Link ar1_turningpts to ar1_bayes and improve exposition (#969)

Redone on top of the numpyro/JAX rewrite of ar1_turningpts (#584).

- Cross-reference ar1_bayes from ar1_turningpts (overview, model setup,
  and posterior section) and add a forward {seealso} link back
- Frame the posterior computation as the "conditioning assumption" of
  ar1_bayes, apt here since the path starts from an atypical y_0
- Match the AR(1) model introduction to the bulleted style of ar1_bayes
  (and write IID rather than i.i.d.)
- Add a Conclusion and interpretation for the original/extended Wecker
  figures, tying back to the two sources of uncertainty

Note: N_t is already defined and figures already use MyST captions in
the rewritten lecture, so those earlier fixes are no longer needed.

Co-authored-by: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

Author: jstac

lectures/ar1_bayes.md

@@ -584,3 +584,7 @@ A simple rule of thumb:
 
 - use the conditioning assumption when early observations look transient or the starting point may be atypical;
 - use the stationary assumption when you are confident the process has been running near its long-run behaviour.
+
+```{seealso}
+{doc}`ar1_turningpts` builds on the posterior computed here to forecast nonlinear sample-path statistics of an AR(1) process, such as the time until the next turning point.
+```

lectures/ar1_turningpts.md

@@ -42,6 +42,10 @@ To investigate sample path properties we'll use a simulation procedure recommend
 
 To acknowledge uncertainty about parameters, we'll deploy `numpyro` to construct a Bayesian joint posterior distribution for unknown parameters.
 
+This lecture builds on {doc}`ar1_bayes`, which studies Bayesian inference for the parameters of exactly this AR(1) model in detail.
+
+We recommend reading that lecture first, since here we treat the construction of the posterior more briefly.
+
 Let's start with some imports.
 
 ```{code-cell} ipython3
@@ -69,14 +73,16 @@ key = random.PRNGKey(0)
 
 ## A Univariate First-Order Autoregressive Process
 
-Consider the univariate AR(1) model: 
+Consider the univariate AR(1) model, which is the same model studied in {doc}`ar1_bayes`:
 
 $$ 
 y_{t+1} = \rho y_t + \sigma \epsilon_{t+1}, \quad t \geq 0 
 $$ (ar1-tp-eq1) 
 
-where the scalars $\rho$ and $\sigma$ satisfy $|\rho| < 1$ and $\sigma > 0$; 
-$\{\epsilon_{t+1}\}$ is a sequence of i.i.d. normal random variables with mean $0$ and variance $1$. 
+where
+
+* the scalars $\rho$ and $\sigma$ satisfy $|\rho| < 1$ and $\sigma > 0$
+* $\{\epsilon_{t+1}\}$ is a sequence of IID normal random variables with mean $0$ and variance $1$.
 
 The initial condition $y_{0}$ is a known number. 
 
@@ -447,7 +453,11 @@ $$
 
 The next code cells use `numpyro` to compute the time $t$ posterior distribution of $\rho, \sigma$.
 
-Note that in defining the likelihood function, we choose to condition on the initial value $y_0$.
+We construct this posterior just as in {doc}`ar1_bayes`, to which we refer for a detailed discussion.
+
+As there, in defining the likelihood function we condition on the initial value $y_0$.
+
+This is the **conditioning assumption** of {doc}`ar1_bayes`, and it is the appropriate choice here because our initial path starts from an atypical value $y_0 = 10$.
 
 ```{code-cell} ipython3
 ---
@@ -666,6 +676,12 @@ plot_Wecker(ar1, initial_path, ax)
 plt.show()
 ```
 
+Apart from the top-left panel, which repeats the initial path and coverage intervals, each panel shows the predictive distribution of one sample path statistic.
+
+These are computed by simulating many future paths with the parameters held fixed at their true values.
+
+The distributions therefore reflect only the uncertainty that comes from future shocks.
+
 ## Extended Wecker Method
 
 Now we apply our "extended" Wecker method based on predictive densities of $y$ defined by
@@ -735,6 +751,10 @@ plot_extended_Wecker(ar1, post_samples, initial_path, ax)
 plt.show()
 ```
 
+The panels show the same statistics, but now each future path is simulated with parameters drawn from the posterior.
+
+These distributions therefore combine two sources of uncertainty: randomness in future shocks and our uncertainty about $(\rho, \sigma)$.
+
 ## Comparison
 
 Finally, we plot both the original Wecker method and the extended method with parameter values drawn from the posterior together to compare the differences that emerge from pretending to know parameter values when they are actually uncertain.
@@ -753,3 +773,23 @@ plot_extended_Wecker(ar1, post_samples, initial_path, ax)
 ax[0, 1].legend(fontsize=10)
 plt.show()
 ```
+
+The two sets of predictive distributions reveal the cost of pretending that we know the parameters.
+
+The extended Wecker method draws $(\rho, \sigma)$ from their posterior each time it simulates a future path.
+
+It therefore layers parameter uncertainty on top of the shock uncertainty already present in the original method.
+
+As a result, its predictive distributions are more dispersed than those computed with the parameters fixed at their true values.
+
+## Conclusion
+
+This lecture combined two tools to forecast nonlinear functions of the future path of an AR(1) process.
+
+Wecker's simulation method let us approximate predictive distributions of sample path statistics, such as the time until the next turning point.
+
+A Bayesian posterior over $(\rho, \sigma)$, computed as in {doc}`ar1_bayes`, let us also account for our uncertainty about the parameters that govern the process.
+
+Putting these together, the extended Wecker method produces predictive distributions that reflect both sources of uncertainty identified at the outset.
+
+Ignoring parameter uncertainty, as the original Wecker method does, yields tighter distributions that overstate how much we really know about these future statistics.