Add new example for Unsupport Distribution (#145)

* add example back in

* reorganize example

docs/examples/unsupported-distributions.md

@@ -3,49 +3,66 @@ comments: true
 ---
 # Unsupported Posterior Predictive Distributions
 
-
-Suppose we want to use the geometric model with a beta prior which doesn't have a 
+Suppose we want to use the Pareto model with a Gamma prior which doesn't have a
 supported distribution for the posterior predictive. 
 
-```python
-from conjugate.distributions import Beta
-from conjugate.models import geometric_beta
-
-prior = Beta(1, 1)
-posterior: Beta = geometric_beta(x_total=12, n=10, prior=prior)
-```
-
 We can get posterior predictive samples by: 
 
 1. Sample from the posterior distribution
 2. Sample from the model distribution using posterior samples
 
-## 1. Using `conjugate-models`
+## Setup
+
+```python
+import numpy as np
+
+from conjugate.distributions import Gamma, Pareto
+from conjugate.models import pareto_gamma
+
+seed = sum(map(ord, "Unsupported Posterior Predictive Distributions"))
+rng = np.random.default_rng(seed)
 
-This is easy to do with this package. 
+n = 10
+
+x_m = 1
+alpha = 2.5
+true_distribution = Pareto(x_m=x_m, alpha=alpha)
+
+data = true_distribution.dist.rvs(size=n, random_state=rng)
+
+prior = Gamma(1, 1)
+posterior: Gamma = pareto_gamma(
+    n=n,
+    ln_x_total=np.log(data).sum(),
+    x_m=x_m,
+    prior=prior,
+)
+```
+
+
+## 1. Using `conjugate-models`
 
 Since the distributions are vectorized, just: 
 
 1. Get the number of samples from the posterior 
 2. Take a single sample from the model distribution
 
 ```python
-from conjugate.distributions import Geometric
-
 n_samples = 1_000
-posterior_samples = posterior.dist.rvs(size=n_samples)
-posterior_predictive_samples = Geometric(p=posterior_samples).dist.rvs()
+
+alpha_samples = posterior.dist.rvs(size=n_samples, random_state=rng)
+posterior_predictive_samples = Pareto(x_m=x_m, alpha=alpha_samples).dist.rvs(random_state=rng)
 ```
 
-## 2. Using `pymc`
+## 2. Using PyMC
 
 Another route would be using PyMC then use the `draw` function. 
 
 ```python 
 import pymc as pm
 
-posterior_dist = pm.Beta.dist(alpha=posterior.alpha, beta=posterior.beta)
-geometric_posterior_predictive = pm.Geometric.dist(posterior_dist)
+posterior_alpha = pm.Gamma.dist(alpha=posterior.alpha, beta=posterior.beta)
+geometric_posterior_predictive = pm.Pareto.dist(m=x_m, alpha=posterior_alpha)
 
 n_samples = 1_000
 posterior_predictive_samples = pm.draw(geometric_posterior_predictive, draws=n_samples)

mkdocs.yml

@@ -49,7 +49,7 @@ nav:
       - Bayesian Update: examples/bayesian-update.md
       - Thompson Sampling: examples/thompson.md
       - Linear Regression: examples/linear-regression.md
-      # - Unsupported Distributions: examples/unsupported-distributions.md
+      - Unsupported Distributions: examples/unsupported-distributions.md
       - Inference in SQL: examples/sql.md
       - Bootstrap Comparison: examples/bootstrap.md
     - Features: