Bayesian Latent Variable Analysis using INLA • INLAvaan

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)
#> ℹ Using MVN log-likelihood.
#> ℹ Finding posterior mode.
#> ✔ Finding posterior mode. [113ms]
#> 
#> ℹ Computing the Hessian.
#> ✔ Computing the Hessian. [357ms]
#> 
#> ℹ Using skew normal approximation.
#> ⠙ Fitting skew normal to 0/31 marginals.
#> ⠹ Fitting skew normal to 2/31 marginals.
#> ⠸ Fitting skew normal to 6/31 marginals.
#> ⠼ Fitting skew normal to 9/31 marginals.
#> ⠴ Fitting skew normal to 13/31 marginals.
#> ⠦ Fitting skew normal to 16/31 marginals.
#> ⠧ Fitting skew normal to 20/31 marginals.
#> ⠇ Fitting skew normal to 23/31 marginals.
#> ⠏ Fitting skew normal to 27/31 marginals.
#> ⠋ Fitting skew normal to 30/31 marginals.
#> ✔ Fitting skew normal to 31/31 marginals. [1.8s]
#> 
#> ℹ Sampling posterior covariances.
#> ✔ Sampling posterior covariances. [127ms]
#> 
#> ⠙ Computing ppp and DIC.
#> ⠹ Computing ppp and DIC.
#> ⠸ Computing ppp and DIC.
#> ⠼ Computing ppp and DIC.
#> ✔ Computing ppp and DIC. [1.1s]
#> 

summary(fit)
#> INLAvaan 0.2.1.9001 ended normally after 77 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)                              0.484 
#> 
#> Information Criteria:
#> 
#>    Deviance (DIC)                             3158.504 
#>    Effective parameters (pD)                    31.109 
#> 
#> Parameter Estimates:
#> 
#>    Marginalisation method                     SKEWNORM
#> 
#> Latent Variables:
#>                    Estimate       SD     2.5%    97.5%    Prior       
#>   ind60 =~                                                            
#>     x1                1.000                                           
#>     x2                2.214    0.145    1.946    2.516    normal(0,10)
#>     x3                1.811    0.152    1.515    2.110    normal(0,10)
#>   dem60 =~                                                            
#>     y1                1.000                                           
#>     y2                1.355    0.206    0.975    1.784    normal(0,10)
#>     y3                1.113    0.156    0.816    1.430    normal(0,10)
#>     y4                1.348    0.163    1.039    1.681    normal(0,10)
#>   dem65 =~                                                            
#>     y5                1.000                                           
#>     y6                1.215    0.179    0.888    1.591    normal(0,10)
#>     y7                1.304    0.164    0.998    1.641    normal(0,10)
#>     y8                1.287    0.165    0.971    1.617    normal(0,10)
#> 
#> Regressions:
#>                    Estimate       SD     2.5%    97.5%    Prior       
#>   dem60 ~                                                             
#>     ind60             1.447    0.378    0.706    2.189    normal(0,10)
#>   dem65 ~                                                             
#>     ind60             0.553    0.241    0.076    1.022    normal(0,10)
#>     dem60             0.860    0.103    0.662    1.066    normal(0,10)
#> 
#> Covariances:
#>                    Estimate       SD     2.5%    97.5%    Prior       
#>  .y1 ~~                                                               
#>    .y5                0.297    0.370    0.021    1.473       beta(1,1)
#>  .y2 ~~                                                               
#>    .y4                0.246    0.687    0.042    2.737       beta(1,1)
#>    .y6                0.340    0.738    0.875    3.766       beta(1,1)
#>  .y3 ~~                                                               
#>    .y7                0.210    0.619   -0.327    2.098       beta(1,1)
#>  .y4 ~~                                                               
#>    .y8                0.103    0.437   -0.462    1.250       beta(1,1)
#>  .y6 ~~                                                               
#>    .y8                0.306    0.563    0.272    2.482       beta(1,1)
#> 
#> Variances:
#>                    Estimate       SD     2.5%    97.5%    Prior       
#>     ind60             0.472    0.094    0.320    0.687      gamma(1,1)
#>    .dem60             3.608    0.830    2.268    5.501     gamma(1,.9)
#>    .dem65             0.354    0.207    0.074    0.859     gamma(1,.5)
#>    .x1                0.086    0.021    0.052    0.133 gamma(1,.5)[sd]
#>    .x2                0.138    0.072    0.037    0.311 gamma(1,.5)[sd]
#>    .x3                0.495    0.098    0.335    0.718 gamma(1,.5)[sd]
#>    .y1                2.082    0.518    1.250    3.266 gamma(1,.5)[sd]
#>    .y2                7.886    1.517    5.405   11.329 gamma(1,.5)[sd]
#>    .y3                5.350    1.059    3.620    7.754 gamma(1,.5)[sd]
#>    .y4                3.349    0.843    2.000    5.281 gamma(1,.5)[sd]
#>    .y5                2.519    0.542    1.640    3.754 gamma(1,.5)[sd]
#>    .y6                5.268    1.004    3.627    7.548 gamma(1,.5)[sd]
#>    .y7                3.611    0.789    2.328    5.403 gamma(1,.5)[sd]
#>    .y8                3.452    0.778    2.197    5.228 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 (2025). INLAvaan: Bayesian structural equation modelling with INLA . R package version 0.2.1.9001. 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 = {2025},
    note = {R package version 0.2.1.9001},
    url = {https://inlavaan.haziqj.ml/},
  }

License

The INLAvaan package is licensed under the GPL-3.

INLAvaan: Bayesian structural equation modelling with INLA
Copyright (C) 2025- 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.