ASRB NET - Agricultural Statistics (and Computer Application / Bioinformatics) · India · Agricultural Scientists Recruitment Board (ICAR)
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)
ASRB NET / ARS interviews and research-institute recruitment
Applied understanding of shrinkage, BLUP, hierarchical variety-trial analysis and reporting of Bayesian results.
How our questions for ASRB NET spread across the units of Bayesian Inference. Difficulty: 5 easy, 28 medium, 3 hard.
| Hierarchical (multilevel) models | 6 | |
| Non-informative and weakly informative priors | 4 | |
| Convergence diagnostics | 4 | |
| Empirical Bayes | 4 | |
| Bayes' theorem for parameters: prior, likelihood and posterior | 3 | |
| Posterior summaries and credible intervals | 3 | |
| Bayesian hypothesis testing and Bayes factors | 3 | |
| Decision theory: loss functions and Bayes risk | 3 | |
| Posterior predictive checks and model comparison (DIC, WAIC) | 3 | |
| Markov chain Monte Carlo: Gibbs sampling and Metropolis-Hastings | 2 | |
| Conjugate families | 1 |
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
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
For a poor seed lot (germination 0.7) against a good one (0.9), 7 of 10 seeds germinate, so the Bayes factor for 'poor' is 4.65. The prior probability of 'poor' is 0.2. What is the posterior probability of 'poor'?
B. 0.538
Prior odds 0.2/0.8 = 0.25; posterior odds 0.25 × 4.65 = 1.162; probability 1.162/(1 + 1.162) = 0.538. 0.823 assumes even prior odds; 0.930 multiplies the probability, not the odds.
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
Trial means vary around μ = 5.0 t/ha with τ = 0.4. A trial with mean 6.2 and standard error 0.6 is analysed with the normal hierarchical model. What is its shrunken estimate?
A. 5.37 t/ha
B = σ²/(σ² + τ²) = 0.36/0.52 = 0.692; estimate = (1 − B) × 6.2 + B × 5.0 = 5.37. 5.96 would apply to a trial with standard error 0.2; 5.60 is a simple average.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
In a hierarchical model, what happens to the group estimates as the between-group variance τ² → 0, and as τ² → ∞?
A. τ² → 0 gives complete pooling; τ² → ∞ gives no pooling
With B = σ²/(σ² + τ²): τ² → 0 makes B → 1 (all groups at μ); τ² → ∞ makes B → 0 (each group keeps its own estimate).
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
How does parametric empirical Bayes estimate the hyperparameters of the prior?
A. From the marginal distribution of the data
EB maximises (or matches moments of) m(x | η) = ∫ f(x | θ)π(θ | η)dθ, so the data choose the prior. Elicitation is a fully Bayesian route.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
District crop-yield estimates from a sample survey are noisy for districts with few crop-cutting experiments. Which model combines them with satellite-based predictions by shrinkage?
C. The Fay-Herriot area-level model
Fay-Herriot shrinks each direct district estimate towards a regression prediction, with weights depending on the estimate's sampling variance; it is the standard area-level small-area model.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
When the number of groups is large, how do empirical Bayes and full hierarchical Bayes estimates of group means compare?
C. They are close, because hyperparameter uncertainty is small
With many groups the hyperparameters are well estimated, so plugging them in loses little. The two differ most with few groups.
Further reading: Gelman et al., Bayesian Data Analysis; Berger, Statistical Decision Theory and Bayesian Analysis; Robert, The Bayesian Choice
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