Title of Lotka-Volterra notebook and slight reformatting

Author: han-ol

examples/Lotka_Volterra_point_estimation_and_expert_stats.ipynb

@@ -5,7 +5,7 @@
    "id": "3cacfcfd-d638-4181-9c16-ba050ab5e367",
    "metadata": {},
    "source": [
-    "# Differential Equation Example: Lotka-Volterra predetor prey dynamics\n",
+    "# Rapid iteration with point estimation and expert statistics for Lotka-Volterra dynamics\n",
     "\n",
     "_Authors: Hans Olischläger_\n",
     "\n",
@@ -541,7 +541,6 @@
     "\n",
     "* Here is a strictly proper scoring rule that is optimal if the estimate, $\\hat \\theta$, is the true **mean** of the posterior:\n",
     "\n",
-    "\n",
     "    $$L(\\hat \\theta, \\theta; k) = | \\theta - \\hat \\theta |^2$$\n",
     "\n",
     "    It is the well known squared error loss!\n",
@@ -550,10 +549,9 @@
     "\n",
     "    $$L(\\hat \\theta, \\theta; k) = | \\theta - \\hat \\theta |$$\n",
     "\n",
-    "\n",
-    "\n",
     "* To estimate **quantiles**, the following is a strictly proper scoring rule:\n",
-    "$$L(\\hat \\theta, \\theta; \\tau) = (\\hat \\theta - \\theta)(\\mathbf{1}_{\\hat \\theta - \\theta > 0} - \\tau)$$\n",
+    "\n",
+    "    $$L(\\hat \\theta, \\theta; \\tau) = (\\hat \\theta - \\theta)(\\mathbf{1}_{\\hat \\theta - \\theta > 0} - \\tau)$$\n",
     "\n",
     "    Here we write an indicator function as $\\mathbf{1}_{\\hat \\theta - \\theta > 0}$ to evaluate to 1 for overestimation (positive $\\hat \\theta - \\theta$) and $0$ otherwise.\n",
     "\n",
@@ -563,8 +561,10 @@
     "\n",
     "\n",
     "* Note, that when approximating the full distribution in BayesFlow we score a **probability estimate** $\\hat p(\\theta|x)$ with the log-score,\n",
-    "$$L(\\hat p(\\theta|x), \\theta) = \\log (\\hat p(\\theta)) $$\n",
-    "which is also a strictly proper scoring rule.\n",
+    "\n",
+    "    $$L(\\hat p(\\theta|x), \\theta) = \\log (\\hat p(\\theta))$$\n",
+    "\n",
+    "    which is also a strictly proper scoring rule.\n",
     "\n",
     "\n",
     "* What if you want to estimate something else? There might just be a loss function that corresponds to the estimator of exactly the quantity you are after.\n",
@@ -1050,6 +1050,7 @@
     "In relation to $q_1$ and $q_2$, where is $x=0$ and $x=1$? These two correspond to location (mean) and scale (standard deviation) of the standard normal.\n",
     "\n",
     "So we solve the equations\n",
+    "\n",
     "$$\n",
     "\\begin{aligned}\n",
     "    0 &= \\tilde q_1 (1-\\alpha) + \\tilde  q_2 \\alpha,\\\\\n",