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3155Posterior Mean After Four Noisy SignalsA latent mean has prior N(10,25). Observations arrive one by one with known noise variance 2=4 and realized values [12, 11, 8, 9]. After processing the full stream sequentially, what are the final posterior mean and variance?统计中等derivation未尝试面试订阅3156Posterior Odds After Three Independent SignalsA binary hypothesis has prior probability 1 2 . Independent signals arrive sequentially with Bayes factors [2, Fraction(1, 2), 3] in favor of the hypothesis. What is the final posterior probability after multiplying all evidence?统计中等derivation未尝试面试订阅3157Posterior Odds Under Mixed EvidenceA binary hypothesis has prior probability 1 3 . Independent signals arrive sequentially with Bayes factors [4, Fraction(1, 5)] in favor of the hypothesis. What is the final posterior probability after multiplying all evidence?统计中等derivation未尝试面试订阅3158Posterior Probability After Bayes Factors 5 and 2A binary hypothesis has prior probability 2 5 . Independent signals arrive sequentially with Bayes factors [5, 2] in favor of the hypothesis. What is the final posterior probability after multiplying all evidence?统计中等derivation未尝试面试订阅3161Trade Only If Posterior Regime Probability Exceeds 70%A latent profitable regime has prior probability 1 2 . Independent signals arrive sequentially; each `H` doubles the odds of the regime and each `T` halves them. After the signal string `HH`, should you act if the required posterior threshold is 7 10 ? Also report the posterior probability.统计中等derivation未尝试面试订阅3162Allocate Capital Only Above a 60% PosteriorA latent profitable regime has prior probability 2 5 . Independent signals arrive sequentially; each `H` doubles the odds of the regime and each `T` halves them. After the signal string `HHT`, should you act if the required posterior threshold is 3 5 ? Also report the posterior probability.统计中等derivation未尝试面试订阅3164Stop Trading Unless the Posterior Stays Above 55%A latent profitable regime has prior probability 3 5 . Independent signals arrive sequentially; each `H` doubles the odds of the regime and each `T` halves them. After the signal string `HTT`, should you act if the required posterior threshold is 11 20 ? Also report the posterior probability.统计中等derivation未尝试面试订阅3165Turn On the Hedge Above 65% PosteriorA latent profitable regime has prior probability 1 4 . Independent signals arrive sequentially; each `H` doubles the odds of the regime and each `T` halves them. After the signal string `HHHT`, should you act if the required posterior threshold is 13 20 ? Also report the posterior probability.统计中等derivation未尝试面试订阅3216Can a 95% Credible Interval Be Read as a 95% Probability Statement?A PM sees a 95% Bayesian credible interval for a strategy's daily edge and says, "So there is a 95% probability the true edge lies in this interval." Is that interpretation correct? Contrast it with the frequentist interpretation of a 95% confidence interval.统计中等essay未尝试面试订阅3217Why Optional Stopping Breaks a Fixed-Horizon p-ValueAn experiment is designed for a fixed horizon, but the desk checks the p-value every day and stops once it drops below 0.05. Why does that invalidate the nominal 5% Type I error guarantee? Why is the Bayesian posterior update answering a different question?统计中等essay未尝试面试订阅3218Small Sample, One Success: Which Lens Shrinks More Naturally?A new quoting rule is tried 5 times and succeeds once. Why does a Bayesian analysis with a skeptical prior naturally shrink the estimated success probability toward a baseline, while a plain frequentist point estimate does not do that unless you add some regularization device on top?统计中等essay未尝试面试订阅3219Posterior Probability of Positive Effect Versus p-ValueIf a PM wants to know, "What is the probability the treatment effect is positive?", why is a Bayesian posterior probability directly aligned with that question while a p-value is not?统计中等essay未尝试面试订阅3220Why a 'Weakly Informative' Prior Still MattersA researcher says, "I used a weakly informative prior, so the Bayesian answer is basically prior-free." Why is that statement too strong, especially in small or noisy samples?统计中等essay未尝试面试订阅3221A Confidence Interval Cannot Price a One-Off Launch DecisionA PM sees a frequentist 95% confidence interval for next-month strategy edge of [-0.1, 0.4] and asks, "So what is the probability the true edge is positive for this launch decision?" Why can't the interval answer that question by itself, and what Bayesian quantity would answer it?统计中等essay未尝试面试订阅3222When a Skeptical Prior Can Widen a Credible IntervalSuppose data are weak and the prior strongly pulls a parameter toward zero but also reflects real uncertainty about how much shrinkage is appropriate. Why can a Bayesian credible interval be wider than a frequentist asymptotic interval in that setting?统计中等essay未尝试面试订阅3223Why Bayes Factors and p-Values Can DisagreeIn a large sample, you can see a small p-value but only weak Bayesian evidence against the null. Why is that not a contradiction?统计中等essay未尝试面试订阅3224Hierarchical Bayes Versus BonferroniYou are screening 200 alphas and most are probably zero. Why does hierarchical Bayes approach this problem differently from Bonferroni-style frequentist correction?统计中等essay未尝试面试订阅3225Posterior Predictive Check Versus Classical Goodness-of-FitWhy does a posterior predictive check answer a different question from a classical goodness-of-fit p-value?统计中等essay未尝试面试订阅3226Decision-Making with Asymmetric LossA one-off event trade has very asymmetric payoffs. Why does Bayesian analysis often feel more natural than a textbook frequentist test when the real goal is a single yes/no decision under asymmetric loss?统计中等essay未尝试面试订阅3227Sequential Updates Without Design LockData trickle in continuously and the desk wants to update beliefs every hour. Why is Bayesian inference naturally sequential, while a frequentist testing workflow often needs more design discipline to preserve its advertised guarantees?统计中等essay未尝试面试订阅