“When Should I Check The Mail?”, Gwern2015-06-21 (R, Bayes, decision theory, survival analysis, tutorial; backlinks)⁠:

Bayesian decision-theoretic analysis of local mail delivery times: modeling deliveries as survival analysis, model comparison, optimizing check times with a loss function, and optimal data collection.

Mail is delivered by the USPS mailman at a regular but not observed time; what is observed is whether the mail has been delivered at a time, yielding somewhat-unusual “interval-censored data”. I describe the problem of estimating when the mailman delivers, write a simulation of the data-generating process, and demonstrate analysis of interval-censored data in R using maximum-likelihood (survival analysis with Gaussian regression using survival library), MCMC (Bayesian model in JAGS), and likelihood-free Bayesian inference (custom ABC, using the simulation). This allows estimation of the distribution of mail delivery times. I compare those estimates from the interval-censored data with estimates from a (smaller) set of exact delivery-times provided by USPS tracking & personal observation, using a multilevel model to deal with heterogeneity apparently due to a change in USPS routes/postmen. Finally, I define a loss function on mail checks, enabling: a choice of optimal time to check the mailbox to minimize loss (exploitation); optimal time to check to maximize information gain (exploration); Thompson sampling (balancing exploration & exploitation indefinitely), and estimates of the value-of-information of another datapoint (to estimate when to stop exploration and start exploitation after a finite amount of data).

• Inference
  • Interval-Censored Data
    • ML
    • MCMC
    • ABC
  • Exact Delivery-Time Data
    • ML
    • MCMC
  • Combined Data
  • Model Checking
    • On Model Uncertainty
    • PPC
    • Crossvalidation
    • Bayesian Model Selection
    • Bayesian Model Averaging
• Decision Theory
  • Optimal Mail Checking Time
    • Defining a Loss Function
    • Finding the Optimum
    • Total Costs
  • Optimal Data-Sampling
    • Reinforcement Learning
    • Expected Information Gains
  • Optimal Sample-Size: Value of Information Metrics
    • EVPI
    • EVSI
• Conclusions
• Appendix
  • Thompson Sampling

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