IIT JAM - Mathematical Statistics (MS) · India · IITs (JAM 2026 organised by IIT Bombay)
Original practice questions written to the published syllabus · not past papersInference through prior and posterior: conjugate analysis, prior choice, credible intervals, Bayes factors and decision theory, hierarchical models, and computation by Markov-chain Monte Carlo, with agricultural uses in variety trials, genomic prediction and small-area estimation.
In the exams: Not an explicit unit in AICE-JRF Ag Stat; PhD course in most programmes; GATE ST Bayes' theorem (light); UPSC ISS decision theory (present)
How our questions for IIT JAM spread across the units of Bayesian Inference. Difficulty: 19 easy, 38 medium, 12 hard.
| Conjugate families | 9 | |
| Posterior summaries and credible intervals | 8 | |
| Bayes' theorem for parameters: prior, likelihood and posterior | 7 | |
| Non-informative and weakly informative priors | 7 | |
| Markov chain Monte Carlo: Gibbs sampling and Metropolis-Hastings | 7 | |
| Convergence diagnostics | 7 | |
| Posterior predictive checks and model comparison (DIC, WAIC) | 7 | |
| Bayesian hypothesis testing and Bayes factors | 6 | |
| Hierarchical (multilevel) models | 4 | |
| Empirical Bayes | 4 | |
| Decision theory: loss functions and Bayes risk | 3 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
A uniform prior is placed on a germination probability p. Which statement is correct?
C. It is not uniform on the log-odds scale
Flatness depends on the parameterisation; transforming p changes the density by the Jacobian. The uniform on (0, 1) is proper, and Jeffreys' prior is beta(1/2, 1/2), not uniform.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
No infected plants are found among 10 tested. With Jeffreys' beta(1/2, 1/2) prior, what is the posterior mean infection rate?
A. 0.0455
The posterior is beta(0.5, 10.5), mean 0.5/11 = 0.0455. The MLE is 0; a uniform prior gives beta(1, 11), mean 1/12.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
Experts give a median and a 90% range for the yield response to a new practice. What does experience with elicited priors suggest?
C. Experts tend to be over-confident, so elicited priors are widened
Studies of elicitation show over-confidence: stated ranges are too narrow. Widening the prior and reporting sensitivity to it are standard practice.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
Before a further season of variety trials, an analyst computes the expected value of sample information (EVSI). How is it used?
D. Run them only if EVSI exceeds their cost
EVSI is the expected improvement in the decision's value from the extra data; it is never negative. Comparing it with the trial cost decides whether the information is worth buying.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
What does the Gelman-Rubin R̂ compare?
A. Between-chain and within-chain variance
R̂ near 1 means the chains, started from dispersed values, agree. Comparing early and late segments of one chain is Geweke's diagnostic.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
Why are several chains started from dispersed values instead of one long chain?
B. Disagreement between chains exposes problems
If chains from different regions have not mixed, their summaries differ, which R̂ and trace plots expose; one chain stuck in one mode can look perfectly stable.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
What does Geweke's diagnostic test?
C. Equality of means in an early and a late segment of one chain
Geweke compares the mean of, say, the first 10% and the last 50% of a chain with a z-score adjusted for autocorrelation; a large |z| suggests the chain had not settled.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
An HMC sampler reports many divergent transitions. What should be done?
B. Investigate and reparameterise, since parts of the posterior are not explored reliably
Divergences indicate regions of high curvature where the sampler fails, which biases estimates; reparameterisation or a smaller step size is needed. R̂ may still look fine.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
A model has log pointwise predictive density lppd = −120 and p_WAIC = 6. What is WAIC on the deviance scale?
A. 252
WAIC = −2(lppd − p_WAIC) = −2(−126) = 252. Omitting the penalty gives 240; adding it with the wrong sign gives 228.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
Among many fields, 40 had exactly one pest colony and 20 had exactly two. Using Robbins' estimator E(θ | x) = (x + 1)f(x + 1)/f(x), what is the empirical Bayes estimate of the colony rate for a field with one colony?
C. 1.0
(1 + 1) × 20/40 = 1.0, using observed frequencies in place of the marginal probabilities. No prior form is assumed.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
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