[' # Priors\n    mu = pm.Normal(\'mu\', mu=0, sigma=10)\n    tau = pm.HalfCauchy(\'tau\', beta=10)\n    \n    # Likelihood\n    theta = pm.Normal(\'theta\', mu=mu, sigma=tau, shape=len(y))\n    y_obs = pm.Normal(\'y_obs\', mu=theta, sigma=sigma, observed=y)\n    \n    # Posterior sampling\n    trace = pm.sample(1000, tune=1000, chains=4, return_inferencedata=True, idata_kwargs={"log_likelihood": True})\n\n# Posterior predictive check\nppc = pm.sample_posterior_predictive(trace, samples=1000, model=m, random_seed=42)\n\n# Plotting the posterior predictive check\nplt.figure(figsize=(10, 6))\nplt.plot(ppc[\'y_obs\'], label=\'Posterior Predictive Check\')\nplt.plot(y, \'o\', label=\'Data\')\nplt.legend()\nplt.title(\'Posterior Predictive Check for 8-Schools Data\')\nplt.show()\n\n# Summary of the posterior distribution\nsummary = az.summary(trace, round_to=2)\nprint(summary)\n\n# Plotting the posterior distribution\naz.plot_trace(trace, var_names=[\'mu\', \'tau\'])\nplt.show()']