Port LKJ example from Pyro (#1065)

* port LKJ example from Pyro

* add code samples in docstring for LKJ and LKJCholesky

* delete LKJ example

* change sample name for correlation matrix from L_omega to corr_mat

* incorporate review comments

Author: irustandi

numpyro/distributions/continuous.py

@@ -545,6 +545,27 @@ class LKJ(TransformedDistribution):
     When ``concentration < 1``, the distribution favors samples with small determinent. This is
     useful when we know a priori that some underlying variables are correlated.
 
+    Sample code for using LKJ in the context of multivariate normal sample::
+
+        def model(y):  # y has dimension N x d
+            d = y.shape[1]
+            N = y.shape[0]
+            # Vector of variances for each of the d variables
+            theta = numpyro.sample("theta", dist.HalfCauchy(jnp.ones(d)))
+
+            concentration = jnp.ones(1)  # Implies a uniform distribution over correlation matrices
+            corr_mat = numpyro.sample("corr_mat", dist.LKJ(d, concentration))
+            sigma = jnp.sqrt(theta)
+            # we can also use a faster formula `cov_mat = jnp.outer(theta, theta) * corr_mat`
+            cov_mat = jnp.matmul(jnp.matmul(jnp.diag(sigma), corr_mat), jnp.diag(sigma))
+
+            # Vector of expectations
+            mu = jnp.zeros(d)
+
+            with numpyro.plate("observations", N):
+                obs = numpyro.sample("obs", dist.MultivariateNormal(mu, covariance_matrix=cov_mat), obs=y)
+            return obs
+
     :param int dimension: dimension of the matrices
     :param ndarray concentration: concentration/shape parameter of the
         distribution (often referred to as eta)
@@ -606,6 +627,28 @@ class LKJCholesky(Distribution):
     (hence small determinent). This is useful when we know a priori that some underlying
     variables are correlated.
 
+    Sample code for using LKJCholesky in the context of multivariate normal sample::
+
+        def model(y):  # y has dimension N x d
+            d = y.shape[1]
+            N = y.shape[0]
+            # Vector of variances for each of the d variables
+            theta = numpyro.sample("theta", dist.HalfCauchy(jnp.ones(d)))
+            # Lower cholesky factor of a correlation matrix
+            concentration = jnp.ones(1)  # Implies a uniform distribution over correlation matrices
+            L_omega = numpyro.sample("L_omega", dist.LKJCholesky(d, concentration))
+            # Lower cholesky factor of the covariance matrix
+            sigma = jnp.sqrt(theta)
+            # we can also use a faster formula `L_Omega = sigma[..., None] * L_omega`
+            L_Omega = jnp.matmul(jnp.diag(sigma), L_omega)
+
+            # Vector of expectations
+            mu = jnp.zeros(d)
+
+            with numpyro.plate("observations", N):
+                obs = numpyro.sample("obs", dist.MultivariateNormal(mu, scale_tril=L_Omega), obs=y)
+            return obs
+
     :param int dimension: dimension of the matrices
     :param ndarray concentration: concentration/shape parameter of the
         distribution (often referred to as eta)