From 1eb1bc2d04f89373225a82b5ce98428026c3250b Mon Sep 17 00:00:00 2001
From: Mitzi Morris <mitzi212@gmail.com>
Date: Tue, 8 Apr 2025 20:52:40 -0400
Subject: [PATCH 1/7] more info on affine xform

---
 src/quarto-config                          |  2 +-
 src/reference-manual/transforms.qmd        |  2 +-
 src/reference-manual/types.qmd             | 21 +++++++++++++++
 src/stan-users-guide/efficiency-tuning.qmd | 30 ++++++++++++++--------
 4 files changed, 43 insertions(+), 12 deletions(-)

diff --git a/src/quarto-config b/src/quarto-config
index 12e4c0771..65904751c 160000
--- a/src/quarto-config
+++ b/src/quarto-config
@@ -1 +1 @@
-Subproject commit 12e4c0771ba11de259bf9298dc9f56fc9dc3e0f0
+Subproject commit 65904751c830555ec8d81d9270e609ec190c425f
diff --git a/src/reference-manual/transforms.qmd b/src/reference-manual/transforms.qmd
index 85083fadd..72d5069cb 100644
--- a/src/reference-manual/transforms.qmd
+++ b/src/reference-manual/transforms.qmd
@@ -330,7 +330,7 @@ $$
 
 The default value for the offset $\mu$ is $0$ and for the multiplier $\sigma$ is
 $1$ in case not both are specified.
-
+For a container variable, the affine transform is applied to each element of that variable.
 
 ### Affine inverse transform {-}
 
diff --git a/src/reference-manual/types.qmd b/src/reference-manual/types.qmd
index 9e23f92e3..c645934f7 100644
--- a/src/reference-manual/types.qmd
+++ b/src/reference-manual/types.qmd
@@ -472,6 +472,27 @@ model {
 }
 ```
 
+For a container variable, the affine transform is applied to each element of that variable.
+As an example, the non-centered parameterization of Neal's Funnel in the
+[Stan User's Guide reparameterization section](https://mc-stan.org/docs/stan-users-guide/reparameterization.html),
+$$
+p(y,x) = \textsf{normal}(y \mid 0,3) \times \prod_{n=1}^9
+\textsf{normal}(x_n \mid 0,\exp(y/2)).
+$$
+can be written as:
+
+```stan
+parameters {
+  real<multiplier=3> y;
+  vector<multiplier=exp(y/2)>[9] x;
+}
+model {
+  y ~ std_normal();
+  x ~ std_normal();
+}
+```
+where the affine transform is applied to every element of vector `x`.
+
 ### Expressions as bounds and offset/multiplier {-}
 
 Bounds (and offset and multiplier)
diff --git a/src/stan-users-guide/efficiency-tuning.qmd b/src/stan-users-guide/efficiency-tuning.qmd
index af26da697..f929dac1d 100644
--- a/src/stan-users-guide/efficiency-tuning.qmd
+++ b/src/stan-users-guide/efficiency-tuning.qmd
@@ -266,8 +266,6 @@ funnel's neck is particularly sharp because of the exponential
 function applied to $y$.  A plot of the log marginal density of $y$
 and the first dimension $x_1$ is shown in the following plot.
 
-
-
 The funnel can be implemented directly in Stan as follows.
 
 ```stan
@@ -327,14 +325,26 @@ model {
 ```
 
 In this second model, the parameters `x_raw` and `y_raw` are
-sampled as independent standard normals, which is easy for Stan.  These
-are then transformed into samples from the funnel.  In this case, the
-same transform may be used to define Monte Carlo samples directly
-based on independent standard normal samples; Markov chain Monte Carlo
-methods are not necessary. If such a reparameterization were used in
-Stan code, it is useful to provide a comment indicating what the
-distribution for the parameter implies for the distribution of the
-transformed parameter.
+sampled as independent standard normals, which is easy for Stan,
+and then transformed into samples from the funnel.
+When this reparameterization is used in Stan code, a comment indicating what the
+distribution for the parameter implies for the distribution of the transformed parameter
+will improve readibility and maintainability.
+
+As of Stan release v2.19.0, this program can be written using Stan's
+[affinely transformed real type](https://mc-stan.org/docs/reference-manual/types.html#affine-transform.section).
+The affine transform on the vector `x` is applied to each element of `x`.
+
+```stan
+parameters {
+  real<multiplier=3> y;
+  vector<multiplier=exp(y/2)>[9] x;
+}
+model {
+  y ~ std_normal(); // implies y ~ normal(0, 3)
+  x ~ std_normal(); // implies x ~ normal(0, exp(y/2))
+}
+```
 
 ### Reparameterizing the Cauchy {-}
 

From 9383f772f71c393b2b8b8ebf74f68f824682bf1c Mon Sep 17 00:00:00 2001
From: Mitzi Morris <mitzi212@gmail.com>
Date: Wed, 9 Apr 2025 13:17:05 -0400
Subject: [PATCH 2/7] remove submodule quarto-config

---
 .gitmodules | 3 ---
 1 file changed, 3 deletions(-)

diff --git a/.gitmodules b/.gitmodules
index 4531c3001..e69de29bb 100644
--- a/.gitmodules
+++ b/.gitmodules
@@ -1,3 +0,0 @@
-[submodule "src/quarto-config"]
-	path = src/quarto-config
-	url = https://github.com/stan-dev/quarto-config

From c9731d238b115b2123ec6769ac8a3eda06c29898 Mon Sep 17 00:00:00 2001
From: Mitzi Morris <mitzi212@gmail.com>
Date: Wed, 9 Apr 2025 13:21:22 -0400
Subject: [PATCH 3/7] fix submodule quarto-config

---
 .gitmodules | 3 +++
 1 file changed, 3 insertions(+)

diff --git a/.gitmodules b/.gitmodules
index e69de29bb..4531c3001 100644
--- a/.gitmodules
+++ b/.gitmodules
@@ -0,0 +1,3 @@
+[submodule "src/quarto-config"]
+	path = src/quarto-config
+	url = https://github.com/stan-dev/quarto-config

From 168a99ec6a0a90996ec4055bc0a84554c2ee3770 Mon Sep 17 00:00:00 2001
From: Mitzi Morris <mitzi212@gmail.com>
Date: Wed, 9 Apr 2025 13:32:40 -0400
Subject: [PATCH 4/7] submodule cleanup

---
 docs/quarto-config/quarto_styles.css | 326 ---------------------------
 src/quarto-config                    |   2 +-
 2 files changed, 1 insertion(+), 327 deletions(-)
 delete mode 100644 docs/quarto-config/quarto_styles.css

diff --git a/docs/quarto-config/quarto_styles.css b/docs/quarto-config/quarto_styles.css
deleted file mode 100644
index 5cd9e2db5..000000000
--- a/docs/quarto-config/quarto_styles.css
+++ /dev/null
@@ -1,326 +0,0 @@
-html {
-  scroll-padding-top: 70px; /* height of header */
-}
-
-/* quarto website styling */
-
-@media (min-width: 1020px) {
-  .navbar-brand-container {
-    margin-right: 1em;
-  }
-}
-
-@media (max-width: 1060px) and (min-width: 991.98px) {
-  #navbarCollapse ul:last-of-type a.nav-link {
-    padding-left: 0.25em;
-    padding-right: 0.25em;
-  }
-
-  .navbar #quarto-search {
-    margin-left: 0.1em;
-  }
-}
-/*Remove top and bottom padding from navbar image*/
-.navbar-brand {
-  padding-top: 0;
-  padding-bottom: 0;
-}
-
-.navbar-logo {
-  max-height: 44px;
-}
-
-.btn-action-primary {
-  color: var(--stan-bg);
-  background-color: var(--stan-secondary) !important;
-}
-
-.btn-action-primary:hover {
-  color: white;
-}
-
-.btn-action {
-  min-width: 165px;
-  border-radius: 30px;
-  border: none;
-}
-
-/* stan-dev custom styling */
-
-a {
-  color: var(--stan-secondary);
-}
-a.nav-link.active {
-  color: var(--stan-secondary);
-}
-a.sidebar-item-text.sidebar-link.active {
-  color: var(--stan-secondary);
-}
-.sidebar.a.active {
-  color: var(--stan-secondary);
-}
-
-div [data-bs-target^="#quarto-sidebar-section"] {
-  font-weight: bold;
-  font-style: italic;
-  color: var(--stan-secondary);
-}
-
-code {
-  white-space: inherit;
-  color: var(--bs-dark-warm);
-  background-color: var(--stan-bg);
-}
-
-code:not(.sourceCode) {
-  color: var(--bs-dark-warm);
-  background-color: var(--stan-bg);
-}
-
-p code {
-  white-space: inherit;
-  color: var(--bs-dark-warm);
-  background-color: var(--stan-bg);
-}
-
-p code:not(.sourceCode) {
-  color: var(--bs-dark-warm);
-  background-color: var(--stan-bg);
-}
-
-li code:not(.sourceCode) {
-  color: var(--bs-dark-warm);
-  background-color: var(--stan-bg);
-}
-
-td code:not(.sourceCode) {
-  color: var(--bs-dark-warm);
-  background-color: var(--stan-bg);
-}
-
-pre {
-  word-break: normal;
-  word-wrap: normal;
-  color: var(--bs-dark-warm);
-  background-color: var(--stan-bg);
-}
-pre code {
-  white-space: inherit;
-  margin: 0;
-  padding: 0;
-  color: var(--bs-dark-warm);
-  background-color: var(--stan-bg);
-}
-
-pre.sourceCode {
-  white-space: inherit;
-  margin: 0;
-  padding: 0;
-  color: var(--bs-dark-warm);
-  background-color: var(--stan-bg);
-}
-
-#quarto-appendix {
-  color: var(--bs-dark-warm);
-  background-color: var(--stan-bg);
-}
-
-.align-equations {
-  text-align: left;
-  margin-left: 2em; /* Adjust as needed */
-}
-
-.book .book-body .page-wrapper .page-inner section.normal pre {
-  font-family: "Lucida Console", Monaco, Menlo, monospaced;
-  margin-left: 2em;
-  margin-below: 0.6em;
-  margin-above: -0.3em;
-  padding: 0;
-}
-
-.sidebar.sidebar-navigation > * {
-  padding-top: 0em;
-}
-
-#quarto-sidebar
-  > div.sidebar-menu-container
-  > ul
-  > li:nth-child(1)
-  > div
-  > a
-  > span {
-  font-weight: bold;
-  font-style: italic;
-  font-size: 1.3em;
-  color: var(--stan-secondary);
-  margin-above: -20em;
-  padding: 0;
-}
-
-/* quarto landing page styling */
-
-.content-block {
-  padding-top: 20px;
-  padding-bottom: 10px;
-  margin-left: 30px;
-  margin-right: 30px;
-}
-
-@media (min-width: 900px) {
-  .content-block {
-    margin-left: 50px;
-    margin-right: 50px;
-  }
-}
-
-@media (min-width: 1200px) {
-  .content-block {
-    max-width: 1100px;
-    margin-left: auto;
-    margin-right: auto;
-  }
-}
-
-.hero-banner {
-  position: relative;
-  display: flex;
-  justify-content: center;
-  color: var(--stan-hero);
-  background-color: var(--stan-hero-bg);
-  border: 10px solid var(--stan-secondary);
-}
-
-.hero-banner h1 {
-  font-size: 2.5rem;
-  margin-top: 1rem;
-}
-
-.hero-banner .content-block {
-  padding-top: 0;
-}
-
-.grid-container {
-  display: grid;
-  grid-template-columns: auto repeat(4, 1fr);
-  gap: 0px;
-  width: 80%;
-}
-.option {
-  border: 1px solid var(--stan-dark);
-  display: flex;
-  justify-content: center;
-  align-items: center;
-  cursor: pointer;
-}
-.option.selected {
-  background-color: var(--stan-highlight);
-  color: var(--stan-dark);
-}
-.option.disabled {
-  cursor: not-allowed;
-  color: var(--stan-highlight);
-}
-.grid-item {
-  padding: 10px;
-  border: 1px solid var(--stan-dark);
-  color: var(--stan-dark);
-}
-
-.flex-grid {
-  display: flex;
-  flex-wrap: wrap;
-  overflow: hidden;
-  margin-left: 1px;
-}
-
-.flex-grid .col {
-  flex: 1 1 100px;
-  margin-top: 0.25rem;
-}
-
-/* when we think we're all on one line, remove the double bars */
-@container (min-width: 500px) {
-  .grid-item + .grid-item {
-    border-left: none;
-  }
-}
-/* otherwise, put a little space */
-@container (max-width: 500px) {
-  .flex-grid {
-    gap: 0.25rem;
-  }
-
-  .flex-grid .col {
-    margin-top: 0;
-  }
-}
-
-.row-label {
-  padding: 10px;
-  color: var(--stan-dark);
-  background-color: var(--stan-bg);
-  font-weight: bold;
-  text-align: left;
-  width: 20%;
-}
-
-/*  on small screens */
-@container (max-width: 700px) {
-  /* Move the label above, rather than to the side */
-  .row-label {
-    flex-basis: 100%;
-  }
-}
-
-.highlight-box {
-  background-color: var(--stan-highlight);
-  padding-left: 10px;
-}
-
-#install-grid,
-#prerequsite-box,
-#install-box {
-  container-type: inline-size;
-}
-
-/* Table Styles */
-table {
-  width: 100%;
-  border-collapse: collapse;
-  border-top: 1px solid var(--stan-dark); /* Adds top border to the table */
-  border-bottom: 1px solid var(--stan-dark); /* Adds bottom border to the table */
-}
-
-th,
-td {
-  border: none; /* Removes borders from table cells */
-  padding: 0.5em; /* Adjusts padding as needed */
-}
-
-thead th {
-  background-color: var(--stan-bg);
-  color: var(--stan-dark);
-}
-
-tbody td:first-child {
-  background-color: var(--stan-bg);
-  color: var(--stan-dark);
-}
-
-/* functions reference */
-.index-container {
-  display: flex;
-  flex-flow: row nowrap;
-  align-items: flex-end;
-  justify-content: space-between;
-}
-
-a.unlink {
-  color: inherit;
-  text-decoration: none;
-}
-
-span.detail {
-  font-size: small;
-  color: var(--quarto-text-muted);
-}
diff --git a/src/quarto-config b/src/quarto-config
index 65904751c..12e4c0771 160000
--- a/src/quarto-config
+++ b/src/quarto-config
@@ -1 +1 @@
-Subproject commit 65904751c830555ec8d81d9270e609ec190c425f
+Subproject commit 12e4c0771ba11de259bf9298dc9f56fc9dc3e0f0

From b3291de3629c69ae552aa55340070eec4967605b Mon Sep 17 00:00:00 2001
From: Mitzi Morris <mitzi212@gmail.com>
Date: Wed, 9 Apr 2025 13:37:59 -0400
Subject: [PATCH 5/7] undelete file

---
 docs/quarto-config/quarto_styles.css | 326 +++++++++++++++++++++++++++
 1 file changed, 326 insertions(+)
 create mode 100644 docs/quarto-config/quarto_styles.css

diff --git a/docs/quarto-config/quarto_styles.css b/docs/quarto-config/quarto_styles.css
new file mode 100644
index 000000000..5cd9e2db5
--- /dev/null
+++ b/docs/quarto-config/quarto_styles.css
@@ -0,0 +1,326 @@
+html {
+  scroll-padding-top: 70px; /* height of header */
+}
+
+/* quarto website styling */
+
+@media (min-width: 1020px) {
+  .navbar-brand-container {
+    margin-right: 1em;
+  }
+}
+
+@media (max-width: 1060px) and (min-width: 991.98px) {
+  #navbarCollapse ul:last-of-type a.nav-link {
+    padding-left: 0.25em;
+    padding-right: 0.25em;
+  }
+
+  .navbar #quarto-search {
+    margin-left: 0.1em;
+  }
+}
+/*Remove top and bottom padding from navbar image*/
+.navbar-brand {
+  padding-top: 0;
+  padding-bottom: 0;
+}
+
+.navbar-logo {
+  max-height: 44px;
+}
+
+.btn-action-primary {
+  color: var(--stan-bg);
+  background-color: var(--stan-secondary) !important;
+}
+
+.btn-action-primary:hover {
+  color: white;
+}
+
+.btn-action {
+  min-width: 165px;
+  border-radius: 30px;
+  border: none;
+}
+
+/* stan-dev custom styling */
+
+a {
+  color: var(--stan-secondary);
+}
+a.nav-link.active {
+  color: var(--stan-secondary);
+}
+a.sidebar-item-text.sidebar-link.active {
+  color: var(--stan-secondary);
+}
+.sidebar.a.active {
+  color: var(--stan-secondary);
+}
+
+div [data-bs-target^="#quarto-sidebar-section"] {
+  font-weight: bold;
+  font-style: italic;
+  color: var(--stan-secondary);
+}
+
+code {
+  white-space: inherit;
+  color: var(--bs-dark-warm);
+  background-color: var(--stan-bg);
+}
+
+code:not(.sourceCode) {
+  color: var(--bs-dark-warm);
+  background-color: var(--stan-bg);
+}
+
+p code {
+  white-space: inherit;
+  color: var(--bs-dark-warm);
+  background-color: var(--stan-bg);
+}
+
+p code:not(.sourceCode) {
+  color: var(--bs-dark-warm);
+  background-color: var(--stan-bg);
+}
+
+li code:not(.sourceCode) {
+  color: var(--bs-dark-warm);
+  background-color: var(--stan-bg);
+}
+
+td code:not(.sourceCode) {
+  color: var(--bs-dark-warm);
+  background-color: var(--stan-bg);
+}
+
+pre {
+  word-break: normal;
+  word-wrap: normal;
+  color: var(--bs-dark-warm);
+  background-color: var(--stan-bg);
+}
+pre code {
+  white-space: inherit;
+  margin: 0;
+  padding: 0;
+  color: var(--bs-dark-warm);
+  background-color: var(--stan-bg);
+}
+
+pre.sourceCode {
+  white-space: inherit;
+  margin: 0;
+  padding: 0;
+  color: var(--bs-dark-warm);
+  background-color: var(--stan-bg);
+}
+
+#quarto-appendix {
+  color: var(--bs-dark-warm);
+  background-color: var(--stan-bg);
+}
+
+.align-equations {
+  text-align: left;
+  margin-left: 2em; /* Adjust as needed */
+}
+
+.book .book-body .page-wrapper .page-inner section.normal pre {
+  font-family: "Lucida Console", Monaco, Menlo, monospaced;
+  margin-left: 2em;
+  margin-below: 0.6em;
+  margin-above: -0.3em;
+  padding: 0;
+}
+
+.sidebar.sidebar-navigation > * {
+  padding-top: 0em;
+}
+
+#quarto-sidebar
+  > div.sidebar-menu-container
+  > ul
+  > li:nth-child(1)
+  > div
+  > a
+  > span {
+  font-weight: bold;
+  font-style: italic;
+  font-size: 1.3em;
+  color: var(--stan-secondary);
+  margin-above: -20em;
+  padding: 0;
+}
+
+/* quarto landing page styling */
+
+.content-block {
+  padding-top: 20px;
+  padding-bottom: 10px;
+  margin-left: 30px;
+  margin-right: 30px;
+}
+
+@media (min-width: 900px) {
+  .content-block {
+    margin-left: 50px;
+    margin-right: 50px;
+  }
+}
+
+@media (min-width: 1200px) {
+  .content-block {
+    max-width: 1100px;
+    margin-left: auto;
+    margin-right: auto;
+  }
+}
+
+.hero-banner {
+  position: relative;
+  display: flex;
+  justify-content: center;
+  color: var(--stan-hero);
+  background-color: var(--stan-hero-bg);
+  border: 10px solid var(--stan-secondary);
+}
+
+.hero-banner h1 {
+  font-size: 2.5rem;
+  margin-top: 1rem;
+}
+
+.hero-banner .content-block {
+  padding-top: 0;
+}
+
+.grid-container {
+  display: grid;
+  grid-template-columns: auto repeat(4, 1fr);
+  gap: 0px;
+  width: 80%;
+}
+.option {
+  border: 1px solid var(--stan-dark);
+  display: flex;
+  justify-content: center;
+  align-items: center;
+  cursor: pointer;
+}
+.option.selected {
+  background-color: var(--stan-highlight);
+  color: var(--stan-dark);
+}
+.option.disabled {
+  cursor: not-allowed;
+  color: var(--stan-highlight);
+}
+.grid-item {
+  padding: 10px;
+  border: 1px solid var(--stan-dark);
+  color: var(--stan-dark);
+}
+
+.flex-grid {
+  display: flex;
+  flex-wrap: wrap;
+  overflow: hidden;
+  margin-left: 1px;
+}
+
+.flex-grid .col {
+  flex: 1 1 100px;
+  margin-top: 0.25rem;
+}
+
+/* when we think we're all on one line, remove the double bars */
+@container (min-width: 500px) {
+  .grid-item + .grid-item {
+    border-left: none;
+  }
+}
+/* otherwise, put a little space */
+@container (max-width: 500px) {
+  .flex-grid {
+    gap: 0.25rem;
+  }
+
+  .flex-grid .col {
+    margin-top: 0;
+  }
+}
+
+.row-label {
+  padding: 10px;
+  color: var(--stan-dark);
+  background-color: var(--stan-bg);
+  font-weight: bold;
+  text-align: left;
+  width: 20%;
+}
+
+/*  on small screens */
+@container (max-width: 700px) {
+  /* Move the label above, rather than to the side */
+  .row-label {
+    flex-basis: 100%;
+  }
+}
+
+.highlight-box {
+  background-color: var(--stan-highlight);
+  padding-left: 10px;
+}
+
+#install-grid,
+#prerequsite-box,
+#install-box {
+  container-type: inline-size;
+}
+
+/* Table Styles */
+table {
+  width: 100%;
+  border-collapse: collapse;
+  border-top: 1px solid var(--stan-dark); /* Adds top border to the table */
+  border-bottom: 1px solid var(--stan-dark); /* Adds bottom border to the table */
+}
+
+th,
+td {
+  border: none; /* Removes borders from table cells */
+  padding: 0.5em; /* Adjusts padding as needed */
+}
+
+thead th {
+  background-color: var(--stan-bg);
+  color: var(--stan-dark);
+}
+
+tbody td:first-child {
+  background-color: var(--stan-bg);
+  color: var(--stan-dark);
+}
+
+/* functions reference */
+.index-container {
+  display: flex;
+  flex-flow: row nowrap;
+  align-items: flex-end;
+  justify-content: space-between;
+}
+
+a.unlink {
+  color: inherit;
+  text-decoration: none;
+}
+
+span.detail {
+  font-size: small;
+  color: var(--quarto-text-muted);
+}

From a5dccff5f5386c6458ebecd4b7d6635199158fee Mon Sep 17 00:00:00 2001
From: Mitzi Morris <mitzi212@gmail.com>
Date: Wed, 9 Apr 2025 15:05:45 -0400
Subject: [PATCH 6/7] changes per code review

---
 src/reference-manual/transforms.qmd        |  1 +
 src/reference-manual/types.qmd             | 36 +++++++++++++++++-----
 src/stan-users-guide/efficiency-tuning.qmd | 30 +++++++++---------
 3 files changed, 43 insertions(+), 24 deletions(-)

diff --git a/src/reference-manual/transforms.qmd b/src/reference-manual/transforms.qmd
index 72d5069cb..9fa0e69b9 100644
--- a/src/reference-manual/transforms.qmd
+++ b/src/reference-manual/transforms.qmd
@@ -330,6 +330,7 @@ $$
 
 The default value for the offset $\mu$ is $0$ and for the multiplier $\sigma$ is
 $1$ in case not both are specified.
+
 For a container variable, the affine transform is applied to each element of that variable.
 
 ### Affine inverse transform {-}
diff --git a/src/reference-manual/types.qmd b/src/reference-manual/types.qmd
index c645934f7..227f64633 100644
--- a/src/reference-manual/types.qmd
+++ b/src/reference-manual/types.qmd
@@ -438,8 +438,8 @@ $1$ and multiplier $2$.
 real<offset=1, multiplier=2> x;
 ```
 
-As an example, we can give `x` a normal distribution with non-centered
-parameterization as follows.
+As an example, we can give `x` a normal distribution with non-centered parameterization.
+In this program, the affine transform is applied to every element of vector `x`.
 
 ```stan
 parameters {
@@ -450,7 +450,7 @@ model {
 }
 ```
 
-Recall that the centered parameterization is achieved with the code
+Recall the Stan code for the centered parameterization of this model.
 
 ```stan
 parameters {
@@ -461,17 +461,37 @@ model {
 }
 ```
 
-or equivalently
+Adding the offset, multiplier transform results in the equivalent non-centered parameterization.
 
 ```stan
 parameters {
-  real<offset=0, multiplier=1> x;
+  real<offset=mu, multiplier=sigma> x;
 }
 model {
   x ~ normal(mu, sigma);
 }
 ```
 
+With the affine transform, the sampling happens in different parameter space.
+The model itself doesn’t change if we make parameter transformation,
+but the geometry of the transformed space can be easier for the sampler.
+The above model is equivalent to the hand-coded non-centered parameterization.
+
+```stan
+parameters {
+  real x_raw;
+}
+transformed parameters {
+  real x = mu + x_raw * sigma;
+}
+model {
+  x_raw ~ std_normal();
+}
+```
+
+Use of the affine transform removes the overhead of declaring additional transformed parameters
+and directly expresses the hierarchical relationship between parameters.
+
 For a container variable, the affine transform is applied to each element of that variable.
 As an example, the non-centered parameterization of Neal's Funnel in the
 [Stan User's Guide reparameterization section](https://mc-stan.org/docs/stan-users-guide/reparameterization.html),
@@ -484,11 +504,11 @@ can be written as:
 ```stan
 parameters {
   real<multiplier=3> y;
-  vector<multiplier=exp(y/2)>[9] x;
+  vector<multiplier=exp(0.5 * y)>[9] x;
 }
 model {
-  y ~ std_normal();
-  x ~ std_normal();
+  y ~ normal(0, 3); 
+  x ~ std_normal(0, 0.5 * y);
 }
 ```
 where the affine transform is applied to every element of vector `x`.
diff --git a/src/stan-users-guide/efficiency-tuning.qmd b/src/stan-users-guide/efficiency-tuning.qmd
index f929dac1d..9af043392 100644
--- a/src/stan-users-guide/efficiency-tuning.qmd
+++ b/src/stan-users-guide/efficiency-tuning.qmd
@@ -293,13 +293,13 @@ inefficient in the body.  This can be seen in the following plots.
 ![](img/funnel-fit.png)
 
 
-Neal's funnel.  (Left) The marginal density of Neal's funnel for the upper-level variable $y$ and one lower-level variable $x_1$ (see the text for the formula).  The blue region has log density greater than -8, the yellow region density greater than -16, and the gray background a density less than -16.
-(Right) 4000 draws are taken from a run of Stan's sampler with default settings.
-Both plots are restricted to the shown window of $x_1$ and $y$ values;
-some draws fell outside of the displayed area as would be expected given
-the density.  The samples are consistent with the marginal density
-$p(y) = \textsf{normal}(y \mid 0,3)$, which has mean 0 and standard
-deviation 3.
+Neal's funnel.  (Left) The marginal density of Neal's funnel for the upper-level variable $y$ and
+one lower-level variable $x_1$ (see the text for the formula).  The blue region has log density
+greater than -8, the yellow region density greater than -16, and the gray background a density less
+than -16.  (Right) 4000 draws are taken from a run of Stan's sampler with default settings.
+Both plots are restricted to the shown window of $x_1$ and $y$ values; some draws fell outside of
+the displayed area as would be expected given the density.  The samples are consistent with the
+marginal density $p(y) = \textsf{normal}(y \mid 0,3)$, which has mean 0 and standard deviation 3.
 :::
 
 In this particular instance, because the analytic form of the density
@@ -312,11 +312,8 @@ parameters {
   vector[9] x_raw;
 }
 transformed parameters {
-  real y;
-  vector[9] x;
-
-  y = 3.0 * y_raw;
-  x = exp(y/2) * x_raw;
+  real y = 3.0 * y_raw;
+  vector[9] x = exp(0.5 * y) * x_raw;
 }
 model {
   y_raw ~ std_normal(); // implies y ~ normal(0, 3)
@@ -327,7 +324,7 @@ model {
 In this second model, the parameters `x_raw` and `y_raw` are
 sampled as independent standard normals, which is easy for Stan,
 and then transformed into samples from the funnel.
-When this reparameterization is used in Stan code, a comment indicating what the
+When this transform is used in Stan code, a comment indicating what the
 distribution for the parameter implies for the distribution of the transformed parameter
 will improve readibility and maintainability.
 
@@ -338,14 +335,15 @@ The affine transform on the vector `x` is applied to each element of `x`.
 ```stan
 parameters {
   real<multiplier=3> y;
-  vector<multiplier=exp(y/2)>[9] x;
+  vector<multiplier=exp(0.5 * y)>[9] x;
 }
 model {
-  y ~ std_normal(); // implies y ~ normal(0, 3)
-  x ~ std_normal(); // implies x ~ normal(0, exp(y/2))
+  y ~ normal(0, 3);
+  x ~ normal(0, 0.5 * y);
 }
 ```
 
+
 ### Reparameterizing the Cauchy {-}
 
 Sampling from heavy tailed distributions such as the Cauchy is

From acba64c7bf256fa11e3f516a8e153a90eef41d95 Mon Sep 17 00:00:00 2001
From: Mitzi Morris <mitzi212@gmail.com>
Date: Thu, 10 Apr 2025 12:23:11 -0400
Subject: [PATCH 7/7] changes per code review

---
 src/reference-manual/types.qmd | 7 +++----
 1 file changed, 3 insertions(+), 4 deletions(-)

diff --git a/src/reference-manual/types.qmd b/src/reference-manual/types.qmd
index 227f64633..df4dd4b22 100644
--- a/src/reference-manual/types.qmd
+++ b/src/reference-manual/types.qmd
@@ -472,10 +472,9 @@ model {
 }
 ```
 
-With the affine transform, the sampling happens in different parameter space.
-The model itself doesn’t change if we make parameter transformation,
-but the geometry of the transformed space can be easier for the sampler.
-The above model is equivalent to the hand-coded non-centered parameterization.
+Sampling is done on the unconstrained parameters.
+After applying the affine transform, the unconstrained parameters are standard normal,
+thus the above model is equivalent to the hand-coded non-centered parameterization.
 
 ```stan
 parameters {