CSIR-UGC NET - Mathematical Sciences (statistics option) · India · National Testing Agency for CSIR
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 CSIR-UGC NET spread across the units of Bayesian Inference. Difficulty: 7 easy, 29 medium, 27 hard.
| Non-informative and weakly informative priors | 7 | |
| Markov chain Monte Carlo: Gibbs sampling and Metropolis-Hastings | 7 | |
| Posterior predictive checks and model comparison (DIC, WAIC) | 7 | |
| Conjugate families | 6 | |
| Bayesian hypothesis testing and Bayes factors | 6 | |
| Convergence diagnostics | 6 | |
| Empirical Bayes | 6 | |
| Bayes' theorem for parameters: prior, likelihood and posterior | 5 | |
| Decision theory: loss functions and Bayes risk | 5 | |
| Hierarchical (multilevel) models | 5 | |
| Posterior summaries and credible intervals | 3 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Counts are Poisson(λ) with a gamma prior on λ. What is the posterior predictive distribution of a new count?
B. Negative binomial
Mixing a Poisson over a gamma distribution for its mean gives a negative binomial; the extra variance reflects uncertainty about λ. Beta-binomial is the predictive for binomial data with a beta prior.
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
What is Lindley's paradox?
D. With large n, a result significant at p = 0.01 can give a Bayes factor favouring the point null
For fixed p and growing n, the data concentrate near θ₀ relative to a diffuse alternative prior, so B₀₁ grows; frequentist and Bayesian conclusions can conflict. Posterior probabilities never exceed 1.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
For random-walk Metropolis in high dimensions, what acceptance rate does theory suggest is efficient?
A. About 0.234
Roberts, Gelman and Gilks (1997) derived 0.234 as optimal in high dimensions; for one-dimensional updates about 0.44 is suggested. Very high rates mean steps too small; very low, steps too large.
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
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
In what situation can DIC be unreliable, even giving a negative effective number of parameters p_D?
A. When the posterior is strongly non-normal or multimodal, as in mixtures
p_D = D̄ − D(θ̄) depends on the deviance at the posterior mean, which can be a poor point for skewed or multimodal posteriors; WAIC and PSIS-LOO avoid this.
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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