[' # Define the prior distributions for the parameters\n    alpha = pm.HalfNormal("alpha", sigma=1.0)\n    beta = pm.Normal("beta", mu=0, sigma=1.0)\n    sigma = pm.HalfNormal("sigma", sigma=1.0)\n    \n    # Define the likelihood function\n    mu = alpha * np.exp(beta * x)\n    likelihood = pm.Poisson("likelihood", mu=mu, 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 distribution of the parameters\naz.plot_trace(trace, var_names=["alpha", "beta", "sigma"])\nplt.show()\n\n# Posterior predictive distribution\nposterior_predictive = pm.sample_posterior_predictive(trace, samples=1000, model=m, random_seed=42)\n\n# Plot the posterior predictive distribution\nplt.figure(figsize=(10, 6))\nplt.errorbar(x, y, yerr=np.sqrt(y), fmt="o", label="Data")\nplt.plot(x, posterior_predictive["likelihood"].mean(axis=0), label="Posterior Predictive")\nplt.xlabel("x")\nplt.ylabel("y")\nplt.legend()\nplt.show()']