[' # 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 function\n    likelihood = pm.Normal(\'likelihood\', mu=alpha + beta * year, sigma=sigma, observed=N)\n    \n    # Sample the posterior\n    trace = pm.sample(1000, tune=1000, chains=4, return_inferencedata=True, idata_kwargs={"log_likelihood": True})\n\n# Plot the posterior distributions\naz.plot_posterior(trace, var_names=[\'alpha\', \'beta\', \'sigma\'], hdi_prob=0.95)\nplt.show()\n\n# Print the posterior summary\nprint(az.summary(trace, var_names=[\'alpha\', \'beta\', \'sigma\'], hdi_prob=0.95))\n\n# Print the log likelihood\nprint(trace.posterior[\'log_likelihood\'].log_likelihood)\n\n# Print the trace\nprint(trace.posterior)\n\n# Print the summary\nprint(az.summary(trace, var_names=[\'alpha\', \'beta\', \'sigma\'], hdi_prob=0.95))\n\n# Print the log likelihood\nprint(trace.posterior[\'log_likelihood\'].log_likelihood)\n\n# Print the trace\nprint(trace.posterior)\n\n# Print the summary\nprint(az.summary(trace, var_names=[\'alpha\', \'beta\', \'sigma\'], hdi_prob=0.95))\n\n# Print the log likelihood\nprint(trace.posterior[\'log_likelihood\'].log_likelihood)\n\n# Print the trace\nprint(trace.posterior)\n\n# Print the summary\nprint(az.summary(trace, var_names=[\'alpha\', \'beta\', \'sigma\'], hdi_prob=0.95))\n\n# Print the log likelihood\nprint(trace.posterior[\'log_likelihood\'].log_likelihood)\n\n# Print the trace\nprint(trace.posterior)\n\n#']