add new slide to elaborate frq-sev in insurance context (#14)

Author: jonsedar

notebooks/000_Intro.ipynb

@@ -46,7 +46,7 @@
         "We seek to create _principled_ models that provide explanatory inference and predictions of Marginal distributions $M$\n",
         "that are jointly coupled by a Latent Copula $C$, using quantified uncertainty to support real-world decision-making.\n",
         "\n",
-        "<img src='../plots/000_jointplot_corr.png' width='480px'/>"
+        "<img src='../plots/000_jointplot_corr.png' width='400px'/>"
       ]
     },
     {
@@ -59,12 +59,12 @@
       "source": [
         "**Motivation:**\n",
         "\n",
-        "+ A classic use-case for this model architecture (in the 2-dimensional setting) is insurance claims frequency and severity\n",
-        "+ The `frequency` of claims and the `severity` of each claim each have marginal distributions and a natural covariance \n",
-        "  $\\Sigma$ between marginals $M_{0}, M_{1}$\n",
-        "+ The joint product `frequency * severity = Loss Cost` i.e. the dollar value of insurable losses\n",
+        "+ A classic use-case for this model architecture (in the 2-dimensional setting) is insurance claims aka incurred loss\n",
+        "+ We decompose the dollar value of claims into two marginal distributions: the `frequency`, and `severity` of \n",
+        "  `expected loss cost`, because these measures are intuitive and can behave differently, with a (highly important)\n",
+        "  degree of covariance $\\Sigma$\n",
         "+ If we use a naive model that doesn't account for the covariance between `frequency` and `severity`, then the model \n",
-        "  predictions for `Loss Cost` can be hugely wrong!"
+        "  predictions for `expected loss cost` can be hugely wrong!"
       ]
     },
     {
@@ -75,7 +75,41 @@
         }
       },
       "source": [
-        "<img src='../plots/000_jointplot_corr.png' width='360px'/>\n",
+        "### Quick Aside on decomposition of claims `frequency` and `severity`\n",
+        "\n",
+        "We can create different decompositions for different purposes, and according to the data available. A very useful one is\n",
+        "shown here: to use the ratio of losses per unit of TIV, and thus generalise to policies of different TIV.\n",
+        "\n",
+        "$$\n",
+        "\\begin{aligned}\n",
+        "frq_{i} &= \\frac{claim\\_ct_{i}}{TIV_{i}} \\\\\n",
+        "sev_{i} &= \\frac{incurred\\_total_{i}}{claim\\_ct_{i}} \\\\\n",
+        "\\\\\n",
+        "\\mathbb{E}_{\\text{loss} \\ i} &= frq_{i} * sev_{i} = \\frac{incurred\\_total_{i}}{TIV_{i}} \\\\\n",
+        "\\end{aligned}\n",
+        "$$\n",
+        "\n",
+        "where:\n",
+        "+ Each policy $i \\in n$ (the dataset of all policies) can have it's own (policy-level) frequency ($frq_{i} \\geq 0$) and \n",
+        "  severity ($sev_{i} \\geq 0$) of claim (and thus policy-level $\\mathbb{E}_{\\text{loss i}} \\geq 0$)\n",
+        "+ Note $frq$ and $sev$ tend to be zero-augmented distributions (where no loss is experienced): this is a very important\n",
+        "  aspect to include in more advanced model architectures\n",
+        "+ $claim\\_ct_{i} \\geq 0$ is the count of claims incurred for policy $i$\n",
+        "+ $TIV_{i} \\gt 0$ is the Total Insured Value (TIV) for policy $i$\n",
+        "+ $incurred\\_total_{i} \\geq 0$ is the total incurred losses for policy $i$\n"
+      ]
+    },
+    {
+      "cell_type": "markdown",
+      "metadata": {
+        "slideshow": {
+          "slide_type": "subslide"
+        }
+      },
+      "source": [
+        "#### Back to this presentation's focus on the copula function\n",
+        "\n",
+        "<img src='../plots/000_jointplot_corr.png' width='300px'/>\n",
         "\n",
         "\n",
         "**Demonstration:**\n",
@@ -89,7 +123,7 @@
         "  + We create a series of principled copula models using advanced architectures and Bayesian inference to fit to the \n",
         "    data and estimate the covariance on $M_{0}, M_{1}$\n",
         "  + The first model is naive and ignores the covariance, the final model is very sophisticated and estimates the covariance\n",
-        "  + We demonstrate **a substantial 32 percentage-point improvement in model accuracy** when using a copula-based model\n",
+        "  + We demonstrate **a substantial 33 percentage-point improvement in model accuracy** when using a copula-based model\n",
         "  + This correct estimation would likely make the difference between profitable pricing / accurate reserving, or greatly loss-making business over a portfolio."
       ]
     },