`vignettes/loo2-elpd.Rmd`

`loo2-elpd.Rmd`

This vignette demonstrates how to do holdout validation and K-fold cross-validation with **loo** for a Stan program.

This vignette uses the same example as in the vignettes *Using the loo package (version >= 2.0.0)* and *Avoiding model refits in leave-one-out cross-validation with moment matching*.

Here is the Stan code for fitting a Poisson regression model:

```
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.

For this approach, the model is first fit to the “train” data and then is evaluated on the held-out “test” data.

The data is divided between train (80% of the data) and test (20%):

```
# Prepare data
data(roaches, package = "rstanarm")
roaches$roach1 <- sqrt(roaches$roach1)
roaches$offset <- log(roaches[,"exposure2"])
# 20% of the data goes to the test set:
roaches$test <- 0
roaches$test[sample(.2 * seq_len(nrow(roaches)))] <- 1
# data to "train" the model
data_train <- list(y = roaches$y[roaches$test == 0],
x = as.matrix(roaches[roaches$test == 0,
c("roach1", "treatment", "senior")]),
N = nrow(roaches[roaches$test == 0,]),
K = 3,
offset = roaches$offset[roaches$test == 0],
beta_prior_scale = 2.5,
alpha_prior_scale = 5.0
)
# data to "test" the model
data_test <- list(y = roaches$y[roaches$test == 1],
x = as.matrix(roaches[roaches$test == 1,
c("roach1", "treatment", "senior")]),
N = nrow(roaches[roaches$test == 1,]),
K = 3,
offset = roaches$offset[roaches$test == 1],
beta_prior_scale = 2.5,
alpha_prior_scale = 5.0
)
```

Next we fit the model to the “test” data in Stan using the **rstan** package:

```
# Compile
stanmodel <- stan_model(model_code = stancode)
# Fit model
fit <- sampling(stanmodel, data = data_train, seed = seed, refresh = 0)
```

We recompute the generated quantities using the posterior draws conditional on the training data, but we now pass in the held-out data to get the log predictive densities for the test data. Because we are using independent data, the log predictive density coincides with the log likelihood of the test data.

```
gen_test <- gqs(stanmodel, draws = as.matrix(fit), data= data_test)
log_pd <- extract_log_lik(gen_test)
```

Now we evaluate the predictive performance of the model on the test data using `elpd()`

.

`(elpd_holdout <- elpd(log_pd))`

```
Computed from 4000 by 52 log-likelihood matrix using the generic elpd function
Estimate SE
elpd -1736.7 288.9
ic 3473.5 577.9
```

When one wants to compare different models, the function `loo_compare()`

can be used to assess the difference in performance.

For this approach the data is divided into folds, and each time one fold is tested while the rest of the data is used to fit the model (see Vehtari et al., 2017).

We use the data that is already pre-processed and we divide it in 10 random folds using `kfold_split_random`

```
# Prepare data
roaches$fold <- kfold_split_random(K = 10, N = nrow(roaches))
```

We now loop over the 10 folds. In each fold we do the following. First, we fit the model to all the observations except the ones belonging to the left-out fold. Second, we compute the log pointwise predictive densities for the left-out fold. Last, we store the predictive density for the observations of the left-out fold in a matrix. The output of this loop is a matrix of the log pointwise predictive densities of all the observations.

```
# Prepare a matrix with the number of post-warmup iterations by number of observations:
log_pd_kfold <- matrix(nrow = 4000, ncol = nrow(roaches))
# Loop over the folds
for(k in 1:10){
data_train <- list(y = roaches$y[roaches$fold != k],
x = as.matrix(roaches[roaches$fold != k,
c("roach1", "treatment", "senior")]),
N = nrow(roaches[roaches$fold != k,]),
K = 3,
offset = roaches$offset[roaches$fold != k],
beta_prior_scale = 2.5,
alpha_prior_scale = 5.0
)
data_test <- list(y = roaches$y[roaches$fold == k],
x = as.matrix(roaches[roaches$fold == k,
c("roach1", "treatment", "senior")]),
N = nrow(roaches[roaches$fold == k,]),
K = 3,
offset = roaches$offset[roaches$fold == k],
beta_prior_scale = 2.5,
alpha_prior_scale = 5.0
)
fit <- sampling(stanmodel, data = data_train, seed = seed, refresh = 0)
gen_test <- gqs(stanmodel, draws = as.matrix(fit), data= data_test)
log_pd_kfold[, roaches$fold == k] <- extract_log_lik(gen_test)
}
```

Now we evaluate the predictive performance of the model on the 10 folds using `elpd()`

.

`(elpd_kfold <- elpd(log_pd_kfold))`

```
Computed from 4000 by 262 log-likelihood matrix using the generic elpd function
Estimate SE
elpd -5552.6 727.7
ic 11105.3 1455.4
```

If one wants to compare several models (with `loo_compare`

), one should use the same folds for all the different models.

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

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.