diff --git a/R/fit.R b/R/fit.R
index 00b239fb8..d44dc246b 100644
--- a/R/fit.R
+++ b/R/fit.R
@@ -787,8 +787,8 @@ CmdStanFit$set("public", name = "constrain_variables", value = constrain_variabl
 #' fit_mcmc <- cmdstanr_example("logistic")
 #' head(fit_mcmc$lp())
 #'
-#' fit_mle <- cmdstanr_example("logistic", method = "optimize")
-#' fit_mle$lp()
+#' fit_optim <- cmdstanr_example("logistic", method = "optimize")
+#' fit_optim$lp()
 #'
 #' fit_vb <- cmdstanr_example("logistic", method = "variational")
 #' plot(fit_vb$lp(), fit_vb$lp_approx())
@@ -1176,13 +1176,13 @@ CmdStanFit$set("public", name = "metric_files", value = metric_files)
 #' fit_vb <- cmdstanr_example("logistic", method = "variational")
 #' fit_vb$time()
 #'
-#' fit_mle <- cmdstanr_example("logistic", method = "optimize", jacobian = TRUE)
-#' fit_mle$time()
+#' fit_optim <- cmdstanr_example("logistic", method = "optimize", jacobian = TRUE)
+#' fit_optim$time()
 #'
-#' # use fit_mle to draw samples from laplace approximation
-#' fit_laplace <- cmdstanr_example("logistic", method = "laplace", mode = fit_mle)
+#' # use fit_optim to draw samples from laplace approximation
+#' fit_laplace <- cmdstanr_example("logistic", method = "laplace", mode = fit_optim)
 #' fit_laplace$time() # just time for drawing sample not for running optimize
-#' fit_laplace$time()$total + fit_mle$time()$total # total time
+#' fit_laplace$time()$total + fit_optim$time()$total # total time
 #' }
 #'
 time <- function() {
@@ -1216,8 +1216,8 @@ CmdStanFit$set("public", name = "time", value = time)
 #' out <- fit_mcmc$output()
 #' str(out)
 #'
-#' fit_mle <- cmdstanr_example("logistic", method = "optimize")
-#' fit_mle$output()
+#' fit_optim <- cmdstanr_example("logistic", method = "optimize")
+#' fit_optim$output()
 #'
 #' fit_vb <- cmdstanr_example("logistic", method = "variational")
 #' fit_vb$output()
@@ -1245,8 +1245,8 @@ CmdStanFit$set("public", name = "output", value = output)
 #' fit_mcmc <- cmdstanr_example("logistic", method = "sample")
 #' str(fit_mcmc$metadata())
 #'
-#' fit_mle <- cmdstanr_example("logistic", method = "optimize")
-#' str(fit_mle$metadata())
+#' fit_optim <- cmdstanr_example("logistic", method = "optimize")
+#' str(fit_optim$metadata())
 #'
 #' fit_vb <- cmdstanr_example("logistic", method = "variational")
 #' str(fit_vb$metadata())
@@ -1972,14 +1972,12 @@ CmdStanMCMC$set("public", name = "num_chains", value = num_chains)
 #'
 #' @description A `CmdStanMLE` object is the fitted model object returned by the
 #'   [`$optimize()`][model-method-optimize] method of a [`CmdStanModel`] object.
-#'   Following CmdStan's terminology, the object contains an MLE if optimization
-#'   was run with `jacobian=FALSE` and a MAP estimate if it was run with
-#'   `jacobian=TRUE`. The name "MLE" is retained for historical reasons. More
-#'   precisely, the estimates correspond to a mode in either the constrained
-#'   parameter space or the unconstrained parameter space, depending on the
-#'   value of `jacobian` (and whether the model has constrained parameters).
-#'   The `jacobian` argument does not control whether prior terms are included;
-#'   all contributions to the Stan program's target are included under either
+#'   The name "MLE" is retained for historical reasons. With
+#'   `jacobian = FALSE`, the point estimate is a mode of the target in the
+#'   constrained parameter space. With `jacobian = TRUE`, it is a mode of the
+#'   corresponding density in the unconstrained parameter space. The `jacobian`
+#'   argument does not control whether prior terms are included; all
+#'   contributions to the Stan program's target are included under either
 #'   setting. See [`$optimize()`][model-method-optimize] and the CmdStan User's
 #'   Guide for more details.
 #'
@@ -2069,19 +2067,18 @@ CmdStanMLE <- R6::R6Class(
 #' @name fit-method-mle
 #' @aliases mle
 #' @description The `$mle()` method is only available for [`CmdStanMLE`]
-#'   objects. It returns the point estimate as a numeric vector with one element
-#'   per variable. The returned vector does *not* include `lp__`, the
-#'   target log density evaluated by Stan, up to an additive constant. `lp__` is
-#'   available via the [`$lp()`][fit-method-lp] method and also included in the
+#'   objects. The method name is retained for historical reasons. It returns the
+#'   point estimate as a numeric vector with one element per variable. The
+#'   returned vector does *not* include `lp__`, the target log density evaluated
+#'   by Stan, up to an additive constant. `lp__` is available via the
+#'   [`$lp()`][fit-method-lp] method and also included in the
 #'   [`$draws()`][fit-method-draws] method.
 #'
-#'   Following CmdStan's terminology, for models with constrained parameters
-#'   that are fit with `jacobian=TRUE`, this point estimate is called a maximum
-#'   a posteriori (MAP) estimate rather than an MLE. More precisely,
-#'   `jacobian=FALSE` finds a mode of the target in the constrained parameter
-#'   space and `jacobian=TRUE` finds a mode in the unconstrained parameter
-#'   space. See [`$optimize()`][model-method-optimize] and the CmdStan User's
-#'   Guide for more details.
+#'   With `jacobian = FALSE`, the point estimate is a mode of the target in the
+#'   constrained parameter space. With `jacobian = TRUE`, it is a mode of the
+#'   corresponding density in the unconstrained parameter space. See
+#'   [`$optimize()`][model-method-optimize] and the CmdStan User's Guide for more
+#'   details.
 #'
 #' @param variables (character vector) The variables (parameters, transformed
 #'   parameters, and generated quantities) to include. If NULL (the default)
diff --git a/R/model.R b/R/model.R
index 876590422..60269d2a8 100644
--- a/R/model.R
+++ b/R/model.R
@@ -1457,25 +1457,19 @@ CmdStanModel$set("public", name = "sample_mpi", value = sample_mpi)
 #' @family CmdStanModel methods
 #'
 #' @description The `$optimize()` method of a [`CmdStanModel`] object runs
-#'   Stan's optimizer. Following CmdStan's terminology, optimization without
-#'   the Jacobian adjustment (the default) returns a maximum likelihood estimate
-#'   (MLE), whereas optimization with the adjustment returns a maximum a
-#'   posteriori (MAP) estimate. More precisely, without the adjustment the
+#'   Stan's optimizer. Without the Jacobian adjustment (the default),
 #'   optimization finds a mode of the target in the original constrained
-#'   parameter space (if the mode exists), whereas with the adjustment it
-#'   finds a mode of the corresponding density in the unconstrained parameter
-#'   space.
+#'   parameter space (if the mode exists). With the adjustment, it finds a mode
+#'   of the corresponding density in the unconstrained parameter space.
 #'
 #'   The `jacobian` argument does not determine whether prior terms are
 #'   included. Every contribution to the Stan program's `target`, including
-#'   prior terms, is included under either setting. The MLE or MAP
-#'   interpretation therefore depends on both the contents of the target and the
-#'   parameterization. The Jacobian adjustment is particularly useful when
-#'   making a distributional approximation in the unconstrained space (see
-#'   [Laplace sampling][model-method-laplace]). If the model has only
-#'   unconstrained parameters, including the Jacobian has no effect. See the
-#'   [CmdStan User's Guide](https://mc-stan.org/docs/cmdstan-guide/index.html)
-#'   for more details.
+#'   prior terms, is included under either setting. The Jacobian adjustment is
+#'   particularly useful when making a distributional approximation in the
+#'   unconstrained space (see [Laplace sampling][model-method-laplace]). If the
+#'   model has only unconstrained parameters, including the Jacobian has no
+#'   effect. See the [CmdStan User's
+#'   Guide](https://mc-stan.org/docs/cmdstan-guide/index.html) for more details.
 #'
 #'   Any argument left as `NULL` will default to the default value used by the
 #'   installed version of CmdStan. See the [CmdStan User’s
@@ -1496,13 +1490,12 @@ CmdStanModel$set("public", name = "sample_mpi", value = sample_mpi)
 #'   running `cmdstanr_example(method="optimize")$metadata()`.
 #' @param jacobian (logical) Whether or not to use the Jacobian adjustment for
 #'   constrained variables. For historical reasons, the default is `FALSE`.
-#'   CmdStan refers to the estimates obtained with `FALSE` and `TRUE` as MLE and
-#'   MAP estimates, respectively. More precisely, `FALSE` finds a mode of the
-#'   target in the constrained parameter space and `TRUE` finds a mode in the
-#'   unconstrained space. This argument does not control whether prior terms are
-#'   included. See the **Description** section and the CmdStan User's Guide for
-#'   more details. For use later with [`$laplace()`][model-method-laplace], the
-#'   `jacobian` argument should typically be set to `TRUE`.
+#'   `FALSE` finds a mode of the target in the constrained parameter space and
+#'   `TRUE` finds a mode in the unconstrained space. This argument does not
+#'   control whether prior terms are included. See the **Description** section
+#'   and the CmdStan User's Guide for more details. For use later with
+#'   [`$laplace()`][model-method-laplace], the `jacobian` argument should
+#'   typically be set to `TRUE`.
 #' @param init_alpha (positive real) The initial step size parameter.
 #' @param tol_obj (positive real) Convergence tolerance on changes in objective function value.
 #' @param tol_rel_obj (positive real) Convergence tolerance on relative changes in objective function value.
@@ -1604,16 +1597,11 @@ CmdStanModel$set("public", name = "optimize", value = optimize)
 #'
 #' @description The `$laplace()` method of a [`CmdStanModel`] object produces a
 #'   sample from a normal approximation centered at the mode of a distribution
-#'   in the unconstrained space. Following CmdStan's terminology, if the mode is
-#'   a maximum a posteriori (MAP) estimate, the samples provide an estimate of
-#'   the mean and standard deviation of the posterior distribution. If the mode
-#'   is a maximum likelihood estimate (MLE), the sample provides an estimate of
-#'   the standard error of the likelihood. Whether the mode is called MAP or MLE
-#'   depends on the value of the `jacobian` argument when running optimization.
-#'   This terminology does not imply that `jacobian` controls whether prior
-#'   terms are included; it controls the parameterization of the density, while
-#'   the Stan program determines the contents of the target. See the
-#'   [CmdStan User’s Guide](https://mc-stan.org/docs/cmdstan-guide/)
+#'   in the unconstrained space. When the mode was found with the Jacobian
+#'   adjustment, the draws provide an estimate of the mean and standard
+#'   deviation of the posterior distribution. See the `jacobian` argument below
+#'   for how this setting relates to the value used when running optimization,
+#'   and the [CmdStan User’s Guide](https://mc-stan.org/docs/cmdstan-guide/)
 #'   for more details.
 #'
 #'   Any argument left as `NULL` will default to the default value used by the
@@ -1639,7 +1627,7 @@ CmdStanModel$set("public", name = "optimize", value = optimize)
 #' @param draws (positive integer) The number of draws to take.
 #' @param jacobian (logical) Whether or not to enable the Jacobian adjustment
 #'   for constrained parameters. The default is `TRUE`. See the
-#'   [Laplace Sampling](https://mc-stan.org/docs/cmdstan-guide/laplace-sampling.html)
+#'   [Laplace Sampling](https://mc-stan.org/docs/cmdstan-guide/laplace_sample_config.html)
 #'   section of the CmdStan User's Guide for more details. If `mode` is not
 #'   `NULL` then the value of `jacobian` must match the value used when
 #'   optimization was originally run so the mode and the Laplace approximation
diff --git a/man/CmdStanMLE.Rd b/man/CmdStanMLE.Rd
index 91c73e561..cf23092c0 100644
--- a/man/CmdStanMLE.Rd
+++ b/man/CmdStanMLE.Rd
@@ -6,14 +6,12 @@
 \description{
 A \code{CmdStanMLE} object is the fitted model object returned by the
 \code{\link[=model-method-optimize]{$optimize()}} method of a \code{\link{CmdStanModel}} object.
-Following CmdStan's terminology, the object contains an MLE if optimization
-was run with \code{jacobian=FALSE} and a MAP estimate if it was run with
-\code{jacobian=TRUE}. The name "MLE" is retained for historical reasons. More
-precisely, the estimates correspond to a mode in either the constrained
-parameter space or the unconstrained parameter space, depending on the
-value of \code{jacobian} (and whether the model has constrained parameters).
-The \code{jacobian} argument does not control whether prior terms are included;
-all contributions to the Stan program's target are included under either
+The name "MLE" is retained for historical reasons. With
+\code{jacobian = FALSE}, the point estimate is a mode of the target in the
+constrained parameter space. With \code{jacobian = TRUE}, it is a mode of the
+corresponding density in the unconstrained parameter space. The \code{jacobian}
+argument does not control whether prior terms are included; all
+contributions to the Stan program's target are included under either
 setting. See \code{\link[=model-method-optimize]{$optimize()}} and the CmdStan User's
 Guide for more details.
 
diff --git a/man/fit-method-lp.Rd b/man/fit-method-lp.Rd
index 4c598a113..25dc27924 100644
--- a/man/fit-method-lp.Rd
+++ b/man/fit-method-lp.Rd
@@ -60,8 +60,8 @@ distribution on the unconstrained space.
 fit_mcmc <- cmdstanr_example("logistic")
 head(fit_mcmc$lp())
 
-fit_mle <- cmdstanr_example("logistic", method = "optimize")
-fit_mle$lp()
+fit_optim <- cmdstanr_example("logistic", method = "optimize")
+fit_optim$lp()
 
 fit_vb <- cmdstanr_example("logistic", method = "variational")
 plot(fit_vb$lp(), fit_vb$lp_approx())
diff --git a/man/fit-method-metadata.Rd b/man/fit-method-metadata.Rd
index e249c2391..2d9802f21 100644
--- a/man/fit-method-metadata.Rd
+++ b/man/fit-method-metadata.Rd
@@ -20,8 +20,8 @@ fitting the model. See \strong{Examples} and \code{\link[=read_cmdstan_csv]{read
 fit_mcmc <- cmdstanr_example("logistic", method = "sample")
 str(fit_mcmc$metadata())
 
-fit_mle <- cmdstanr_example("logistic", method = "optimize")
-str(fit_mle$metadata())
+fit_optim <- cmdstanr_example("logistic", method = "optimize")
+str(fit_optim$metadata())
 
 fit_vb <- cmdstanr_example("logistic", method = "variational")
 str(fit_vb$metadata())
diff --git a/man/fit-method-mle.Rd b/man/fit-method-mle.Rd
index 072d8503c..1cb0c063f 100644
--- a/man/fit-method-mle.Rd
+++ b/man/fit-method-mle.Rd
@@ -17,19 +17,18 @@ A numeric vector. See \strong{Examples}.
 }
 \description{
 The \verb{$mle()} method is only available for \code{\link{CmdStanMLE}}
-objects. It returns the point estimate as a numeric vector with one element
-per variable. The returned vector does \emph{not} include \code{lp__}, the
-target log density evaluated by Stan, up to an additive constant. \code{lp__} is
-available via the \code{\link[=fit-method-lp]{$lp()}} method and also included in the
+objects. The method name is retained for historical reasons. It returns the
+point estimate as a numeric vector with one element per variable. The
+returned vector does \emph{not} include \code{lp__}, the target log density evaluated
+by Stan, up to an additive constant. \code{lp__} is available via the
+\code{\link[=fit-method-lp]{$lp()}} method and also included in the
 \code{\link[=fit-method-draws]{$draws()}} method.
 
-Following CmdStan's terminology, for models with constrained parameters
-that are fit with \code{jacobian=TRUE}, this point estimate is called a maximum
-a posteriori (MAP) estimate rather than an MLE. More precisely,
-\code{jacobian=FALSE} finds a mode of the target in the constrained parameter
-space and \code{jacobian=TRUE} finds a mode in the unconstrained parameter
-space. See \code{\link[=model-method-optimize]{$optimize()}} and the CmdStan User's
-Guide for more details.
+With \code{jacobian = FALSE}, the point estimate is a mode of the target in the
+constrained parameter space. With \code{jacobian = TRUE}, it is a mode of the
+corresponding density in the unconstrained parameter space. See
+\code{\link[=model-method-optimize]{$optimize()}} and the CmdStan User's Guide for more
+details.
 }
 \examples{
 \dontrun{
diff --git a/man/fit-method-output.Rd b/man/fit-method-output.Rd
index bfa8a9eda..a4bc4b287 100644
--- a/man/fit-method-output.Rd
+++ b/man/fit-method-output.Rd
@@ -32,8 +32,8 @@ fit_mcmc$output(1)
 out <- fit_mcmc$output()
 str(out)
 
-fit_mle <- cmdstanr_example("logistic", method = "optimize")
-fit_mle$output()
+fit_optim <- cmdstanr_example("logistic", method = "optimize")
+fit_optim$output()
 
 fit_vb <- cmdstanr_example("logistic", method = "variational")
 fit_vb$output()
diff --git a/man/fit-method-time.Rd b/man/fit-method-time.Rd
index c2a3011ab..7e5dfe489 100644
--- a/man/fit-method-time.Rd
+++ b/man/fit-method-time.Rd
@@ -39,13 +39,13 @@ fit_mcmc$time()
 fit_vb <- cmdstanr_example("logistic", method = "variational")
 fit_vb$time()
 
-fit_mle <- cmdstanr_example("logistic", method = "optimize", jacobian = TRUE)
-fit_mle$time()
+fit_optim <- cmdstanr_example("logistic", method = "optimize", jacobian = TRUE)
+fit_optim$time()
 
-# use fit_mle to draw samples from laplace approximation
-fit_laplace <- cmdstanr_example("logistic", method = "laplace", mode = fit_mle)
+# use fit_optim to draw samples from laplace approximation
+fit_laplace <- cmdstanr_example("logistic", method = "laplace", mode = fit_optim)
 fit_laplace$time() # just time for drawing sample not for running optimize
-fit_laplace$time()$total + fit_mle$time()$total # total time
+fit_laplace$time()$total + fit_optim$time()$total # total time
 }
 
 }
diff --git a/man/model-method-laplace.Rd b/man/model-method-laplace.Rd
index aab83fe96..e0b6ab415 100644
--- a/man/model-method-laplace.Rd
+++ b/man/model-method-laplace.Rd
@@ -166,7 +166,7 @@ taken to run the \verb{$optimize()} method.}
 
 \item{jacobian}{(logical) Whether or not to enable the Jacobian adjustment
 for constrained parameters. The default is \code{TRUE}. See the
-\href{https://mc-stan.org/docs/cmdstan-guide/laplace-sampling.html}{Laplace Sampling}
+\href{https://mc-stan.org/docs/cmdstan-guide/laplace_sample_config.html}{Laplace Sampling}
 section of the CmdStan User's Guide for more details. If \code{mode} is not
 \code{NULL} then the value of \code{jacobian} must match the value used when
 optimization was originally run so the mode and the Laplace approximation
@@ -200,16 +200,11 @@ A \code{\link{CmdStanLaplace}} object.
 \description{
 The \verb{$laplace()} method of a \code{\link{CmdStanModel}} object produces a
 sample from a normal approximation centered at the mode of a distribution
-in the unconstrained space. Following CmdStan's terminology, if the mode is
-a maximum a posteriori (MAP) estimate, the samples provide an estimate of
-the mean and standard deviation of the posterior distribution. If the mode
-is a maximum likelihood estimate (MLE), the sample provides an estimate of
-the standard error of the likelihood. Whether the mode is called MAP or MLE
-depends on the value of the \code{jacobian} argument when running optimization.
-This terminology does not imply that \code{jacobian} controls whether prior
-terms are included; it controls the parameterization of the density, while
-the Stan program determines the contents of the target. See the
-\href{https://mc-stan.org/docs/cmdstan-guide/}{CmdStan User’s Guide}
+in the unconstrained space. When the mode was found with the Jacobian
+adjustment, the draws provide an estimate of the mean and standard
+deviation of the posterior distribution. See the \code{jacobian} argument below
+for how this setting relates to the value used when running optimization,
+and the \href{https://mc-stan.org/docs/cmdstan-guide/}{CmdStan User’s Guide}
 for more details.
 
 Any argument left as \code{NULL} will default to the default value used by the
diff --git a/man/model-method-optimize.Rd b/man/model-method-optimize.Rd
index b176b3783..0bd0f8a17 100644
--- a/man/model-method-optimize.Rd
+++ b/man/model-method-optimize.Rd
@@ -162,13 +162,12 @@ running \code{cmdstanr_example(method="optimize")$metadata()}.}
 
 \item{jacobian}{(logical) Whether or not to use the Jacobian adjustment for
 constrained variables. For historical reasons, the default is \code{FALSE}.
-CmdStan refers to the estimates obtained with \code{FALSE} and \code{TRUE} as MLE and
-MAP estimates, respectively. More precisely, \code{FALSE} finds a mode of the
-target in the constrained parameter space and \code{TRUE} finds a mode in the
-unconstrained space. This argument does not control whether prior terms are
-included. See the \strong{Description} section and the CmdStan User's Guide for
-more details. For use later with \code{\link[=model-method-laplace]{$laplace()}}, the
-\code{jacobian} argument should typically be set to \code{TRUE}.}
+\code{FALSE} finds a mode of the target in the constrained parameter space and
+\code{TRUE} finds a mode in the unconstrained space. This argument does not
+control whether prior terms are included. See the \strong{Description} section
+and the CmdStan User's Guide for more details. For use later with
+\code{\link[=model-method-laplace]{$laplace()}}, the \code{jacobian} argument should
+typically be set to \code{TRUE}.}
 
 \item{init_alpha}{(positive real) The initial step size parameter.}
 
@@ -211,25 +210,18 @@ A \code{\link{CmdStanMLE}} object.
 }
 \description{
 The \verb{$optimize()} method of a \code{\link{CmdStanModel}} object runs
-Stan's optimizer. Following CmdStan's terminology, optimization without
-the Jacobian adjustment (the default) returns a maximum likelihood estimate
-(MLE), whereas optimization with the adjustment returns a maximum a
-posteriori (MAP) estimate. More precisely, without the adjustment the
+Stan's optimizer. Without the Jacobian adjustment (the default),
 optimization finds a mode of the target in the original constrained
-parameter space (if the mode exists), whereas with the adjustment it
-finds a mode of the corresponding density in the unconstrained parameter
-space.
+parameter space (if the mode exists). With the adjustment, it finds a mode
+of the corresponding density in the unconstrained parameter space.
 
 The \code{jacobian} argument does not determine whether prior terms are
 included. Every contribution to the Stan program's \code{target}, including
-prior terms, is included under either setting. The MLE or MAP
-interpretation therefore depends on both the contents of the target and the
-parameterization. The Jacobian adjustment is particularly useful when
-making a distributional approximation in the unconstrained space (see
-\link[=model-method-laplace]{Laplace sampling}). If the model has only
-unconstrained parameters, including the Jacobian has no effect. See the
-\href{https://mc-stan.org/docs/cmdstan-guide/index.html}{CmdStan User's Guide}
-for more details.
+prior terms, is included under either setting. The Jacobian adjustment is
+particularly useful when making a distributional approximation in the
+unconstrained space (see \link[=model-method-laplace]{Laplace sampling}). If the
+model has only unconstrained parameters, including the Jacobian has no
+effect. See the \href{https://mc-stan.org/docs/cmdstan-guide/index.html}{CmdStan User's Guide} for more details.
 
 Any argument left as \code{NULL} will default to the default value used by the
 installed version of CmdStan. See the \href{https://mc-stan.org/docs/cmdstan-guide/}{CmdStan User’s Guide} for more details on the
diff --git a/vignettes/cmdstanr.Rmd b/vignettes/cmdstanr.Rmd
index dd6d42cd2..86c287b5e 100644
--- a/vignettes/cmdstanr.Rmd
+++ b/vignettes/cmdstanr.Rmd
@@ -324,33 +324,34 @@ are called in a similar way to the `$sample()` method demonstrated above.
 
 ### Optimization
 
-Following CmdStan's terminology, we can find the (penalized) maximum likelihood
-estimate (MLE) using [`$optimize()`](https://mc-stan.org/cmdstanr/reference/model-method-optimize.html).
+We can find a mode of the target using
+[`$optimize()`](https://mc-stan.org/cmdstanr/reference/model-method-optimize.html).
+The point estimate is extracted with the `$mle()` method (the method name is
+retained for historical reasons).
 
 ```{r optimize}
-fit_mle <- mod$optimize(data = data_list, seed = 123)
-fit_mle$print() # includes lp__ (log prob calculated by Stan program)
-fit_mle$mle("theta")
+fit_optim <- mod$optimize(data = data_list, seed = 123)
+fit_optim$print() # includes lp__ (log prob calculated by Stan program)
+fit_optim$mle("theta")
 ```
 
-Here's a plot comparing the penalized MLE to the posterior distribution of
+Here's a plot comparing the optimized point estimate to the posterior of
 `theta`.
 
-```{r plot-mle, message = FALSE}
+```{r plot-optim, message = FALSE}
 mcmc_hist(fit$draws("theta")) +
-  vline_at(fit_mle$mle("theta"), size = 1.5)
+  vline_at(fit_optim$mle("theta"), size = 1.5)
 ```
 
 For optimization, by default the mode is calculated without the Jacobian
-adjustment for constrained variables. Including the Jacobian adjustment by setting `jacobian=TRUE` produces a maximum a posteriori (MAP) estimate. More precisely, the default
-finds a mode of the target in the constrained parameter space, whereas
-`jacobian=TRUE` finds a mode of the corresponding density in the unconstrained
-space. See the
-[Maximum Likelihood Estimation](https://mc-stan.org/docs/cmdstan-guide/maximum-likelihood-estimation.html)
+adjustment for constrained variables, which finds a mode of the target in the
+constrained parameter space, whereas `jacobian = TRUE` finds a mode of the
+corresponding density in the unconstrained space. See the
+[Jacobian adjustments](https://mc-stan.org/docs/cmdstan-guide/optimize_config.html#jacobian-adjustments)
 section of the CmdStan User's Guide for more details.
 
-```{r optimize-map}
-fit_map <- mod$optimize(
+```{r optimize-jacobian}
+fit_optim_jacobian <- mod$optimize(
   data = data_list,
   jacobian = TRUE,
   seed = 123
@@ -361,23 +362,21 @@ fit_map <- mod$optimize(
 
 The [`$laplace()`](https://mc-stan.org/cmdstanr/reference/model-method-laplace.html)
 method produces a sample from a normal approximation centered at the mode of a
-distribution in the unconstrained space. Following CmdStan's terminology, if
-the mode is a MAP estimate, the draws provide an estimate of the mean and
-standard deviation of the posterior distribution. If the mode is an MLE, the
-draws provide an estimate of the standard error of the likelihood. Whether
-the mode is called MAP or MLE depends on the value of the `jacobian` argument
-when running optimization. The Laplace `jacobian` setting must match the setting
-used for optimization so that both use the same target density. See the
-[Laplace Sampling](https://mc-stan.org/docs/cmdstan-guide/laplace-sampling.html)
+distribution in the unconstrained space. When the mode was found with the
+Jacobian adjustment, the draws provide an estimate of the mean and standard
+deviation of the posterior distribution. The Laplace `jacobian` setting must
+match the setting used for optimization so that both use the same target
+density. See the
+[Laplace Sampling](https://mc-stan.org/docs/cmdstan-guide/laplace_sample_config.html)
 chapter of the CmdStan User's Guide for more details.
 
-Here we pass in the `fit_map` object from above as the `mode` argument. If
-`mode` is omitted then optimization will be run internally before taking draws
-from the normal approximation.
+Here we pass in the `fit_optim_jacobian` object from above as the `mode`
+argument. If `mode` is omitted then optimization will be run internally before
+taking draws from the normal approximation.
 
 ```{r laplace}
 fit_laplace <- mod$laplace(
-    mode = fit_map,
+    mode = fit_optim_jacobian,
     draws = 4000,
     data = data_list,
     seed = 123,