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Efficient approximate Bayesian inference for Structural Equation Models.

While Markov Chain Monte Carlo (MCMC) methods remain the gold standard for exact Bayesian inference, they can be prohibitively slow for iterative model development. {INLAvaan} offers a rapid alternative for latent variable analysis, delivering Bayesian results at (or near) the speed of frequentist estimators. It achieves this through a custom, ground-up implementation of the Integrated Nested Laplace Approximation (INLA), engineered specifically for the lavaan modelling framework.

A familiar interface

{INLAvaan} is designed to fit seamlessly into your existing workflow. If you are familiar with the (b)lavaan syntax, you can begin using {INLAvaan} immediately.

As a first impression of the package, consider the canonical example of SEM applied to the Industrialisation and Political Democracy data set of Bollen (1989)1:

model <- "
  # Latent variable definitions
     ind60 =~ x1 + x2 + x3
     dem60 =~ y1 + y2 + y3 + y4
     dem65 =~ y5 + y6 + y7 + y8

  # Latent regressions
    dem60 ~ ind60
    dem65 ~ ind60 + dem60

  # Residual correlations
    y1 ~~ y5
    y2 ~~ y4 + y6
    y3 ~~ y7
    y4 ~~ y8
    y6 ~~ y8
  
  # Custom priors on latent variances
    ind60 ~~ prior('gamma(1, 1)')*ind60
    dem60 ~~ prior('gamma(1,.9)')*dem60
    dem65 ~~ prior('gamma(1,.5)')*dem65
"
utils::data("PoliticalDemocracy", package = "lavaan")

fit <- asem(model, PoliticalDemocracy, marginal_method = "skewnorm", vb_correction = FALSE)
#> ℹ Finding posterior mode.
#> ✔ Finding posterior mode. [34ms]
#> 
#> ℹ Computing the Hessian.
#> ✔ Computing the Hessian. [113ms]
#> 
#> ⠙ Fitting skew normal to 0/31 marginals.
#> ⠹ Fitting skew normal to 5/31 marginals.
#> ⠸ Fitting skew normal to 16/31 marginals.
#> ⠼ Fitting skew normal to 27/31 marginals.
#> ✔ Fitting skew normal to 31/31 marginals. [612ms]
#> 
#> ℹ Sampling covariances and defined parameters.
#> ✔ Sampling covariances and defined parameters. [64ms]
#> 
#> ⠙ Computing ppp and DIC.
#> ⠹ Computing ppp and DIC.
#> ✔ Computing ppp and DIC. [202ms]
#> 

summary(fit)
#> INLAvaan 0.2.1.9006 ended normally after 71 iterations
#> 
#>   Estimator                                      BAYES
#>   Optimization method                           NLMINB
#>   Number of model parameters                        31
#> 
#>   Number of observations                            75
#> 
#> Model Test (User Model):
#> 
#>    Marginal log-likelihood                   -1641.277 
#>    PPP (Chi-square)                              0.174 
#> 
#> Information Criteria:
#> 
#>    Deviance (DIC)                             3224.940 
#>    Effective parameters (pD)                    64.327 
#> 
#> Parameter Estimates:
#> 
#>    Marginalisation method                     SKEWNORM
#>    VB correction                                 FALSE
#> 
#> Latent Variables:
#>                    Estimate       SD     2.5%    97.5%     NMAD    Prior       
#>   ind60 =~                                                                     
#>     x1                1.000                                                    
#>     x2                2.214    0.145    1.946    2.516    0.005    normal(0,10)
#>     x3                1.824    0.154    1.534    2.140    0.007    normal(0,10)
#>   dem60 =~                                                                     
#>     y1                1.000                                                    
#>     y2                1.355    0.206    0.976    1.784    0.010    normal(0,10)
#>     y3                1.120    0.160    0.824    1.450    0.008    normal(0,10)
#>     y4                1.368    0.173    1.057    1.736    0.009    normal(0,10)
#>   dem65 =~                                                                     
#>     y5                1.000                                                    
#>     y6                1.213    0.178    0.886    1.586    0.015    normal(0,10)
#>     y7                1.318    0.169    1.015    1.676    0.011    normal(0,10)
#>     y8                1.316    0.176    1.003    1.692    0.012    normal(0,10)
#> 
#> Regressions:
#>                    Estimate       SD     2.5%    97.5%     NMAD    Prior       
#>   dem60 ~                                                                      
#>     ind60             1.466    0.381    0.738    2.234    0.003    normal(0,10)
#>   dem65 ~                                                                      
#>     ind60             0.566    0.242    0.100    1.050    0.002    normal(0,10)
#>     dem60             0.868    0.106    0.671    1.089    0.021    normal(0,10)
#> 
#> Covariances:
#>                    Estimate       SD     2.5%    97.5%     NMAD    Prior       
#>  .y1 ~~                                                                        
#>    .y5                0.294    0.383    0.027    1.529    0.003       beta(1,1)
#>  .y2 ~~                                                                        
#>    .y4                0.242    0.692    0.003    2.715    0.004       beta(1,1)
#>    .y6                0.343    0.702    0.779    3.536    0.019       beta(1,1)
#>  .y3 ~~                                                                        
#>    .y7                0.209    0.619   -0.255    2.174    0.005       beta(1,1)
#>  .y4 ~~                                                                        
#>    .y8                0.099    0.463   -0.496    1.324    0.007       beta(1,1)
#>  .y6 ~~                                                                        
#>    .y8                0.310    0.576    0.258    2.520    0.005       beta(1,1)
#> 
#> Variances:
#>                    Estimate       SD     2.5%    97.5%     NMAD    Prior       
#>     ind60             0.469    0.092    0.317    0.674    0.003      gamma(1,1)
#>    .dem60             3.536    0.809    5.322    2.161    0.009     gamma(1,.9)
#>    .dem65             0.350    0.202    4.581    0.057    0.035     gamma(1,.5)
#>    .x1                0.085    0.020    0.196    0.050    0.004 gamma(1,.5)[sd]
#>    .x2                0.127    0.065    1.425    0.021    0.042 gamma(1,.5)[sd]
#>    .x3                0.493    0.096    0.333    0.708    0.003 gamma(1,.5)[sd]
#>    .y1                2.037    0.498    4.655    1.195    0.008 gamma(1,.5)[sd]
#>    .y2                7.791    1.444    5.347   10.985    0.004 gamma(1,.5)[sd]
#>    .y3                5.321    1.037    3.601    7.651    0.001 gamma(1,.5)[sd]
#>    .y4                3.246    0.791    7.739    1.885    0.010 gamma(1,.5)[sd]
#>    .y5                2.495    0.524    3.667    1.622    0.007 gamma(1,.5)[sd]
#>    .y6                5.211    0.960    3.591    7.341    0.002 gamma(1,.5)[sd]
#>    .y7                3.577    0.764    5.283    2.304    0.009 gamma(1,.5)[sd]
#>    .y8                3.381    0.736    5.005    2.130    0.005 gamma(1,.5)[sd]

Validation against MCMC

Computation speed is valuable only when accuracy is preserved. Our method yields posterior distributions that are visually and numerically comparable to those obtained via MCMC (e.g., via {blavaan}/Stan), but at a fraction of the computational cost.

The figure below illustrates the posterior density overlap for the example above. The percentages refer to (one minus) the Jensen-Shannon divergence, which gives a measure of similarity between two probability distributions.

# install.packages("blavaan")
library(blavaan)
fit_blav <- bsem(model, PoliticalDemocracy)
res <- INLAvaan:::compare_mcmc(fit_blav, INLAvaan = fit)
print(res$p_compare)

Installation

Install the development version of {INLAvaan} from GitHub using:

# install.packages("pak")
pak::pak("haziqj/INLAvaan")

Optionally2, you may wish to install INLA. Following the official instructions given here, install the package by running this command in R:

install.packages(
  "INLA",
  repos = c(getOption("repos"), 
            INLA = "https://inla.r-inla-download.org/R/stable"), 
  dep = TRUE
)

Citation

To cite package {INLAvaan} in publications use:

Jamil, H (2026). INLAvaan: Bayesian structural equation modelling with INLA. R package version 0.2.1.9006. URL: https://inlavaan.haziqj.ml/

A BibTeX entry for LaTeX users is:

@Manual{,
    title = {INLAvaan: Bayesian structural equation modelling with INLA},
    author = {Haziq Jamil},
    year = {2026},
    note = {R package version 0.2.1.9006},
    url = {https://inlavaan.haziqj.ml/},
  }

License

The {INLAvaan} package is licensed under the GPL-3.

INLAvaan: Bayesian Latent Variable Analysis using INLA
Copyright (C) 2026 Haziq Jamil

This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <http://www.gnu.org/licenses/>.

By using this package, you agree to comply with both licenses: the GPL-3 license for the software and the CC BY 4.0 license for the data.

  1. Bollen, K. A. (1989). Structural equations with latent variables (pp. xiv, 514). John Wiley & Sons. https://doi.org/10.1002/9781118619179↩︎

  2. R-INLA dependency has been removed temporarily for v0.2-0.↩︎