题目2421 · 机器学习
Bayes Act Under Weighted Squared Loss 1
Suppose the loss is L(a,Y)=W(Y-a)^2 where W>0 is observed at prediction time. What predictor minimizes E[L(a,Y)|X,W]?
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中文题目题目3232 · 统计
BIC Versus Bayes Factor
Why is BIC often described as an approximation to Bayesian model comparison rather than the same thing as a Bayes factor?
打开 →题目3238 · 统计
Empirical Bayes Sits Between the Camps
Why does empirical Bayes often feel like it sits between Bayesian and frequentist workflows?
打开 →题目3224 · 统计
Hierarchical Bayes Versus Bonferroni
You are screening 200 alphas and most are probably zero. Why does hierarchical Bayes approach this problem differently from Bonferroni-style frequentist correction?
打开 →题目3158 · 统计
Posterior Probability After Bayes Factors 5 and 2
A binary hypothesis has prior probability $\frac{2}{5}$. Independent signals arrive sequentially with Bayes factors [5, 2] in favor of the hypothesis. What is the final posterior probability after multiplying all evidence?
打开 →题目3223 · 统计
Why Bayes Factors and p-Values Can Disagree
In a large sample, you can see a small p-value but only weak Bayesian evidence against the null. Why is that not a contradiction?
打开 →题目4146 · 机器学习
Naive Bayes Posterior 1
A generative regime model assigns posterior probability P(trend|x)=0.7 to the trend regime. If the next-day expected payoff is 12 bps in trend and -4 bps in mean reversion, what conditional expected payoff E[r|x] does the model imply?
打开 →题目4148 · 机器学习
Naive Bayes Posterior 3
A generative regime model assigns posterior probability P(trend|x)=0.6 to the trend regime. If the next-day expected payoff is 0.015 return units in trend and -0.01 return units in mean reversion, what conditional expected payoff E[r|x] does the model imply?
打开 →题目4149 · 机器学习
Naive Bayes Posterior 4
A generative regime model assigns posterior probability P(trend|x)=0.4 to the trend regime. If the next-day expected payoff is 3 return units in trend and 1 return units in mean reversion, what conditional expected payoff E[r|x] does the model imply?
打开 →题目4150 · 机器学习
Naive Bayes Posterior 5
A generative regime model assigns posterior probability P(trend|x)=0.8 to the trend regime. If the next-day expected payoff is -2 bps in trend and 5 bps in mean reversion, what conditional expected payoff E[r|x] does the model imply?
打开 →题目033 · 概率
Three-Regime Bayesian Updating from Daily Returns
A portfolio manager models the market as being in one of three regimes, each equally likely a priori: - **Bull**: the stock goes up on any given day with probability $\frac{4}{5}$. - **Neutral**: the stock goes up with probability $\frac{1}{2}$. - **Bear**: the stock goes up wit
打开 →题目2441 · 机器学习
Weighted Log-Loss Bayes Probability Numerically 20
If p = 0.3, alpha = 4, and beta = 1 in weighted log-loss, what Bayes probability q* is optimal?
打开 →题目2430 · 机器学习
Why the Weighted-Brier Bayes Act Is Still a Weighted Mean 7
For binary Y and weighted squared loss alpha Y (1-q)^2 + beta (1-Y) q^2, derive the Bayes probability q as a function of p=P(Y=1|X).
打开 →题目2432 · 机器学习
Squared-Loss Bayes Act for a Discrete Target
Y takes values 1, 1, 4, and 7 with equal probability. What constant forecast minimizes expected squared loss?
打开 →题目2423 · 机器学习
Weighted Log-Loss Bayes Probability 3
For binary Y with P(Y=1|X)=p, consider weighted log-loss L(q,Y) = -alpha Y ln q - beta (1-Y) ln(1-q). What probability q minimizes the conditional expected loss?
打开 →题目2443 · 机器学习
Weighted Log-Loss Moves the Bayes Probability Toward the Costlier Class 16
Why does class-weighted log-loss shift the optimal reported probability toward the class with the larger weight?
打开 →题目4161 · 机器学习
Why Naive Bayes Can Work Despite Wrong Independence
You have only a few hundred labeled observations, but domain knowledge gives a plausible class-conditional structure and you also have many unlabeled feature vectors. Would you start with a generative or a discriminative model first?
打开 →题目3221 · 统计
A Confidence Interval Cannot Price a One-Off Launch Decision
A 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
打开 →题目3216 · 统计
Can 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.
打开 →题目3226 · 统计
Decision-Making with Asymmetric Loss
A 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?
打开 →题目3233 · 统计
Frequentist Risk Does Not Condition on a Prior
What is the key conceptual difference between minimizing Bayes risk under a prior and minimizing frequentist risk or regret uniformly over parameter values?
打开 →题目4151 · 机器学习
Generative Classification with a Missing Feature 1
A two-feature naive Bayes model was trained generatively, but at prediction time X2 is missing. Prior P(Y=1)=0.5, P(X1=1|Y=1)=0.8, P(X1=1|Y=0)=0.3, P(X2=1|Y=1)=0.75, P(X2=1|Y=0)=0.4. You only observe X1=1. What posterior P(Y=1|X1) should the generative model use?
打开 →题目3234 · 统计
Partial Pooling Versus Separate Fits
Why does a hierarchical Bayesian model for many related assets often produce more stable estimates than fitting each asset separately and then testing them one by one?
打开 →题目5892 · 机器学习
Posterior from a Generative Gaussian Model
A generative classifier models one feature as Gaussian within each class with equal variance: x|Y=0 ~ N(0,1), x|Y=1 ~ N(2,1), and class prior P(Y=1)=0.5. Using Bayes' rule to convert this generative description into the discriminative posterior, compute P(Y=1|x=1.5).
打开 →题目3156 · 统计
Posterior Odds After Three Independent Signals
A binary hypothesis has prior probability $\frac{1}{2}$. 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?
打开 →题目3157 · 统计
Posterior Odds Under Mixed Evidence
A binary hypothesis has prior probability $\frac{1}{3}$. 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?
打开 →题目3219 · 统计
Posterior Probability of Positive Effect Versus p-Value
If 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?
打开 →题目3235 · 统计
Predictive Interval Versus Interval for the Mean
Why is a Bayesian posterior predictive interval for next week's count answering a different question from a frequentist confidence interval for the underlying mean count?
打开 →题目3228 · 统计
Regularization as an Implicit Prior
Why do people say that ridge or lasso regularization has a Bayesian interpretation, even if the optimization is carried out in a frequentist workflow?
打开 →题目3227 · 统计
Sequential Updates Without Design Lock
Data 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?
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