[' # Define the prior distributions for the hyperparameters\n    alpha = pm.Gamma(\'alpha\', alpha=1, beta=1)\n    beta = pm.Gamma(\'beta\', alpha=1, beta=1)\n    sigma = pm.HalfNormal(\'sigma\', sigma=1)\n    \n    # Define the likelihood function\n    likelihood = pm.Poisson(\'likelihood\', mu=alpha * np.exp(beta * x), 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# Compute the posterior predictive distribution\nposterior_predictive = pm.sample_posterior_predictive(trace, samples=1000, model=m)\n\n# Plot the posterior predictive distribution\nplt.figure(figsize=(10, 6))\nplt.scatter(x, y, color=\'blue\', label=\'Data\')\nplt.plot(x, posterior_predictive[\'likelihood\'].mean(axis=0), color=\'red\', label=\'Posterior Predictive Mean\')\nplt.xlabel(\'x\')\nplt.ylabel(\'y\')\nplt.title(\'Posterior Predictive Distribution\')\nplt.legend()\nplt.show()\n\n# Compute the log likelihood\nlog_likelihood = pm.logp(m, trace[\'alpha\'].mean(), trace[\'beta\'].mean(), trace[\'sigma\'].mean(), y, x)\n\n# Print the log likelihood\nprint("Log Likelihood:", log_likelihood)']