Merge pull request #865 from stan-dev/bugfix/844-Neals-funnel-refs

Neal's funnel density and non-centered sample plot side-by-side with single caption explaining both.

Author: mitzimorris

src/stan-users-guide/efficiency-tuning.qmd

@@ -267,9 +267,6 @@ 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 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.
-
-![Neal's funnel density](img/funnel.png)
 
 The funnel can be implemented directly in Stan as follows.
 
@@ -289,16 +286,23 @@ the neck of the funnel, where $y$ is small and thus $x$ is constrained
 to be near 0.  This is due to the fact that the density's scale
 changes with $y$, so that a step size that works well in the body will
 be too large for the neck, and a step size that works in the neck will be
-inefficient in the body.  This can be seen in the following plot.
+inefficient in the body.  This can be seen in the following plots.
+
+::: {#fig-funnel layout="[55, 45]"}
+
+![](img/funnel.png)
+
+![](img/funnel-fit.png)
+
 
-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
+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.
-
-![](img/funnel-fit.png)
+:::
 
 In this particular instance, because the analytic form of the density
 from which samples are drawn is known, the model can be converted to

src/stan-users-guide/regression.qmd

@@ -875,7 +875,8 @@ y_n     & \sim \textsf{normal}(x_n \beta, \sigma) \\
 In this case, as $\tau \rightarrow 0$ and $\beta_k \rightarrow 0$, the
 posterior density
 $$ p(\beta,\tau,\sigma|y,x) \propto p(y|x,\beta,\tau,\sigma) $$
-grows without bound.  See the [plot of Neal's funnel density](#funnel.figure), which has similar behavior.
+grows without bound.
+See the [Neal's funnel density](efficiency-tuning.qmd#funnel.figure), which has similar behavior.
 
 There is obviously no MLE estimate for $\beta,\tau,\sigma$ in such a
 case, and therefore the model must be modified if posterior modes are