[' # Define priors for the model parameters\n    alpha = pm.Normal(\'alpha\', mu=0, sigma=10)\n    beta = pm.Normal(\'beta\', mu=0, sigma=10)\n    sigma = pm.HalfNormal(\'sigma\', sigma=10)\n    \n    # Define the likelihood function\n    likelihood = pm.Normal(\'likelihood\', mu=alpha + beta * X, sigma=sigma, observed=y)\n    \n    # Sample the posterior distribution\n    trace = pm.sample(1000, tune=1000, chains=4, return_inferencedata=True, idata_kwargs={"log_likelihood": True})\n\n# Posterior analysis\naz.plot_posterior(trace, var_names=[\'alpha\', \'beta\', \'sigma\'], round_to=2)\nplt.show()\n\n# Extract the posterior samples\nposterior_samples = trace.posterior\n\n# Calculate the posterior predictive distribution\nposterior_predictive = pm.sample_posterior_predictive(trace, samples=1000, model=m, var_names=[\'likelihood\'])\n\n# Calculate the mean and standard deviation of the posterior predictive distribution\nposterior_predictive_mean = np.mean(posterior_predictive[\'likelihood\'], axis=0)\nposterior_predictive_std = np.std(posterior_predictive[\'likelihood\'], axis=0)\n\n# Plot the posterior predictive distribution\nplt.figure(figsize=(10, 6))\nplt.plot(X, posterior_predictive_mean, color=\'blue\', alpha=0.5)\nplt.fill_between(X, posterior_predictive_mean - posterior_predictive_std, posterior_predictive_mean + posterior_predictive_std, color=\'blue\', alpha=0.3)\nplt.scatter(X, y, color=\'red\', label=\'Data\')\nplt.xlabel(\'X\')\nplt.ylabel(\'y\')\nplt.title(\'Posterior Predictive Distribution\')\nplt.legend()\nplt.show()\n\n# Calculate the mean and standard deviation of the posterior predictive distribution\nposterior_predictive_mean = np.mean(poster']