[' # Define the priors\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\n    mu = alpha + beta * X\n    y_obs = pm.Normal(\'y_obs\', mu=mu, sigma=sigma, observed=y)\n    \n    # Sample the posterior\n    trace = pm.sample(1000, tune=1000, chains=4, return_inferencedata=True, idata_kwargs={"log_likelihood": True})\n\n# Print the summary of the posterior\nprint(trace.posterior.summary())\n\n# Plot the posterior distributions\naz.plot_posterior(trace, var_names=[\'alpha\', \'beta\', \'sigma\'])\nplt.show()\n\n# Print the log likelihood\nprint(trace.log_likelihood)\n\n# Print the trace\nprint(trace)\n\n# Print the posterior\nprint(trace.posterior)\n\n# Print the summary of the posterior\nprint(trace.posterior.summary())\n\n# Print the trace\nprint(trace)\n\n# Print the posterior\nprint(trace.posterior)\n\n# Print the summary of the posterior\nprint(trace.posterior.summary())\n\n# Print the trace\nprint(trace)\n\n# Print the posterior\nprint(trace.posterior)\n\n# Print the summary of the posterior\nprint(trace.posterior.summary())\n\n# Print the trace\nprint(trace)\n\n# Print the posterior\nprint(trace.posterior)\n\n# Print the summary of the posterior\nprint(trace.posterior.summary())\n\n# Print the trace\nprint(trace)\n\n# Print the posterior\nprint(trace.posterior)\n\n# Print the summary of the posterior\nprint(trace.posterior.summary())\n\n# Print the trace\nprint(trace)\n\n# Print the posterior\nprint(trace.posterior)\n\n# Print the summary of the posterior\nprint(trace.posterior.summary())\n\n# Print the trace\nprint(trace)\n\n# Print the posterior\nprint(trace.posterior)\n\n# Print']