This vignette demonstrates how to write a Stan program that computes and stores the pointwise log-likelihood required for using the **loo** package. The other vignettes included with the package demonstrate additional functionality.

Some sections from this vignette are excerpted from our papers

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., Gelman, A., and Gabry, J. (2017). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.04544.

which provide important background for understanding the methods implemented in the package.

This example comes from a survey of residents from a small area in Bangladesh that was affected by arsenic in drinking water. Respondents with elevated arsenic levels in their wells were asked if they were interested in getting water from a neighbor’s well, and a series of logistic regressions were fit to predict this binary response given various information about the households (Gelman and Hill, 2007). Here we fit a model for the well-switching response given two predictors: the arsenic level of the water in the resident’s home, and the distance of the house from the nearest safe well.

The sample size in this example is \(N=3020\), which is not huge but is large enough that it is important to have a computational method for LOO that is fast for each data point. On the plus side, with such a large dataset, the influence of any given observation is small, and so the computations should be stable.

Here is the Stan code for fitting the logistic regression model, which we save in a file called `logistic.stan`

:

```
data {
int<lower=0> N; // number of data points
int<lower=0> P; // number of predictors (including intercept)
matrix[N,P] X; // predictors (including 1s for intercept)
int<lower=0,upper=1> y[N]; // binary outcome
}
parameters {
vector[P] beta;
}
model {
beta ~ normal(0, 1);
y ~ bernoulli_logit(X * beta);
}
generated quantities {
vector[N] log_lik;
for (n in 1:N) {
log_lik[n] = bernoulli_logit_lpmf(y[n] | X[n] * beta);
}
}
```

We have defined the log likelihood as a vector named `log_lik`

in the generated quantities block so that the individual terms will be saved by Stan. After running Stan, `log_lik`

can be extracted (using the `extract_log_lik`

function provided in the **loo** package) as an \(S \times N\) matrix, where \(S\) is the number of simulations (posterior draws) and \(N\) is the number of data points.

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

```
library("rstan")
# Prepare data
url <- "http://stat.columbia.edu/~gelman/arm/examples/arsenic/wells.dat"
wells <- read.table(url)
wells$dist100 <- with(wells, dist / 100)
X <- model.matrix(~ dist100 + arsenic, wells)
standata <- list(y = wells$switch, X = X, N = nrow(X), P = ncol(X))
# Fit model
fit_1 <- stan("logistic.stan", data = standata)
print(fit_1, pars = "beta")
```

```
mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
beta[1] 0.00 0 0.08 -0.16 -0.05 0.00 0.05 0.15 1964 1
beta[2] -0.89 0 0.10 -1.09 -0.96 -0.89 -0.82 -0.68 2048 1
beta[3] 0.46 0 0.04 0.38 0.43 0.46 0.49 0.54 2198 1
```

We can then use the **loo** package to compute the efficient PSIS-LOO approximation to exact LOO-CV:

```
library("loo")
# Extract pointwise log-likelihood and compute LOO
log_lik_1 <- extract_log_lik(fit_1, merge_chains = FALSE)
# as of loo v2.0.0 we can optionally provide relative effective sample sizes
# when calling loo, which allows for better estimates of the PSIS effective
# sample sizes and Monte Carlo error
r_eff <- relative_eff(exp(log_lik_1))
loo_1 <- loo(log_lik_1, r_eff = r_eff, cores = 2)
print(loo_1)
```

```
Computed from 4000 by 3020 log-likelihood matrix
Estimate SE
elpd_loo -1968.5 15.6
p_loo 3.2 0.1
looic 3937.0 31.2
------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.
```

The printed output from the `loo`

function shows the estimates \(\widehat{\mbox{elpd}}_{\rm loo}\) (expected log predictive density), \(\widehat{p}_{\rm loo}\) (effective number of parameters), and \({\rm looic} =-2\, \widehat{\mbox{elpd}}_{\rm loo}\) (the LOO information criterion).

The line at the bottom of the printed output provides information about the reliability of the LOO approximation (the interpretation of the \(k\) parameter is explained in the PSIS-LOO section in `help("loo-package")`

and in greater detail in Vehtari, Gelman, and Gabry (2017)). In this case the message tells us that all of the estimates for \(k\) are fine.

To compare this model to an alternative model for the same data we can use the `compare`

function in the **loo** package. First we’ll fit a second model to the well-switching data, using `log(arsenic)`

instead of `arsenic`

as a predictor:

```
standata$X[, "arsenic"] <- log(standata$X[, "arsenic"])
fit_2 <- stan(fit = fit_1, data = standata)
log_lik_2 <- extract_log_lik(fit_2, merge_chains = FALSE)
r_eff_2 <- relative_eff(exp(log_lik_2))
loo_2 <- loo(log_lik_2, r_eff = r_eff_2, cores = 2)
print(loo_2)
```

```
Computed from 4000 by 3020 log-likelihood matrix
Estimate SE
elpd_loo -1952.3 16.2
p_loo 3.1 0.1
looic 3904.6 32.4
------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.
```

We can now compare the models on LOO using the `compare`

function:

This new object, `comp`

, contains the estimated difference of expected leave-one-out prediction errors between the two models, along with the standard error:

```
elpd_diff se
16.2 4.4
```

The positive difference in `elpd`

(and its scale relative to the standard error) indicates a preference for the *second* model.

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

Stan Development Team (2017). *The Stan C++ Library, Version 2.17.0.* http://mc-stan.org

Stan Development Team (2018) *RStan: the R interface to Stan, Version 2.17.3.* http://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. online, arXiv preprint arXiv:1507.04544.

Vehtari, A., Gelman, A., and Gabry, J. (2017). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.