[' # Define priors for the 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 * year, sigma=sigma, observed=N)\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_trace(trace)\nplt.show()\n\n# Posterior Predictive Checks\nposterior_predictive = pm.sample_posterior_predictive(trace, samples=1000, model=m)\nplt.plot(year, N, \'o\', label=\'Observed Data\')\nplt.plot(year, posterior_predictive[\'likelihood\'].mean(axis=0), label=\'Posterior Predictive\')\nplt.legend()\nplt.show()\n\n# Summary of Posterior Distributions\naz.summary(trace)\n\n# Additional Analysis\naz.plot_posterior(trace)\nplt.show()\n\n# Summary of Posterior Predictive Distribution\naz.plot_ppc(trace, kind=\'kde\')\nplt.show()\n\n# Summary of Posterior Predictive Distribution\naz.plot_ppc(trace, kind=\'hist\')\nplt.show()']