Chapter 14

Chapter 14: Prior Elicitation and Sensitivity Analysis

The choice of prior is a fundamental part of Bayesian modeling. It is crucial to check how sensitive our conclusions are to that choice. A sensitivity analysis involves fitting the model with several different, plausible priors and comparing the results.

14.1 Code Example: Prior Sensitivity

Let's revisit our coin flip example. What if we had used a much stronger prior, believing the coin was almost certainly fair?
Python

# Data: 7 heads in 10 tosses

# Model 1: Weak prior Beta(2, 2)
with pm.Model() as weak_prior_model:
    p = pm.Beta('p', 2.0, 2.0)
    y = pm.Binomial('y', n=10, p=p, observed=7)
    trace_weak = pm.sample(1000, tune=500)

# Model 2: Strong prior Beta(50, 50) - strong belief in fairness
with pm.Model() as strong_prior_model:
    p = pm.Beta('p', 50.0, 50.0)
    y = pm.Binomial('y', n=10, p=p, observed=7)
    trace_strong = pm.sample(1000, tune=500)

# Compare the posteriors
az.plot_posterior(trace_weak, var_names=['p'], color='blue', hdi_prob=0.94)
plt.suptitle("Posterior with WEAK Prior")
az.plot_posterior(trace_strong, var_names=['p'], color='green', hdi_prob=0.94)
plt.suptitle("Posterior with STRONG Prior")
plt.show()
You'll see the posterior from the weak prior is centered around the data's estimate (0.7). The posterior from the strong prior is still centered very close to 0.5; the 10 data points were not enough to overwhelm the very strong prior belief. This shows how influential priors can be, especially with limited data.