add spaces, increase indentation, and fix number order to Pareto note…

…book (#651)

* add spaces, increase indentation, and fix number order

* explicit with 6

docs/source/notebooks/clv/pareto_nbd.ipynb

@@ -476,27 +476,32 @@
    "id": "e10b0672-8967-4ecc-9870-f8c08133f9ee",
    "metadata": {},
    "source": [
-    "# Model Definition\n",
+    "## Model Definition\n",
     "The Pareto/NBD model is based on the following assumptions for each customer:\n",
     "1. Customers are active for an unobserved period of time, then become permanently inactive.\n",
     "   \n",
-    "### Purchasing Process\n",
+    "#### Purchasing Process\n",
     "\n",
     "2. While active, the the number of transactions made by a customer follows a Poisson process with transaction rate $\\lambda$:\n",
+    "   \n",
     "   $$P(X(t)=x|\\lambda) = \\frac{(\\lambda t)^{x}e^{-\\lambda t}}{x!}, x=0,1,2,...$$\n",
+    "   \n",
     "   This is equivalent to assuming time between transactions is exponentially distributed with transaction rate $\\lambda$:\n",
+    "   \n",
     "   $$f(t_{j}-t_{j-1}| \\lambda) = \\lambda e^{-\\lambda (t_{j} - t_{j - 1})}, \\quad t_{j} \\geq t_{j - 1} \\geq 0$$\n",
+    "   \n",
     "   Where $t$ is the time period of the $j$th purchase.\n",
     "3. Heterogeneity in $\\lambda$ follows a Gamma distribution with shape parameter $r$ and scale parameter $\\alpha$:\n",
     "\n",
     "    $$g(\\lambda|r, \\alpha) = \\frac{\\alpha^{r}\\lambda^{r - 1}e^{-\\lambda \\alpha}}{\\Gamma(r)}$$\n",
-    "### Dropout Process\n",
-    "5. The duration of a customer's unobserved active lifetime is exponentially distributed with dropout rate $\\mu$.\n",
+    "#### Dropout Process\n",
+    "4. The duration of a customer's unobserved active lifetime is exponentially distributed with dropout rate $\\mu$.\n",
     "\n",
-    "6. Heterogeneity in $\\mu$ also follows a Gamma distribution with shape parameter $s$ and scale parameter $\\beta$:\n",
+    "5. Heterogeneity in $\\mu$ also follows a Gamma distribution with shape parameter $s$ and scale parameter $\\beta$:\n",
     "\n",
     "    $$g(\\mu|s, \\beta) = \\frac{\\beta^{s}\\mu^{s - 1}e^{-\\mu \\beta}}{\\Gamma(s)}$$\n",
-    "7. Transaction rate $\\lambda$ and time until dropout $\\mu$ vary independently for each customer.\n",
+    "   \n",
+    "6. Transaction rate $\\lambda$ and time until dropout $\\mu$ vary independently for each customer.\n",
     "\n",
     "If we take the expectation across the distributions of $\\lambda$ and $\\mu$, we can derive a likelihood function to estimate parameters $r$, $\\alpha$, $s$, and $\\beta$ across the customer population. For more details on the `ParetoNBD` likelihood please refer to the [docs](https://www.pymc-marketing.io/en/stable/api/generated/pymc_marketing.clv.distributions.ParetoNBD.html#pymc_marketing.clv.distributions.ParetoNBD)."
    ]
@@ -506,15 +511,15 @@
    "id": "bee69f5b-1b9e-4aa4-bdd4-5358c866453c",
    "metadata": {},
    "source": [
-    "# Model Fitting"
+    "## Model Fitting"
    ]
   },
   {
    "cell_type": "markdown",
    "id": "325d5448",
    "metadata": {},
    "source": [
-    "## `lifetimes` Benchmark Model\n",
+    "### `lifetimes` Benchmark Model\n",
     "\n",
     "Let's time travel back to July 2020 and use the old `lifetimes` library to fit a Pareto/NBD model with Maximum Likelihood Estimation (MLE). The `Nelder-Mead` optimizer from `scipy.optimize` is ran under the hood to estimate scalar values for $r$, $\\alpha$, $s$, and $\\beta$."
    ]
@@ -772,7 +777,7 @@
    "id": "512e5ef6-8fac-43fa-8d54-d6cfa14f64a6",
    "metadata": {},
    "source": [
-    "### Prior and Posterior Predictive Checks\n",
+    "#### Prior and Posterior Predictive Checks\n",
     "PPCs allow us to check the efficacy of our priors, and the peformance of the fitted posteriors. PPCs aren't usually an option with MAP fitted models, but here we're actually sampling from the latent $\\lambda$ and $\\mu$ Gamma distributions, so PPCs are possible for `ParetoNBDModel` regardless of the fit method!\n",
     "\n",
     "Let's see how the model performs in a *prior* predictive check, where we sample from the default priors before fitting the model: "
@@ -1118,7 +1123,7 @@
    "id": "25724a17-538a-4ec5-9df8-8dd28b547a86",
    "metadata": {},
    "source": [
-    "# Full Bayesian Inference"
+    "## Full Bayesian Inference"
    ]
   },
   {
@@ -3484,7 +3489,7 @@
    "id": "2ec3ff94-83fe-4a6c-99ef-e25e0ee12cef",
    "metadata": {},
    "source": [
-    "After fititing, models can be persisted for later use:"
+    "After fitting, models can be persisted for later use:"
    ]
   },
   {
@@ -3519,7 +3524,7 @@
    "id": "370d109c",
    "metadata": {},
    "source": [
-    "# Predictive Methods\n",
+    "## Predictive Methods\n",
     "\n",
     "The Pareto/NBD model supports a variety of predictive methods:\n",
     "\n",
@@ -4095,7 +4100,7 @@
    "id": "7d2b668e-a53d-487b-9b05-0090a2dfa2e3",
    "metadata": {},
    "source": [
-    "# Time-Invariant Covariates"
+    "## Time-Invariant Covariates"
    ]
   },
   {
@@ -4436,7 +4441,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.14"
+   "version": "3.10.9"
   },
   "nteract": {
    "version": "[email protected]"