pareto_k: num [1:32] 0 0 0 0 0 0 0 0 0 0 ...
#>   ..$ n_eff   : num [1:32] 901 923 930 896 896 ...
#>   ..$ r_eff   : num [1:32] 0.933 0.939 0.968 0.913 0.911 ...
#>  - attr(*, "norm_const_log")= num [1:32] 9.28 9.04 9.24 9.09 9 ...
#>  - attr(*, "tail_len")= num [1:32] 99 98 97 100 100 102 99 100 103 98 ...
#>  - attr(*, "r_eff")= num [1:32] 0.933 0.939 0.968 0.913 0.911 ...
#>  - attr(*, "dims")= int [1:2] 1000 32
#>  - attr(*, "method")= chr "tis"
#>  - attr(*, "class")= chr [1:3] "tis" "importance_sampling" "list"

# extract smoothed weights
lw <- weights(tis_result) # default args are log=TRUE, normalize=TRUE
ulw <- weights(tis_result, normalize=FALSE) # unnormalized log-weights

w <- weights(tis_result, log=FALSE) # normalized weights (not log-weights)
uw <- weights(tis_result, log=FALSE, normalize = FALSE) # unnormalized weights