Bayesian Problem: The Quality Control Dilemma
Background
A pharmaceutical company produces a critical medication. Historical data suggests that the production process has typically maintained a defect rate of around 2%. However, recent changes in the manufacturing process have raised concerns about whether this rate has increased.
The Data
In a recent batch of 100 units:
- 5 units were found to be defective
- 95 units met quality standards
Prior Information
Based on historical process data:
$$\theta \sim \text{Beta}(4, 196)$$Where θ represents the defect probability.
This prior encodes both the historical mean (≈ 0.02) and our uncertainty about the true rate.
Your Tasks
- Compute the Posterior:
- Derive the posterior distribution for θ
- Calculate the posterior mean and 95% credible interval
- Decision Analysis:
- What is the probability that the true defect rate exceeds 3%?
- Should the company investigate the process? (Consider the costs: false alarms vs. missed problems)
- Prior Sensitivity:
- How would your conclusions change with a more diffuse prior?
- Try Beta(2, 98) as an alternative prior
Key Questions to Consider
- How does the conjugate prior property of the Beta distribution help us here?
- What role does sample size play in the strength of our conclusions?
- How would you explain your findings to:
- A process engineer?
- A regulatory body?
- Senior management?
Helpful Formulas
Beta-Binomial conjugate update:
$$\text{Beta}(\alpha, \beta) + \text{Bin}(n,k) \rightarrow \text{Beta}(\alpha + k, \beta + n – k)$$Posterior mean:
$$E[\theta|x] = \frac{\alpha + k}{\alpha + \beta + n}$$Extension Challenges
- What if we had sequential data from multiple batches?
- How would you model time-dependency in the defect rate?
- What if the cost of false positives and false negatives were asymmetric?
Learning Objectives
This problem helps develop:
- Understanding of conjugate prior relationships
- Practical interpretation of credible intervals
- Decision-making under uncertainty
- Sensitivity analysis skills