When computing ELPD-based LOO-CV for a Bayesian model we need to compute the log leave-one-out predictive densities \(\log{p(y_i | y_{-i})}\) for every response value \(y_i, \: i = 1, \ldots, N\), where \(y_{-i}\) denotes all response values except observation \(i\). To obtain \(p(y_i | y_{-i})\), we need to have access to the pointwise likelihood \(p(y_i\,|\, y_{-i}, \theta)\) and integrate over the model parameters \(\theta\):

\[ p(y_i\,|\,y_{-i}) = \int p(y_i\,|\, y_{-i}, \theta) \, p(\theta\,|\, y_{-i}) \,d \theta \]

Here, \(p(\theta\,|\, y_{-i})\) is the leave-one-out posterior distribution for \(\theta\), that is, the posterior distribution for \(\theta\) obtained by fitting the model while holding out the \(i\)th observation (we will later show how refitting the model to data \(y_{-i}\) can be avoided).

If the observation model is formulated directly as the product of the pointwise observation models, we call it a *factorized* model. In this case, the likelihood is also the product of the pointwise likelihood contributions \(p(y_i\,|\, y_{-i}, \theta)\). To better illustrate possible structures of the observation models, we formally divide \(\theta\) into two parts, observation-specific latent variables \(f = (f_1, \ldots, f_N)\) and hyperparameters \(\psi\), so that \(p(y_i\,|\, y_{-i}, \theta) = p(y_i\,|\, y_{-i}, f_i, \psi)\). Depending on the model, one of the two parts of \(\theta\) may also be empty. In very simple models, such as linear regression models, latent variables are not explicitly presented and response values are conditionally independent given \(\psi\), so that \(p(y_i\,|\, y_{-i}, f_i, \psi) = p(y_i \,|\, \psi)\). The full likelihood can then be written in the familiar form

\[ p(y \,|\, \psi) = \prod_{i=1}^N p(y_i \,|\, \psi), \]

where \(y = (y_1, \ldots, y_N)\) denotes the vector of all responses. When the likelihood factorizes this way, the conditional pointwise log-likelihood can be obtained easily by computing \(p(y_i\,|\, \psi)\) for each \(i\) with computational cost \(O(n)\).

Yet, there are several reasons why a *non-factorized* observation model may be necessary or preferred. In non-factorized models, the joint likelihood of the response values \(p(y \,|\, \theta)\) is not factorized into observation-specific components, but rather given directly as one joint expression. For some models, an analytic factorized formulation is simply not available in which case we speak of a *non-factorizable* model. Even in models whose observation model can be factorized in principle, it may still be preferable to use a non-factorized form for reasons of efficiency and numerical stability (Bürkner et al. 2020).

Whether a non-factorized model is used by necessity or for efficiency and stability, it comes at the cost of having no direct access to the leave-one-out predictive densities and thus to the overall leave-one-out predictive accuracy. In theory, we can express the observation-specific likelihoods in terms of the joint likelihood via

\[ p(y_i \,|\, y_{i-1}, \theta) = \frac{p(y \,|\, \theta)}{p(y_{-i} \,|\, \theta)} = \frac{p(y \,|\, \theta)}{\int p(y \,|\, \theta) \, d y_i}, \]

but the expression on the right-hand side may not always have an analytical solution. Computing \(\log p(y_i \,|\, y_{-i}, \theta)\) for non-factorized models is therefore often impossible, or at least inefficient and numerically unstable. However, there is a large class of multivariate normal and Student-\(t\) models for which there are efficient analytical solutions available.

More details can be found in our paper about LOO-CV for non-factorized models (Bürkner, Gabry, & Vehtari, 2020), which is available as a preprint on arXiv (https://arxiv.org/abs/1810.10559).

In this vignette, we will focus on non-factorized multivariate normal models. Based on results of Sundararajan and Keerthi (2001), Bürkner et al. (2020) show that, for multivariate normal models with coriance matrix \(C\), the LOO predictive mean and standard deviation can be computed as follows:

\[\begin{align} \mu_{\tilde{y},-i} &= y_i-\bar{c}_{ii}^{-1} g_i \nonumber \\ \sigma_{\tilde{y},-i} &= \sqrt{\bar{c}_{ii}^{-1}}, \end{align}\] where \(g_i\) and \(\bar{c}_{ii}\) are \[\begin{align} g_i &= \left[C^{-1} y\right]_i \nonumber \\ \bar{c}_{ii} &= \left[C^{-1}\right]_{ii}. \end{align}\]

Using these results, the log predictive density of the \(i\)th observation is then computed as

\[ \log p(y_i \,|\, y_{-i},\theta) = - \frac{1}{2}\log(2\pi) - \frac{1}{2}\log \sigma^2_{-i} - \frac{1}{2}\frac{(y_i-\mu_{-i})^2}{\sigma^2_{-i}}. \]

Expressing this same equation in terms of \(g_i\) and \(\bar{c}_{ii}\), the log predictive density becomes:

\[ \log p(y_i \,|\, y_{-i},\theta) = - \frac{1}{2}\log(2\pi) + \frac{1}{2}\log \bar{c}_{ii} - \frac{1}{2}\frac{g_i^2}{\bar{c}_{ii}}. \] (Note that Vehtari et al. (2016) has a typo in the corresponding Equation 34.)

From these equations we can now derive a recipe for obtaining the conditional pointwise log-likelihood for *all* models that can be expressed conditionally in terms of a multivariate normal with invertible covariance matrix \(C\).

The above LOO equations for multivariate normal models are conditional on parameters \(\theta\). Therefore, to obtain the leave-one-out predictive density \(p(y_i \,|\, y_{-i})\) we need to integrate over \(\theta\),

\[ p(y_i\,|\,y_{-i}) = \int p(y_i\,|\,y_{-i}, \theta) \, p(\theta\,|\,y_{-i}) \,d\theta. \]

Here, \(p(\theta\,|\,y_{-i})\) is the leave-one-out posterior distribution for \(\theta\), that is, the posterior distribution for \(\theta\) obtained by fitting the model while holding out the \(i\)th observation.

To avoid the cost of sampling from \(N\) leave-one-out posteriors, it is possible to take the posterior draws \(\theta^{(s)}, \, s=1,\ldots,S\), from the posterior \(p(\theta\,|\,y)\), and then approximate the above integral using integrated importance sampling (Vehtari et al., 2016, Section 3.6.1):

\[ p(y_i\,|\,y_{-i}) \approx \frac{ \sum_{s=1}^S p(y_i\,|\,y_{-i},\,\theta^{(s)}) \,w_i^{(s)}}{ \sum_{s=1}^S w_i^{(s)}}, \]

where \(w_i^{(s)}\) are importance weights. First we compute the raw importance ratios

\[ r_i^{(s)} \propto \frac{1}{p(y_i \,|\, y_{-i}, \,\theta^{(s)})}, \]

and then stabilize them using Pareto smoothed importance sampling (PSIS, Vehtari et al, 2019) to obtain the weights \(w_i^{(s)}\). The resulting approximation is referred to as PSIS-LOO (Vehtari et al, 2017).

In order to validate the approximate LOO procedure, and also in order to allow exact computations to be made for a small number of leave-one-out folds for which the Pareto \(k\) diagnostic (Vehtari et al, 2019) indicates an unstable approximation, we need to consider how we might to do *exact* leave-one-out CV for a non-factorized model. In the case of a Gaussian process that has the marginalization property, we could just drop the one row and column of \(C\) corresponding to the held out out observation. This does not hold in general for multivariate normal models, however, and to keep the original prior we may need to maintain the full covariance matrix \(C\) even when one of the observations is left out.

The solution is to model \(y_i\) as a missing observation and estimate it along with all of the other model parameters. For a conditional multivariate normal model, \(\log p(y_i\,|\,y_{-i})\) can be computed as follows. First, we model \(y_i\) as missing and denote the corresponding parameter \(y_i^{\mathrm{mis}}\). Then, we define

\[ y_{\mathrm{mis}(i)} = (y_1, \ldots, y_{i-1}, y_i^{\mathrm{mis}}, y_{i+1}, \ldots, y_N). \] to be the same as the full set of observations \(y\), except replacing \(y_i\) with the parameter \(y_i^{\mathrm{mis}}\).

Second, we compute the LOO predictive mean and standard deviations as above, but replace \(y\) with \(y_{\mathrm{mis}(i)}\) in the computation of \(\mu_{\tilde{y},-i}\):

\[ \mu_{\tilde{y},-i} = y_{{\mathrm{mis}}(i)}-\bar{c}_{ii}^{-1}g_i, \]

where in this case we have

\[ g_i = \left[ C^{-1} y_{\mathrm{mis}(i)} \right]_i. \]

The conditional log predictive density is then computed with the above \(\mu_{\tilde{y},-i}\) and the left out observation \(y_i\):

\[ \log p(y_i\,|\,y_{-i},\theta) = - \frac{1}{2}\log(2\pi) - \frac{1}{2}\log \sigma^2_{\tilde{y},-i} - \frac{1}{2}\frac{(y_i-\mu_{\tilde{y},-i})^2}{\sigma^2_{\tilde{y},-i}}. \]

Finally, the leave-one-out predictive distribution can then be estimated as

\[ p(y_i\,|\,y_{-i}) \approx \sum_{s=1}^S p(y_i\,|\,y_{-i}, \theta_{-i}^{(s)}), \]

where \(\theta_{-i}^{(s)}\) are draws from the posterior distribution \(p(\theta\,|\,y_{\mathrm{mis}(i)})\).

A common non-factorized multivariate normal model is the simultaneously autoregressive (SAR) model, which is frequently used for spatially correlated data. The lagged SAR model is defined as

\[ y = \rho Wy + \eta + \epsilon \] or equivalently \[ (I - \rho W)y = \eta + \epsilon, \] where \(\rho\) is the spatial correlation parameter and \(W\) is a user-defined weight matrix. The matrix \(W\) has entries \(w_{ii} = 0\) along the diagonal and the off-diagonal entries \(w_{ij}\) are larger when areas \(i\) and \(j\) are closer to each other. In a linear model, the predictor term \(\eta\) is given by \(\eta = X \beta\) with design matrix \(X\) and regression coefficients \(\beta\). However, since the above equation holds for arbitrary \(\eta\), these results are not restricted to linear models.

If we have \(\epsilon \sim {\mathrm N}(0, \,\sigma^2 I)\), it follows that \[
(I - \rho W)y \sim {\mathrm N}(\eta, \sigma^2 I),
\] which corresponds to the following log PDF coded in **Stan**:

```
/**
* Normal log-pdf for spatially lagged responses
*
* @param y Vector of response values.
* @param mu Mean parameter vector.
* @param sigma Positive scalar residual standard deviation.
* @param rho Positive scalar autoregressive parameter.
* @param W Spatial weight matrix.
*
* @return A scalar to be added to the log posterior.
*/
normal_lagsar_lpdf(vector y, vector mu, real sigma,
real
real rho, matrix W) {= rows(y);
int N = 1 / square(sigma);
real inv_sigma2 = -rho * W;
matrix[N, N] W_tilde
vector[N] half_pred;
for (n in 1:N) W_tilde[n,n] += 1;
= W_tilde * (y - mdivide_left(W_tilde, mu));
half_pred
0.5 * log_determinant(crossprod(W_tilde) * inv_sigma2) -
return 0.5 * dot_self(half_pred) * inv_sigma2;
}
```

For the purpose of computing LOO-CV, it makes sense to rewrite the SAR model in slightly different form. Conditional on \(\rho\), \(\eta\), and \(\sigma\), if we write

\[\begin{align} y-(I-\rho W)^{-1}\eta &\sim {\mathrm N}(0, \sigma^2(I-\rho W)^{-1}(I-\rho W)^{-T}), \end{align}\] or more compactly, with \(\widetilde{W}=(I-\rho W)\), \[\begin{align} y-\widetilde{W}^{-1}\eta &\sim {\mathrm N}(0, \sigma^2(\widetilde{W}^{T}\widetilde{W})^{-1}), \end{align}\]

then this has the same form as the zero mean Gaussian process from above. Accordingly, we can compute the leave-one-out predictive densities with the equations from Sundararajan and Keerthi (2001), replacing \(y\) with \((y-\widetilde{W}^{-1}\eta)\) and taking the covariance matrix \(C\) to be \(\sigma^2(\widetilde{W}^{T}\widetilde{W})^{-1}\).

In order to demonstrate how to carry out the computations implied by these equations, we will first fit a lagged SAR model to data on crime in 49 different neighborhoods of Columbus, Ohio during the year 1980. The data was originally described in Aneslin (1988) and ships with the **spdep** R package.

In addition to the **loo** package, for this analysis we will use the **brms** interface to Stan to generate a Stan program and fit the model, and also the **bayesplot** and **ggplot2** packages for plotting.

```
library("loo")
library("brms")
library("bayesplot")
library("ggplot2")
color_scheme_set("brightblue")
theme_set(theme_default())
<- 10001
SEED set.seed(SEED) # only sets seed for R (seed for Stan set later)
# loads COL.OLD data frame and COL.nb neighbor list
data(oldcol, package = "spdep")
```

The three variables in the data set relevant to this example are:

`CRIME`

: the number of residential burglaries and vehicle thefts per thousand households in the neighbood`HOVAL`

: housing value in units of $1000 USD`INC`

: household income in units of $1000 USD

`str(COL.OLD[, c("CRIME", "HOVAL", "INC")])`

```
'data.frame': 49 obs. of 3 variables:
$ CRIME: num 18.802 32.388 38.426 0.178 15.726 ...
$ HOVAL: num 44.6 33.2 37.1 75 80.5 ...
$ INC : num 21.23 4.48 11.34 8.44 19.53 ...
```

We will also use the object `COL.nb`

, which is a list containing information about which neighborhoods border each other. From this list we will be able to construct the weight matrix to used to help account for the spatial dependency among the observations.

A model predicting `CRIME`

from `INC`

and `HOVAL`

, while accounting for the spatial dependency via an SAR structure, can be specified in **brms** as follows.

```
<- brm(
fit ~ INC + HOVAL + sar(COL.nb, type = "lag"),
CRIME data = COL.OLD,
data2 = list(COL.nb = COL.nb),
chains = 4,
seed = SEED
)
```

The code above fits the model in **Stan** using a log PDF equivalent to the `normal_lagsar_lpdf`

function we defined above. In the summary output below we see that both higher income and higher housing value predict lower crime rates in the neighborhood. Moreover, there seems to be substantial spatial correlation between adjacent neighborhoods, as indicated by the posterior distribution of the `lagsar`

parameter.

```
<- as.matrix(fit, pars = "lagsar")
lagsar <- quantile(lagsar, probs = c(0.25, 0.5, 0.75))
estimates mcmc_hist(lagsar) +
vline_at(estimates, linetype = 2, size = 1) +
ggtitle("lagsar: posterior median and 50% central interval")
```

After fitting the model, the next step is to compute the pointwise log-likelihood values needed for approximate LOO-CV. To do this we will use the recipe laid out in the previous sections.

```
<- as.data.frame(fit)
posterior <- fit$data$CRIME
y <- length(y)
N <- nrow(posterior)
S <- yloo <- sdloo <- matrix(nrow = S, ncol = N)
loglik
for (s in 1:S) {
<- posterior[s, ]
p <- p$b_Intercept + p$b_INC * fit$data$INC + p$b_HOVAL * fit$data$HOVAL
eta <- diag(N) - p$lagsar * spdep::nb2mat(COL.nb)
W_tilde <- t(W_tilde) %*% W_tilde / p$sigma^2
Cinv <- Cinv %*% (y - solve(W_tilde, eta))
g <- diag(Cinv)
cbar <- y - g / cbar
yloo[s, ] <- sqrt(1 / cbar)
sdloo[s, ] <- dnorm(y, yloo[s, ], sdloo[s, ], log = TRUE)
loglik[s, ]
}
# use loo for psis smoothing
<- -loglik
log_ratios <- psis(log_ratios) psis_result
```

The quality of the PSIS-LOO approximation can be investigated graphically by plotting the Pareto-k estimate for each observation. Ideally, they should not exceed \(0.5\), but in practice the algorithm turns out to be robust up to values of \(0.7\) (Vehtari et al, 2017, 2019). In the plot below, we see that the fourth observation is problematic and so may reduce the accuracy of the LOO-CV approximation.

`plot(psis_result, label_points = TRUE)`

We can also check that the conditional leave-one-out predictive distribution equations work correctly, for instance, using the last posterior draw:

```
<- yloo[S, ]
yloo_sub <- sdloo[S, ]
sdloo_sub <- data.frame(
df y = y,
yloo = yloo_sub,
ymin = yloo_sub - sdloo_sub * 2,
ymax = yloo_sub + sdloo_sub * 2
)ggplot(data=df, aes(x = y, y = yloo, ymin = ymin, ymax = ymax)) +
geom_errorbar(
width = 1,
color = "skyblue3",
position = position_jitter(width = 0.25)
+
) geom_abline(color = "gray30", size = 1.2) +
geom_point()
```

Finally, we use PSIS-LOO to approximate the expected log predictive density (ELPD) for new data, which we will validate using exact LOO-CV in the upcoming section.

`<- loo(loglik)) (psis_loo `

```
Computed from 4000 by 49 log-likelihood matrix
Estimate SE
elpd_loo -187.2 11.1
p_loo 8.3 5.4
looic 374.3 22.1
------
Monte Carlo SE of elpd_loo is NA.
Pareto k diagnostic values:
Count Pct. Min. n_eff
(-Inf, 0.5] (good) 47 95.9% 2240
(0.5, 0.7] (ok) 1 2.0% 819
(0.7, 1] (bad) 0 0.0% <NA>
(1, Inf) (very bad) 1 2.0% 15
See help('pareto-k-diagnostic') for details.
```

Exact LOO-CV for the above example is somewhat more involved, as we need to re-fit the model \(N\) times and each time model the held-out data point as a parameter. First, we create an empty dummy model that we will update below as we loop over the observations.

```
# see help("mi", "brms") for details on the mi() usage
<- brm(
fit_dummy | mi() ~ INC + HOVAL + sar(COL.nb, type = "lag"),
CRIME data = COL.OLD,
data2 = list(COL.nb = COL.nb),
chains = 0
)
```

Next, we fit the model \(N\) times, each time leaving out a single observation and then computing the log predictive density for that observation. For obvious reasons, this takes much longer than the approximation we computed above, but it is necessary in order to validate the approximate LOO-CV method. Thanks to the PSIS-LOO approximation, in general doing these slow exact computations can be avoided.

```
<- 500
S <- vector("list", N)
res <- matrix(nrow = S, ncol = N)
loglik for (i in seq_len(N)) {
<- COL.OLD
dat_mi $CRIME[i] <- NA
dat_mi<- update(fit_dummy, newdata = dat_mi,
fit_i # just for vignette
chains = 1, iter = S * 2)
<- as.data.frame(fit_i)
posterior <- sdloo <- rep(NA, S)
yloo for (s in seq_len(S)) {
<- posterior[s, ]
p <- y
y_miss_i <- p$Ymi
y_miss_i[i] <- p$b_Intercept + p$b_INC * fit_i$data$INC + p$b_HOVAL * fit_i$data$HOVAL
eta <- diag(N) - p$lagsar * spdep::nb2mat(COL.nb)
W_tilde <- t(W_tilde) %*% W_tilde / p$sigma^2
Cinv <- Cinv %*% (y_miss_i - solve(W_tilde, eta))
g <- diag(Cinv);
cbar <- y_miss_i[i] - g[i] / cbar[i]
yloo[s] <- sqrt(1 / cbar[i])
sdloo[s] <- dnorm(y[i], yloo[s], sdloo[s], log = TRUE)
loglik[s, i]
}<- rnorm(S, yloo, sdloo)
ypred <- data.frame(y = c(posterior$Ymi, ypred))
res[[i]] $type <- rep(c("pp", "loo"), each = S)
res[[i]]$obs <- i
res[[i]]
}<- do.call(rbind, res) res
```

A first step in the validation of the pointwise predictive density is to compare the distribution of the implied response values for the left-out observation to the distribution of the \(y_i^{\mathrm{mis}}\) posterior-predictive values estimated as part of the model. If the pointwise predictive density is correct, the two distributions should match very closely (up to sampling error). In the plot below, we overlay these two distributions for the first four observations and see that they match very closely (as is the case for all \(49\) observations of in this example).

```
<- res[res$obs %in% 1:4, ]
res_sub ggplot(res_sub, aes(y, fill = type)) +
geom_density(alpha = 0.6) +
facet_wrap("obs", scales = "fixed", ncol = 4)
```

In the final step, we compute the ELPD based on the exact LOO-CV and compare it to the approximate PSIS-LOO result computed earlier.

```
<- function(x) {
log_mean_exp # more stable than log(mean(exp(x)))
<- max(x)
max_x + log(sum(exp(x - max_x))) - log(length(x))
max_x
}<- apply(loglik, 2, log_mean_exp)
exact_elpds <- sum(exact_elpds)
exact_elpd round(exact_elpd, 1)
```

`[1] -189.3`

The results of the approximate and exact LOO-CV are similar but not as close as we would expect if there were no problematic observations. We can investigate this issue more closely by plotting the approximate against the exact pointwise ELPD values.

```
<- data.frame(
df approx_elpd = psis_loo$pointwise[, "elpd_loo"],
exact_elpd = exact_elpds
)ggplot(df, aes(x = approx_elpd, y = exact_elpd)) +
geom_abline(color = "gray30") +
geom_point(size = 2) +
geom_point(data = df[4, ], size = 3, color = "red3") +
xlab("Approximate elpds") +
ylab("Exact elpds") +
coord_fixed(xlim = c(-16, -3), ylim = c(-16, -3))
```

In the plot above the fourth data point —the observation flagged as problematic by the PSIS-LOO approximation— is colored in red and is the clear outlier. Otherwise, the correspondence between the exact and approximate values is strong. In fact, summing over the pointwise ELPD values and leaving out the fourth observation yields practically equivalent results for approximate and exact LOO-CV:

```
<- c(
without_pt_4 approx = sum(psis_loo$pointwise[-4, "elpd_loo"]),
exact = sum(exact_elpds[-4])
)round(without_pt_4, 1)
```

```
approx exact
-173.1 -173.0
```

From this we can conclude that the difference we found when including *all* observations does not indicate a bug in our implementation of the approximate LOO-CV but rather a violation of its assumptions.

So far, we have specified the models in brms and only used Stan implicitely behind the scenes. This allowed us to focus on the primary purpose of validating approximate LOO-CV for non-factorized models. However, we would also like to show how everything can be set up in Stan directly. The Stan code brms generates is human readable and so we can use it to learn some of the essential aspects of Stan and the particular model we are implementing. The Stan program below is a slightly modified version of the code extracted via `stancode(fit_dummy)`

:

```
// generated with brms 2.2.0
functions {/**
* Normal log-pdf for spatially lagged responses
*
* @param y Vector of response values.
* @param mu Mean parameter vector.
* @param sigma Positive scalar residual standard deviation.
* @param rho Positive scalar autoregressive parameter.
* @param W Spatial weight matrix.
*
* @return A scalar to be added to the log posterior.
*/
normal_lagsar_lpdf(vector y, vector mu, real sigma,
real
real rho, matrix W) {= rows(y);
int N = 1 / square(sigma);
real inv_sigma2 = -rho * W;
matrix[N, N] W_tilde
vector[N] half_pred;for (n in 1:N) W_tilde[n, n] += 1;
= W_tilde * (y - mdivide_left(W_tilde, mu));
half_pred 0.5 * log_determinant(crossprod(W_tilde) * inv_sigma2) -
return 0.5 * dot_self(half_pred) * inv_sigma2;
}
}
data {<lower=1> N; // total number of observations
int// response variable
vector[N] Y; <lower=0> Nmi; // number of missings
int<lower=1> Jmi[Nmi]; // positions of missings
int<lower=1> K; // number of population-level effects
int// population-level design matrix
matrix[N, K] X; // spatial weight matrix
matrix[N, N] W; // should the likelihood be ignored?
int prior_only;
}
transformed data {= K - 1;
int Kc - 1] Xc; // centered version of X
matrix[N, K - 1] means_X; // column means of X before centering
vector[K for (i in 2:K) {
- 1] = mean(X[, i]);
means_X[i - 1] = X[, i] - means_X[i - 1];
Xc[, i
}
}
parameters {// estimated missings
vector[Nmi] Ymi; // population-level effects
vector[Kc] b; // temporary intercept
real temp_Intercept; <lower=0> sigma; // residual SD
real<lower=0,upper=1> lagsar; // SAR parameter
real
}
transformed parameters {
}
model {= Y;
vector[N] Yl = Xc * b + temp_Intercept;
vector[N] mu = Ymi;
Yl[Jmi] // priors including all constants
+= student_t_lpdf(temp_Intercept | 3, 34, 17);
target += student_t_lpdf(sigma | 3, 0, 17)
target - 1 * student_t_lccdf(0 | 3, 0, 17);
// likelihood including all constants
if (!prior_only) {
+= normal_lagsar_lpdf(Yl | mu, sigma, lagsar, W);
target
}
}
generated quantities {// actual population-level intercept
= temp_Intercept - dot_product(means_X, b);
real b_Intercept }
```

Here we want to focus on two aspects of the Stan code. First, because there is no built-in function in Stan that calculates the log-likelihood for the lag-SAR model, we define a new `normal_lagsar_lpdf`

function in the `functions`

block of the Stan program. This is the same function we showed earlier in the vignette and it can be used to compute the log-likelihood in an efficient and numerically stable way. The `_lpdf`

suffix used in the function name informs Stan that this is a log probability density function.

Second, this Stan program nicely illustrates how to set up missing value imputation. Instead of just computing the log-likelihood for the observed responses `Y`

, we define a new variable `Yl`

which is equal to `Y`

if the reponse is observed and equal to `Ymi`

if the response is missing. The latter is in turn defined as a parameter and thus estimated along with all other paramters of the model. More details about missing value imputation in Stan can be found in the *Missing Data & Partially Known Parameters* section of the Stan manual.

The Stan code extracted from brms is not only helpful when learning Stan, but can also drastically speed up the specification of models that are not support by brms. If brms can fit a model similar but not identical to the desired model, we can let brms generate the Stan program for the similar model and then mold it into the program that implements the model we actually want to fit. Rather than calling `stancode()`

, which requires an existing fitted model object, we recommend using `make_stancode()`

and specifying the `save_model`

argument to write the Stan program to a file. The corresponding data can be prepared with `make_standata()`

and then manually amended if needed. Once the code and data have been edited, they can be passed to RStan’s `stan()`

function via the `file`

and `data`

arguments.

In summary, we have shown how to set up and validate approximate and exact LOO-CV for non-factorized multivariate normal models using Stan with the **brms** and **loo** packages. Although we focused on the particular example of a spatial SAR model, the presented recipe applies more generally to models that can be expressed in terms of a multivariate normal likelihood.

Anselin L. (1988). *Spatial econometrics: methods and models*. Dordrecht: Kluwer Academic.

Bürkner P. C., Gabry J., & Vehtari A. (2020). Efficient leave-one-out cross-validation for Bayesian non-factorized normal and Student-t models. ArXiv preprint.

Sundararajan S. & Keerthi S. S. (2001). Predictive approaches for choosing hyperparameters in Gaussian processes. *Neural Computation*, 13(5), 1103–1118.

Vehtari A., Mononen T., Tolvanen V., Sivula T., & Winther O. (2016). Bayesian leave-one-out cross-validation approximations for Gaussian latent variable models. *Journal of Machine Learning Research*, 17(103), 1–38. Online.

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