Chapter 5: Loss Functions and Decision Theory

Bayesian inference is a complete framework for rational decision-making under uncertainty. This is formalized through Bayesian decision theory.

5.1 The Principle of Minimizing Posterior Expected Loss

The core idea is to:
Define a loss function L(θ, a) that quantifies the cost of taking action a when the true parameter value is θ.
Choose the action a that has the minimum expected loss, where the expectation is taken over the posterior distribution of θ.
Different loss functions lead to different optimal estimators:
Squared Error Loss: (θ - a)². The optimal action is the posterior mean.
Absolute Error Loss: |θ - a|. The optimal action is the posterior median.
0-1 Loss: 0 if correct, 1 if incorrect. The optimal action is the posterior mode (MAP).

5.2 Code Example: A Simple Decision Problem

Imagine we're the coin manufacturer. If the coin's bias p is greater than 0.6, we must recall the batch, which costs $1000. If p is less than or equal to 0.6 but we recall it anyway, it costs us $200 (a wasted recall). If p is greater than 0.6 and we don't recall it, it costs us $5000 (reputation damage).
Python

# Continuing from the coin flip MCMC trace...

# Get the posterior samples for 'p'
p_samples = trace.posterior['p'].values.flatten()

# Define loss for the action "Recall"
def loss_recall(p):
    if p > 0.6:
        return 0  # Correct decision
    else:
        return 200 # Wasted recall

# Define loss for the action "Do Not Recall"
def loss_no_recall(p):
    if p > 0.6:
        return 5000 # Big mistake
    else:
        return 0 # Correct decision

# Calculate expected loss for each action by averaging over posterior samples
expected_loss_recall = np.mean([loss_recall(p) for p in p_samples])
expected_loss_no_recall = np.mean([loss_no_recall(p) for p in p_samples])

print(f"Expected Loss if we RECALL: ${expected_loss_recall:.2f}")
print(f"Expected Loss if we DO NOT RECALL: ${expected_loss_no_recall:.2f}")

if expected_loss_recall < expected_loss_no_recall:
    print("\nOptimal decision: RECALL the batch.")
else:
    print("\nOptimal decision: DO NOT RECALL the batch.")