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Sampling from the Generative Model

Source: vignettes/articles/sampling.qmd

Purpose

After fitting a Bayesian SEM with INLAvaan, we often want to draw samples from the model itself—either using posterior or prior parameter values. The sampling() function does exactly this: it propagates parameter draws through the full generative chain

𝜽⏟parameters→𝜼⏟latent variables→𝒚*⏟observed variables \underbrace{\boldsymbol\theta}_{\text{parameters}} \;\longrightarrow\; \underbrace{\boldsymbol\eta}_{\text{latent variables}} \;\longrightarrow\; \underbrace{\mathbf{y}^*}_{\text{observed variables}}

producing samples that are not tied to any individual observation. This is distinct from predict(), which returns individual-specific factor scores 𝜼∣𝒚,𝜽\boldsymbol\eta \mid \mathbf{y}, \boldsymbol\theta.

Typical use cases include posterior predictive checks (PPCs) and prior predictive checks; the predictive checks article walks through that workflow.

sampling() versus simulate()

INLAvaan offers two ways to draw data from the generative model, and the difference is what varies between draws:

One draw is Returns Use for
sampling() one 𝜽(s)\boldsymbol\theta^{(s)} → one observation 𝒚*(s)\mathbf{y}^{*(s)} a matrix (or list) distributions of quantities: parameters, implied moments
simulate() one 𝜽(s)\boldsymbol\theta^{(s)} → a whole dataset of nn rows a list of data frames replicate datasets: PPC overlays, test statistics, SBC

sampling() refreshes 𝜽\boldsymbol\theta at every draw, pooling parameter uncertainty and sampling variability into a single marginal predictive distribution. simulate() holds each 𝜽(s)\boldsymbol\theta^{(s)} fixed for a whole dataset, so variation across replicates reflects parameter uncertainty while variation within a replicate reflects sampling variability—which is what makes replicates exchangeable with the observed data, and hence the right tool for predictive checks and simulation-based calibration. Both functions accept prior = TRUE. The rest of this article covers sampling().

The generative model

Let 𝜽(s)\boldsymbol\theta^{(s)} (s=1,…,Ss = 1, \dots, S) denote one parameter draw. From this draw the SEM matrices 𝚲\boldsymbol\Lambda, 𝚿\boldsymbol\Psi, 𝑩\mathbf{B}, 𝜶\boldsymbol\alpha, 𝝂\boldsymbol\nu, and 𝚯\boldsymbol\Theta are constructed. The generative chain is:

1. Latent variables. 𝜼(s)∼𝒩((𝑰−𝑩)−1𝜶,𝚽),𝚽=(𝑰−𝑩)−1𝚿[(𝑰−𝑩)−1]′. \boldsymbol\eta^{(s)} \sim \mathcal{N}\!\bigl( (\mathbf{I} - \mathbf{B})^{-1}\boldsymbol\alpha,\; \boldsymbol\Phi \bigr), \qquad \boldsymbol\Phi = (\mathbf{I} - \mathbf{B})^{-1}\boldsymbol\Psi\, [(\mathbf{I} - \mathbf{B})^{-1}]'.

2. Observed variables. 𝒚*(s)∼𝒩(𝚲𝜼(s)+𝝂,𝚯). \mathbf{y}^{*(s)} \sim \mathcal{N}\!\bigl( \boldsymbol\Lambda\,\boldsymbol\eta^{(s)} + \boldsymbol\nu,\; \boldsymbol\Theta \bigr).

When prior = FALSE (default), 𝜽(s)\boldsymbol\theta^{(s)} comes from the posterior; when prior = TRUE, each parameter is drawn independently from its prior.

Quick start 

dat <- lavaan::HolzingerSwineford1939
mod <- "
  visual  =~ x1 + x2 + x3
  textual =~ x4 + x5 + x6
  speed   =~ x7 + x8 + x9
"
fit <- acfa(mod, dat, verbose = FALSE)

Parameter samples

The default type "lavaan" returns an S×pS \times p matrix of lavaan-side (constrained) parameter draws:

theta_post <- sampling(fit, type = "lavaan", nsamp = 2000)
dim(theta_post)
#> [1] 2000   21
head(theta_post[, 1:4])
#>      visual=~x2 visual=~x3 textual=~x5 textual=~x6
#> [1,]  0.6217858  0.9061140   1.1243541   0.9963907
#> [2,]  0.5512358  0.8050023   1.2136311   0.9163359
#> [3,]  0.5589911  0.7004874   0.9801615   0.9248339
#> [4,]  0.5567474  0.7638705   1.1415682   1.0361746
#> [5,]  0.5948947  1.0233424   1.0731154   0.8664639
#> [6,]  0.3630961  0.7067411   1.0889999   0.8696213

Latent and observed samples 

eta  <- sampling(fit, type = "latent",   nsamp = 1000)
ystar <- sampling(fit, type = "observed", nsamp = 1000)
dim(eta)    # 1000 x 3 (one per latent factor)
#> [1] 1000    3
dim(ystar)  # 1000 x 9 (one per observed indicator)
#> [1] 1000    9

Everything at once 

all_samps <- sampling(fit, type = "all", nsamp = 1000)
names(all_samps)
#> [1] "lavaan"   "theta"    "latent"   "observed" "implied"

Comparing samples to the fitted marginals

INLAvaan approximates each marginal posterior with a skew-normal density. We can overlay the sampling() histogram (drawn via the copula method) on top of the fitted density curve stored in pdf_data to verify that they agree.

# 1. Draw copula samples
samp_cop <- sampling(fit, type = "lavaan", nsamp = 1000, samp_copula = TRUE)

# 2. Retrieve the fitted skew-normal densities
int <- fit@external$inlavaan_internal
pdf_data <- int$pdf_data

# 3. Build a long data frame for ggplot
par_names <- colnames(samp_cop)
hist_df <- data.frame(
  param = rep(par_names, each = nrow(samp_cop)),
  value = as.vector(samp_cop)
)
hist_df$param <- factor(hist_df$param, levels = par_names)

curve_df <- do.call(rbind, Map(function(nm, df) {
  data.frame(param = nm, x = df$x, y = df$y)
}, par_names, pdf_data[par_names]))
curve_df$param <- factor(curve_df$param, levels = par_names)

# 4. Plot
ggplot(hist_df, aes(x = value)) +
  geom_histogram(aes(y = after_stat(density)), bins = 50,
                 fill = "grey40", alpha = 0.5) +
  geom_line(data = curve_df, aes(x = x, y = y),
            colour = "#00A6AA", linewidth = 0.8) +
  facet_wrap(~param, scales = "free") +
  labs(x = "Parameter value", y = "Density",
       title = "Copula samples vs. fitted skew-normal posterior") +
  theme_minimal(base_size = 11)
Figure 1: Copula samples (histogram) versus the fitted skew-normal marginal (red curve) for all free parameters.

Copula vs. Gaussian sampling

By default, sampling() uses the copula method, which respects the skew-normal marginals. Setting samp_copula = FALSE uses the multivariate Gaussian (Laplace) approximation instead. The difference is most visible for parameters with asymmetric posteriors, such as variance components.

samp_gauss <- sampling(fit, type = "lavaan", nsamp = 5000, samp_copula = FALSE)

cop_df <- data.frame(
  param = rep(par_names, each = nrow(samp_cop)),
  value = as.vector(samp_cop),
  method = "Copula"
)
gauss_df <- data.frame(
  param = rep(par_names, each = nrow(samp_gauss)),
  value = as.vector(samp_gauss),
  method = "Gaussian"
)
both_df <- rbind(cop_df, gauss_df)
both_df$param <- factor(both_df$param, levels = par_names)
both_df$method <- factor(both_df$method, levels = c("Copula", "Gaussian"))

ggplot(both_df, aes(x = value, fill = method)) +
  geom_histogram(aes(y = after_stat(density)), bins = 50,
                 alpha = 0.45, position = "identity") +
  facet_wrap(~param, scales = "free") +
  scale_fill_manual(values = c(Copula = "#00A6AA", Gaussian = "#F18F00")) +
  labs(x = "Parameter value", y = "Density", fill = "Method",
       title = "Copula vs. Gaussian posterior sampling") +
  theme_minimal(base_size = 11) +
  theme(legend.position = "top")
Figure 2: Comparison of marginal histograms from copula sampling (blue) and Gaussian sampling (orange) for all parameters.

For symmetric posteriors (e.g., factor loadings), the two methods are nearly identical. For positively-skewed posteriors (e.g., residual variances), the copula method captures the asymmetry while the Gaussian approximation is symmetric by construction.

Prior sampling

Both sampling() and simulate() accept prior = TRUE: each free parameter is drawn independently from its prior (the data play no role) and propagated through the generative chain. Draws that imply a non-positive-definite covariance matrix are rejected and redrawn, so the exact prior is preserved. What these draws are useful for—and how to read them when the model has no mean structure—is the subject of the predictive checks article.