Avoiding model refits in leave-one-out cross-validation with moment matching

Topi Paananen, Paul Bürkner, Aki Vehtari and Jonah Gabry

2020-12-07

Introduction

This vignette demonstrates how to improve the Monte Carlo sampling accuracy of leave-one-out cross-validation with the loo package and Stan. The loo package automatically monitors the sampling accuracy using Pareto \(k\) diagnostics for each observation. Here, we present a method for quickly improving the accuracy when the Pareto diagnostics indicate problems. This is done by performing some additional computations using the existing posterior sample. If successful, this will decrease the Pareto \(k\) values, making the model assessment more reliable. loo also stores the original Pareto \(k\) values with the name influence_pareto_k which are not changed. They can be used as a diagnostic of how much each observation influences the posterior distribution.

The methodology presented is based on the paper

More information about the Pareto \(k\) diagnostics is given in the following papers

Example: Eradication of Roaches

We will use the same example as in the vignette Using the loo package (version >= 2.0.0). See the demo for a description of the problem and data. We will use the same Poisson regression model as in the case study.

Coding the Stan model

Here is the Stan code for fitting the Poisson regression model, which we will use for modeling the number of roaches.

stancode <- "
data {
  int<lower=1> K;
  int<lower=1> N;
  matrix[N,K] x;
  int y[N];
  vector[N] offset;

  real beta_prior_scale;
  real alpha_prior_scale;
}
parameters {
  vector[K] beta;
  real intercept;
}
model {
  y ~ poisson(exp(x * beta + intercept + offset));
  beta ~ normal(0,beta_prior_scale);
  intercept ~ normal(0,alpha_prior_scale);
}
generated quantities {
  vector[N] log_lik;
  for (n in 1:N)
    log_lik[n] = poisson_lpmf(y[n] | exp(x[n] * beta + intercept + offset[n]));
}
"

Following the usual approach recommended in Writing Stan programs for use with the loo package, we compute the log-likelihood for each observation in the generated quantities block of the Stan program.

Setup

In addition to loo, we load the rstan package for fitting the model, and the rstanarm package for the data.

library("rstan")
library("loo")
seed <- 9547
set.seed(seed)

Fitting the model with RStan

Next we fit the model in Stan using the rstan package:

# Prepare data
data(roaches, package = "rstanarm")
roaches$roach1 <- sqrt(roaches$roach1)
y <- roaches$y
x <- roaches[,c("roach1", "treatment", "senior")]
offset <- log(roaches[,"exposure2"])
n <- dim(x)[1]
k <- dim(x)[2]

standata <- list(N = n, K = k, x = as.matrix(x), y = y, offset = offset, beta_prior_scale = 2.5, alpha_prior_scale = 5.0)

# Compile
stanmodel <- stan_model(model_code = stancode)

# Fit model
fit <- sampling(stanmodel, data = standata, seed = seed, refresh = 0)
print(fit, pars = "beta")
Inference for Stan model: 73ed397b74a7b4968fc8fce8b1300023.
4 chains, each with iter=2000; warmup=1000; thin=1; 
post-warmup draws per chain=1000, total post-warmup draws=4000.

         mean se_mean   sd  2.5%   25%   50%   75% 97.5% n_eff Rhat
beta[1]  0.16       0 0.00  0.16  0.16  0.16  0.16  0.16  2618    1
beta[2] -0.57       0 0.02 -0.62 -0.58 -0.57 -0.55 -0.52  2006    1
beta[3] -0.31       0 0.03 -0.38 -0.34 -0.31 -0.29 -0.25  2408    1

Samples were drawn using NUTS(diag_e) at Mon Dec  7 14:04:39 2020.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).

Let us now evaluate the predictive performance of the model using loo().

loo1 <- loo(fit)
Warning: Some Pareto k diagnostic values are too high. See help('pareto-k-diagnostic') for details.
loo1

Computed from 4000 by 262 log-likelihood matrix

         Estimate     SE
elpd_loo  -5459.4  694.7
p_loo       251.7   53.7
looic     10918.9 1389.3
------
Monte Carlo SE of elpd_loo is NA.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     235   89.7%   270       
 (0.5, 0.7]   (ok)        10    3.8%   65        
   (0.7, 1]   (bad)        9    3.4%   23        
   (1, Inf)   (very bad)   8    3.1%   1         
See help('pareto-k-diagnostic') for details.

The loo() function output warnings that there are some observations which are highly influential, and thus the accuracy of importance sampling is compromised as indicated by the large Pareto \(k\) diagnostic values (> 0.7). As discussed in the vignette Using the loo package (version >= 2.0.0), this may be an indication of model misspecification. Despite that, it is still beneficial to be able to evaluate the predictive performance of the model accurately.

Moment matching correction for importance sampling

To improve the accuracy of the loo() result above, we could perform leave-one-out cross-validation by explicitly leaving out single observations and refitting the model using MCMC repeatedly. However, the Pareto \(k\) diagnostics indicate that there are 19 observations which are problematic. This would require 19 model refits which may require a lot of computation time.

Instead of refitting with MCMC, we can perform a faster moment matching correction to the importance sampling for the problematic observations. This can be done with the loo_moment_match() function in the loo package, which takes our existing loo object as input and modifies it. The moment matching requires some evaluations of the model posterior density. For models fitted with rstan, this can be conveniently done by using the existing stanfit object.

First, we show how the moment matching can be used for a model fitted using rstan. It only requires setting the argument moment_match to TRUE in the loo() function. Optionally, you can also set the argument k_threshold which determines the Pareto \(k\) threshold, above which moment matching is used. By default, it operates on all observations whose Pareto \(k\) value is larger than 0.7.

# available in rstan >= 2.21
loo2 <- loo(fit, moment_match = TRUE)
Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.
loo2

Computed from 4000 by 262 log-likelihood matrix

         Estimate     SE
elpd_loo  -5478.5  700.2
p_loo       253.5   57.5
looic     10957.1 1400.3
------
Monte Carlo SE of elpd_loo is 0.4.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     250   95.4%   127       
 (0.5, 0.7]   (ok)        12    4.6%   65        
   (0.7, 1]   (bad)        0    0.0%   <NA>      
   (1, Inf)   (very bad)   0    0.0%   <NA>      

All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.

After the moment matching, all observations have the diagnostic Pareto \(k\) less than 0.7, meaning that the estimates are now reliable. The total elpd_loo estimate also changed from -5457.8 to -5478.5, showing that before moment matching, loo() overestimated the predictive performance of the model.

The updated Pareto \(k\) values stored in loo2$diagnostics$pareto_k are considered algorithmic diagnostic values that indicate the sampling accuracy. The original Pareto \(k\) values are stored in loo2$pointwise[,"influence_pareto_k"] and these are not modified by the moment matching. These can be considered as diagnostics for how big influence each observation has on the posterior distribution. In addition to the Pareto \(k\) diagnostics, moment matching also updates the effective sample size estimates.

Using loo_moment_match() directly

The moment matching can also be performed by explicitly calling the function loo_moment_match(). This enables its use also for models that are not using rstan or another package with built-in support for loo_moment_match(). To use loo_moment_match(), the user must give the model object x, the loo object, and 5 helper functions as arguments to loo_moment_match(). The helper functions are

Next, we show how the helper functions look like for RStan objects, and show an example of using loo_moment_match() directly. For stanfit objects from rstan objects, the functions look like this:

# create a named list of draws for use with rstan methods
.rstan_relist <- function(x, skeleton) {
  out <- utils::relist(x, skeleton)
  for (i in seq_along(skeleton)) {
    dim(out[[i]]) <- dim(skeleton[[i]])
  }
  out
}

# rstan helper function to get dims of parameters right
.create_skeleton <- function(pars, dims) {
  out <- lapply(seq_along(pars), function(i) {
    len_dims <- length(dims[[i]])
    if (len_dims < 1) return(0)
    return(array(0, dim = dims[[i]]))
  })
  names(out) <- pars
  out
}

# extract original posterior draws
post_draws_stanfit <- function(x, ...) {
  as.matrix(x)
}

# compute a matrix of log-likelihood values for the ith observation
# matrix contains information about the number of MCMC chains
log_lik_i_stanfit <- function(x, i, parameter_name = "log_lik", ...) {
  loo::extract_log_lik(x, parameter_name, merge_chains = FALSE)[, , i]
}

# transform parameters to the unconstraint space
unconstrain_pars_stanfit <- function(x, pars, ...) {
  skeleton <- .create_skeleton(x@sim$pars_oi, x@par_dims[x@sim$pars_oi])
  upars <- apply(pars, 1, FUN = function(theta) {
    rstan::unconstrain_pars(x, .rstan_relist(theta, skeleton))
  })
  # for one parameter models
  if (is.null(dim(upars))) {
    dim(upars) <- c(1, length(upars))
  }
  t(upars)
}

# compute log_prob for each posterior draws on the unconstrained space
log_prob_upars_stanfit <- function(x, upars, ...) {
  apply(upars, 1, rstan::log_prob, object = x,
        adjust_transform = TRUE, gradient = FALSE)
}

# compute log_lik values based on the unconstrained parameters
log_lik_i_upars_stanfit <- function(x, upars, i, parameter_name = "log_lik",
                                  ...) {
  S <- nrow(upars)
  out <- numeric(S)
  for (s in seq_len(S)) {
    out[s] <- rstan::constrain_pars(x, upars = upars[s, ])[[parameter_name]][i]
  }
  out
}

Using these function, we can call loo_moment_match() to update the existing loo object.

loo3 <- loo::loo_moment_match.default(
  x = fit,
  loo = loo1,
  post_draws = post_draws_stanfit,
  log_lik_i = log_lik_i_stanfit,
  unconstrain_pars = unconstrain_pars_stanfit,
  log_prob_upars = log_prob_upars_stanfit,
  log_lik_i_upars = log_lik_i_upars_stanfit
)
Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.
loo3

Computed from 4000 by 262 log-likelihood matrix

         Estimate     SE
elpd_loo  -5478.5  700.2
p_loo       253.5   57.5
looic     10957.1 1400.3
------
Monte Carlo SE of elpd_loo is 0.4.

Pareto k diagnostic values:
                         Count Pct.    Min. n_eff
(-Inf, 0.5]   (good)     250   95.4%   127       
 (0.5, 0.7]   (ok)        12    4.6%   65        
   (0.7, 1]   (bad)        0    0.0%   <NA>      
   (1, Inf)   (very bad)   0    0.0%   <NA>      

All Pareto k estimates are ok (k < 0.7).
See help('pareto-k-diagnostic') for details.

As expected, the result is identical to the previous result of loo2 <- loo(fit, moment_match = TRUE).

References

Gelman, A., and Hill, J. (2007). Data Analysis Using Regression and Multilevel Hierarchical Models. Cambridge University Press.

Stan Development Team (2020) RStan: the R interface to Stan, Version 2.21.1 https://mc-stan.org

Paananen, T., Piironen, J., Buerkner, P.-C., Vehtari, A. (2020). Implicitly Adaptive Importance Sampling. arXiv preprint arXiv:1906.08850.

Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. Links: published | arXiv preprint.

Vehtari, A., Simpson, D., Gelman, A., Yao, Y., and Gabry, J. (2019). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.