docs: improve narrative flow, correct mistakes, and clarify heading levels (#1168)

* docs: improve narrative flow and clarify heading levels

- Replace bare demonstratives and vague transitions with explicit noun phrases across chapters and subfiles
- Demote section headings inside theorem/definition/example divs to level 4 (####) to preserve document outline structure and numbering
- Fix slidebreak shortcode usage in probability subfile

Co-authored-by: dem-extra1 <112029334+dem-extra1@users.noreply.github.com>

* docs: improve narrative flow, clarify heading levels, and fix semantic line breaks

- Replace bare demonstratives and vague transitions with explicit noun phrases across chapters and subfiles
- Demote section headings inside theorem/definition/example divs to level 4 (####) to preserve document outline structure and numbering
- Fix slidebreak shortcode usage in probability subfile
- Enforce semantic line breaks after semicolons to pass check-new-line-breaks CI check

Co-authored-by: dem-extra1 <112029334+dem-extra1@users.noreply.github.com>

---------

Co-authored-by: google-labs-jules[bot] <161369871+google-labs-jules[bot]@users.noreply.github.com>
Co-authored-by: Claude <noreply@anthropic.com>

Author: 3 people

_subfiles/Linear-models-overview/_exr-prac-linreg-score.qmd

@@ -6,7 +6,7 @@
 
 Using the log-likelihood from @exr-prac-linreg-lik,
 derive the score equations for $\beta_0$ and $\beta_1$.
-That is, compute
+Specifically, compute
 $\frac{\partial \ell}{\partial \beta_0} = 0$
 and
 $\frac{\partial \ell}{\partial \beta_1} = 0$,
@@ -68,7 +68,7 @@ $$
 \sum_{i=1}^n x_i y_i = \beta_0\sum_{i=1}^n x_i + \beta_1\sum_{i=1}^n x_i^2
 $$
 
-These equations are the **normal equations**,
+These score equations are the **normal equations**,
 whose solution gives the OLS/MLE estimators
 $\hat\beta_0$ and $\hat\beta_1$.
 

_subfiles/Linear-models-overview/_sec_linreg_gof.qmd

@@ -16,7 +16,7 @@ AIC and BIC can be used to compare both nested and non-nested models.
 ### Formulas
 
 :::{#def-aic}
-### Akaike Information Criterion (AIC)
+#### Akaike Information Criterion (AIC)
 
 $$\text{AIC} = -2 \ell(\hat\theta) + 2 p$$
 
@@ -30,7 +30,7 @@ and $n$ is the number of observations.
 ---
 
 :::{#def-bic}
-### Bayesian Information Criterion (BIC)
+#### Bayesian Information Criterion (BIC)
 
 $$\text{BIC} = -2 \ell(\hat\theta) + p \log(n)$$
 

_subfiles/inference/_def-inference.qmd

@@ -1,5 +1,5 @@
 :::{#def-inference}
-### Statistical inference
+#### Statistical inference
 
 Statistical inference is the process of analyzing data in order 
 to learn about

_subfiles/intro-to-survival-analysis/_exr_survival_prediction_intervals.qmd

@@ -1,6 +1,6 @@
 :::{#exr-compute-surv-pred}
 
-## Construct a prediction interval
+#### Construct a prediction interval
 
 ```{r}
 #| include: false
@@ -51,7 +51,7 @@ Left to the reader for now.
 ---
 
 :::{#exr-explain-surv-pred}
-## Explain the results
+#### Explain the results
 
 Why do time-to-event models have such wide predictive intervals?
 

_subfiles/logistic-regression/_exm_wcgs_coef_interp.qmd

@@ -145,7 +145,7 @@ $$
 \ea
 $$
 
-That is,
+In terms of finite differences,
 $\beta_{A}$ is the difference in 
 log-odds of experiencing CHD
 per one-year difference in age 
@@ -193,7 +193,7 @@ $$\b_P = \eta(P = 1, A = 0) - \eta(P=0, A = 0)$${#eq-wcgs-intxn-betaP}
 -   $\text{e}^{\beta_{P}}$ is the ratio of the odds (aka "the odds ratio")
     of experiencing CHD, 
     for a `r age_offset`-year old individual with type A personality 
-    vs a `r age_offset`-year old individual with type B personality; that is,
+    vs a `r age_offset`-year old individual with type B personality; specifically,
 
 $$
 \expf{\beta_{P}}
@@ -266,7 +266,7 @@ Accordingly,
 the odds ratio of experiencing CHD per one-year difference in age
 differs by a factor of $\text{e}^{\beta_{PA}}$
 for participants with type A personality compared to
-participants with type B personality; that is,
+participants with type B personality; specifically,
 
 $$
 \ba

_subfiles/logistic-regression/_sec_compare_probs.qmd

@@ -85,7 +85,7 @@ $$\rho(\pi_1,\pi_2) = \frac{\pi_1}{\pi_2}$$
 :::{#sol-exr-name-RR}
 
 :::{#def-RR}
-### Relative risk ratios
+#### Relative risk ratios
 
 The ratio of two probabilities $\pi_1$ and $\pi_2$,
 

_subfiles/math-prereqs/_cor_fubini_rect.qmd

@@ -1,6 +1,6 @@
 :::{#cor-fubini-rect}
 
-### Continuous functions on a rectangle (corollary of @thm-fubini)
+#### Continuous functions on a rectangle (corollary of @thm-fubini)
 
 If $f : [a, b] \times [c, d] \to \mathbb{R}$ is **continuous** on the
 closed bounded rectangle $[a, b] \times [c, d]$, then:

_subfiles/math-prereqs/_fubini_intro_thm.qmd

@@ -13,7 +13,7 @@ follow from a stated theorem rather than from an aside.
 
 :::{#thm-fubini}
 
-### Fubini's theorem (Riemann version)
+#### Fubini's theorem (Riemann version)
 
 Let $f$ be **continuous** on a plane region $R \subseteq \mathbb{R}^2$.
 

_subfiles/math-prereqs/_sec_linear_algebra.qmd

@@ -96,7 +96,7 @@ is the sum of all entries of $\vx$.
 ---
 
 :::{#def-indicator-vector}
-### Indicator vector / standard basis vector
+#### Indicator vector / standard basis vector
 
 The $j$-th **indicator vector** (or **standard basis vector**) $\vecf{e}_j$
 of length $p$ has a $1$ in position $j$ and $0$s elsewhere:

_subfiles/math-prereqs/_thm_fubini_tonelli.qmd

@@ -1,6 +1,6 @@
 :::{#thm-fubini-tonelli}
 
-### Fubini–Tonelli theorem (measure-theoretic form)
+#### Fubini–Tonelli theorem (measure-theoretic form)
 
 Let $(\Omega_1, \mathcal F_1, \mu_1)$ and $(\Omega_2, \mathcal F_2, \mu_2)$
 be **σ-finite** measure spaces,