# Introduction to BayesRTMB

# What is BayesRTMB?

BayesRTMB is a package for specifying and estimating statistical models in R, using RTMB as the computational engine.

In BayesRTMB, standard analyses can start from wrapper functions such as `rtmb_lm()` and `rtmb_glmer()`. When needed, you can also use `rtmb_code()` to write custom models with a Stan-like feel.

The basic idea of BayesRTMB is that MCMC, MAP estimation, variational inference, and frequentist analysis can all be called from a single model object.

    fit_mcmc <- mdl$sample()
    fit_map <- mdl$optimize()
    fit_vb <- mdl$variational()
    fit_freq <- mdl$classic()

This page introduces where BayesRTMB fits and the overall picture that is useful to know when first using the package. More detailed model-writing rules and individual analysis examples are covered in the other vignettes.

## Links to Articles

The following articles are available.

- English Introduction
- Quick Start
- Wrapper Functions
- Hierarchical Models and GLMMs
- Writing Model Codes
- RTMB Internals and Inference Algorithms
- Analysis Reference

# What BayesRTMB Can Do

The main features of BayesRTMB are as follows.

- Write models directly in R
  With `rtmb_code()`, you can specify models by separating them into blocks such as `parameters`, `transform`, `model`, and `generate`.

- Start analysis immediately from wrapper functions
  Regression, generalized linear models, mixed models, t tests, correlations, factor analysis, IRT, latent rank theory, multidimensional unfolding, and other analyses can be run from specialized wrapper functions.

- Use multiple inference methods from the same model
  MCMC, MAP estimation, variational inference, and frequentist analysis can all be called from the same `RTMB_Model` object.

- Handle random effects efficiently
  Parameters declared with `random = TRUE` can be marginalized by Laplace approximation in MAP estimation.

- Inspect the model created by wrappers
  `print_code()` lets you inspect the `rtmb_code()` object internally generated by a wrapper function.

- Support model comparison
  MCMC results can be used to compute marginal likelihoods and Bayes factors by bridge sampling.

# Two Entry Points

BayesRTMB has two broad entry points.

The first is a set of wrapper functions for standard analyses. Usually, this is the easiest place to start.

    data(debate)

    mdl <- rtmb_lm(sat ~ talk + perf, data = debate)
    fit <- mdl$optimize()
    fit

The second is `rtmb_code()`, which is used when you want to write the model yourself. This is useful when a model is difficult to express with wrappers or when you want to try a new model.

    dat <- list(
      Y = debate$sat,
      X = data.matrix(debate[c("talk", "perf")])
    )

    code <- rtmb_code(
      setup = {
        N <- length(Y)
        P <- ncol(X)
      },
      parameters = {
        alpha = Dim()
        beta = Dim(P)
        sigma = Dim(lower = 0)
      },
      transform = {
        mu <- alpha + X %*% beta
      },
      model = {
        Y ~ normal(mu, sigma)
        alpha ~ normal(0, 10)
        beta ~ normal(0, 10)
        sigma ~ exponential(1)
      }
    )

    mdl <- rtmb_model(dat, code)

In the `model` block, you can use sampling syntax like this.

    Y ~ normal(mu, sigma)
    beta ~ normal(0, 10)
    sigma ~ exponential(1)

Internally, this is converted into code that adds log densities.

# Starting from Wrapper Functions

First, let us look at an example using a wrapper function. Here we use the `debate` data and consider a linear regression model in which satisfaction `sat` is explained by `talk` and `perf`.

    data(debate)

    mdl_lm <- rtmb_lm(
      sat ~ talk + perf,
      data = debate
    )

At this point, estimation has not yet been run. `mdl_lm` is an `RTMB_Model` object that stores the model definition and data.

To run MAP estimation, use `optimize()`.

    fit_map <- mdl_lm$optimize()
    fit_map

    ## Starting RTMB optimization...
    ##
    ## Call:
    ## MAP Estimation via RTMB
    ##
    ## Negative Log-Posterior: 395.58
    ## Approx. Log Marginal Likelihood (Laplace): -404.61
    ##
    ## Point Estimates and 95% Wald CI:
    ##  variable  Estimate  Std. Error  Lower 95%  Upper 95%
    ## Intercept   1.83366     0.21105    1.42000    2.24732
    ## b[talk]     0.28694     0.05291    0.18323    0.39064
    ## b[perf]     0.15632     0.02987    0.09777    0.21486
    ## sigma       0.90453     0.03693    0.83497    0.97988

If you want to estimate the model as a frequentist linear regression, use `classic()`. `classic()` is available for wrapper models constructed with a flat prior.

    fit_freq <- mdl_lm$classic()
    fit_freq

    ## Starting RTMB optimization...
    ##
    ## Call:
    ## Classical estimation via lm
    ##
    ## Log-Likelihood: -402.220, AIC: 812.440, BIC: 827.255
    ##
    ## Point Estimates and Confidence Intervals:
    ##           Estimate Std. Error Lower 95% Upper 95%  df  t value     Pr
    ## Intercept  1.83366    0.21212   1.41621   2.25110 297  8.64454 <.0001 ***
    ## b[talk]    0.28694    0.05318   0.18229   0.39159 297  5.39580 <.0001 ***
    ## b[perf]    0.15632    0.03002   0.09724   0.21539 297  5.20703 <.0001 ***
    ## sigma      0.90909    0.03730   0.83856   0.98554 297     NA

You can inspect the model code generated by a wrapper function with `print_code()`.

    mdl_lm$print_code()

    ## === RTMB Model Code ===
    ##
    ## rtmb_code(
    ##   setup = {
    ##     mf <- model.frame(sat ~ talk + perf, na.action = na.omit)
    ##     Y <- model.response(mf)
    ##     X <- model.matrix(sat ~ talk + perf, mf)[, -1, drop = FALSE]
    ##     N <- nrow(mf)
    ##     K <- ncol(X)
    ##   },
    ##   parameters = {
    ##     Intercept <- Dim(1)
    ##     b <- Dim(K)
    ##     sigma <- Dim(1, lower = 0)
    ##   },
    ##   model = {
    ##     # Transform
    ##     eta <- Intercept + X %*% b
    ##     # Likelihood (Data)
    ##     Y ~ normal(eta, sigma)
    ##     # Priors
    ##   }
    ## )

`print_code()` displays blocks such as `setup`, `parameters`, `transform`, and `model` so that users can reproduce the same model with `rtmb_model()`.

# Writing Your Own Model

When you want to write a more flexible model, use `rtmb_code()`.

The main blocks of `rtmb_code()` are as follows.

- `setup`: preprocess quantities needed from the data
- `parameters`: declare parameters to be estimated
- `transform`: create derived quantities and linear predictors from parameters
- `model`: write the likelihood and prior distributions
- `generate`: compute predictions and generated quantities

For example, a regression model with simple normal and exponential priors close to `prior_normal()` can be written as follows.

    dat <- list(
      Y = as.numeric(debate$sat),
      X = data.matrix(debate[c("talk", "perf", "skill")])
    )

    code_reg <- rtmb_code(
      setup = {
        N <- length(Y)
        P <- ncol(X)
      },
      parameters = {
        alpha = Dim()
        beta = Dim(P)
        sigma = Dim(lower = 0)
      },
      transform = {
        mu <- alpha + X %*% beta
      },
      model = {
        Y ~ normal(mu, sigma)
        alpha ~ normal(0, 10)
        beta ~ normal(0, 2.5)
        sigma ~ exponential(1)
      },
      generate = {
        Y_rep <- rnorm(length(Y), mu, sigma)
      }
    )

    mdl_reg <- rtmb_model(dat, code_reg)

Writing preprocessing in `setup` makes the `parameters` and `model` blocks easier to read. It also makes the processing needed to reproduce the model visible when `print_code()` is displayed.

# Choosing an Inference Method

BayesRTMB is built primarily around Bayesian estimation. For that reason, this section presents the methods in the order MCMC, MAP estimation, variational inference, and frequentist analysis.

Method | Main purpose
--- | ---
`$sample()` | Run MCMC sampling with NUTS
`$optimize()` | Run fast MAP estimation or maximum likelihood estimation
`$variational()` | Approximate the posterior distribution with ADVI
`$classic()` | Treat a flat-prior wrapper model as a frequentist analysis

## MCMC

`sample()` runs MCMC sampling with NUTS. Use it when you want to inspect the full posterior distribution or when MAP estimation alone is not enough to represent uncertainty.

    set.seed(1)

    fit_mcmc <- mdl_lm$sample()

    fit_mcmc$summary()

    ##  variable     mean    sd      map     q2.5    q97.5  ess_bulk  ess_tail  rhat
    ## lp         -397.79  1.54  -396.75  -401.63  -395.94       859      1236  1.01
    ## Intercept     1.82  0.22     1.78     1.38     2.26       629      1023  1.01
    ## b[talk]       0.29  0.05     0.28     0.18     0.40       665      1205  1.00
    ## b[perf]       0.16  0.03     0.15     0.10     0.22       671       990  1.01
    ## sigma         0.91  0.04     0.91     0.84     1.00       934      1254  1.01

## MAP Estimation

`optimize()` performs point estimation by maximizing the posterior distribution or likelihood. It is useful for checking a model and for initial exploration of complex models.

    fit_map <- mdl_lm$optimize()
    fit_map$summary()

    ## Starting RTMB optimization...
    ##
    ## Call:
    ## MAP Estimation via RTMB
    ##
    ## Negative Log-Posterior: 395.58
    ## Approx. Log Marginal Likelihood (Laplace): -404.61
    ##
    ## Point Estimates and 95% Wald CI:
    ##  variable  Estimate  Std. Error  Lower 95%  Upper 95%
    ## Intercept   1.83366     0.21105    1.42000    2.24732
    ## b[talk]     0.28694     0.05291    0.18323    0.39064
    ## b[perf]     0.15632     0.02987    0.09777    0.21486
    ## sigma       0.90453     0.03693    0.83497    0.97988

With `se_method = "sampling"`, standard errors and intervals for generated quantities can be calculated by propagating uncertainty through approximate normal samples.

    fit_map <- mdl_lm$optimize(se_method = "sampling")

## Variational Inference

`variational()` approximates the posterior distribution using ADVI. It is useful when you want rough results faster than MCMC. However, it can underestimate posterior uncertainty, so caution is needed when using it for final uncertainty assessment.

    fit_vb <- mdl_lm$variational(
      method = "meanfield",
      iter = 3000,
      num_estimate = 4
    )

## Frequentist Analysis

`classic()` returns frequentist estimation results for flat-prior models constructed by wrapper functions.

    fit_freq <- mdl_lm$classic()
    fit_freq$summary()

    ## Starting RTMB optimization...
    ##
    ## Call:
    ## Classical estimation via lm
    ##
    ## Log-Likelihood: -402.220, AIC: 812.440, BIC: 827.255
    ##
    ## Point Estimates and Confidence Intervals:
    ##           Estimate Std. Error Lower 95% Upper 95%  df  t value     Pr
    ## Intercept  1.83366    0.21212   1.41621   2.25110 297  8.64454 <.0001 ***
    ## b[talk]    0.28694    0.05318   0.18229   0.39159 297  5.39580 <.0001 ***
    ## b[perf]    0.15632    0.03002   0.09724   0.21539 297  5.20703 <.0001 ***
    ## sigma      0.90909    0.03730   0.83856   0.98554 297     NA

For `lm`- and `glm`-type models, test statistics are displayed in a form close to the usual notation. For mixed models, degrees-of-freedom corrections and Laplace approximation are used when needed.

# Random Effects and Laplace Approximation

In hierarchical and mixed models, parameters can be declared with `random = TRUE`.

    Y <- debate$sat
    group <- as.integer(factor(debate$group))
    G <- length(unique(group))

    dat_icc <- list(Y = Y, group = group, G = G)

    code_icc <- rtmb_code(
      parameters = {
        mu = Dim()
        sigma = Dim(lower = 0)
        tau = Dim(lower = 0)
        r = Dim(G, random = TRUE)
      },
      model = {
        Y ~ normal(mu + r[group] * tau, sigma)
        r ~ normal(0, 1)
        tau ~ exponential(1)
        sigma ~ exponential(1)
      },
      generate = {
        icc <- tau^2 / (tau^2 + sigma^2)
      }
    )

    mdl_icc <- rtmb_model(dat_icc, code_icc)
    fit_icc <- mdl_icc$optimize(laplace = TRUE)

In MAP estimation, `laplace = TRUE` is the default. Parameters declared with `random = TRUE` are marginalized by Laplace approximation.

With wrapper functions, a mixed model can be written as follows.

    mdl_hlm <- rtmb_glmer(
      sat ~ talk + perf + (1 | group),
      data = debate,
      family = "gaussian"
    )

    fit_hlm <- mdl_hlm$optimize()

# Model Comparison (Bridge Sampling / WAIC)

## Marginal Likelihood and Bayes Factor by Bridge Sampling

For MCMC results, bridge sampling can be used to compute the marginal likelihood.

    set.seed(1)

    fit_mcmc$bridgesampling()

For example, the output is displayed as follows.

    ## Bridge Sampling Converged: LogML = -404.591 (Error = 0.0032, ESS = 604.0)

The returned value is a numeric log marginal likelihood with `error` and `ess` attributes.

Use `bayes_factor()` when you want to compute a Bayes factor.

    bf <- fit_mcmc$bayes_factor(fixed = list("b[talk]" = 0))
    bf

    ## --- Bayes Factor Analysis (Bridge Sampling) ---
    ## Bayes Factor (BF12) : 142963.4
    ## Log Bayes Factor    : 11.8703 (Approx. Error = 0.0040)
    ## Evidence            : Decisive evidence for Model 1
    ## Comparison model    : Parameters fixed at list("b[talk]" = 0)

## Model Comparison by WAIC

In BayesRTMB, if you store the pointwise log likelihood for each observation in a variable named `log_lik` within the `generate` block of `rtmb_code()`, you can calculate the WAIC (Widely Applicable Information Criterion) using the `$WAIC()` method after estimation.

For example, define `log_lik` in the `generate` block of a custom model as follows:

      generate = {
        # Compute the pointwise log likelihood and assign it to log_lik (specify sum = FALSE)
        log_lik <- normal_lpdf(Y, mu, sigma, sum = FALSE)
      }

Note: When using wrapper functions (e.g., `rtmb_lm()`), you can specify `WAIC = TRUE` in the arguments to automatically generate `log_lik` internally.

    mdl_lm <- rtmb_lm(sat ~ talk + perf, data = debate, WAIC = TRUE)
    fit_mcmc <- mdl_lm$sample()

    # Compute WAIC
    fit_mcmc$WAIC()

The output will be displayed as follows:

    ## WAIC
    ##   WAIC (elpd scale): -400.07398
    ##   -2 WAIC: 800.14796
    ##   p_waic: 4.70921
    ##   lppd: -395.36477
    ##   nobs: 300, draws: 4000

The `$WAIC()` method is available for results from MCMC (`MCMC_Fit`), variational inference (`VB_Fit`), and MAP estimation (`MAP_Fit`) with `se_method = "sampling"`.

# Next Steps

Once you have a general picture of BayesRTMB, the next step is to read the articles that match your purpose.

1. Quick Start
   Walk through a basic model and learn the flow of `rtmb_code()` and `rtmb_model()`.

2. Wrapper Functions
   Learn how to run standard analyses such as regression, generalized linear models, mixed models, and t tests with wrapper functions.

3. Hierarchical Models and GLMMs
   Learn how to use `rtmb_glmer()` for hierarchical models, GLMMs, residual correlation, ordinal models, priors, and visualization.

4. Writing Model Codes
   Learn how to write your own models using the `setup`, `parameters`, `transform`, `model`, and `generate` blocks.

5. RTMB Internals and Inference Algorithms
   Learn about internal processing such as Laplace approximation, constrained parameters, MCMC, and variational inference.

6. Analysis Reference
   Check fit-object methods, model-comparison tools, fixed parameters, distributions, and AD-taping notes.