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3440Why Entropy Is Interpreted as Average SurpriseWhy is entropy often described as the average surprise of a source?数学中等essay未尝试面试订阅3441Base-Rate Forecast Penalty on a 70% EventA binary event truly occurs with probability 0.7. A forecast uses probability 0.55 instead of the true probability. What is the expected extra log-loss in bits relative to a calibrated forecast?数学困难derivation未尝试面试订阅3446Mutual Information of a Signal With Noise Variance FourA latent Gaussian alpha X ~ N(0,1) is observed through Y = X + eps with eps ~ N(0,4). How many bits of mutual information does Y carry about X?数学中等derivation未尝试面试订阅3447Mutual Information When Signal Variance Is Two and Noise Is SixA latent Gaussian factor has variance 2 and observation noise has variance 6. How many bits of mutual information does one observation carry about the factor?数学困难derivation未尝试面试订阅3448Incremental Information From a Second Independent SensorA latent X ~ N(0,1) is observed by two conditionally independent sensors with noise variances 4 and 9. How many extra bits do you gain by adding the second sensor after already seeing the first?数学中等derivation未尝试面试订阅3449Noise Variance Needed for 0.25 Bits of InformationA latent X ~ N(0,1) is observed as Y = X + eps with eps ~ N(0,sigma 2). What sigma 2 gives mutual information exactly 0.25 bits?数学中等derivation未尝试面试订阅3450Second Sensor Noise Required for 0.4 Bits TotalA latent X ~ N(0,1) is already observed by one sensor with noise variance 3. A second independent sensor with noise variance v will be added. What v makes the total mutual information 0.4 bits?数学中等derivation未尝试面试订阅3451Mutual Information of an Asymmetric Binary AlertA latent regime R is 1 with probability 0.3. An alert Y has hit rate 0.8 and false-alarm rate 0.2. How many bits of mutual information does Y carry about R?数学简单derivation未尝试面试订阅3452Posterior-vs-Prior KL After a High-Quality Alert FiresA regime prior is 0.1. An alert fires with hit rate 0.9 and false-alarm rate 0.05. If the alert fires, what is the KL divergence in bits between the posterior on the regime and the prior?数学简单derivation未尝试面试订阅3454KL Between Signal Distributions Under the Two RegimesA binary signal Y has P(Y=1|R=1) = 0.8 and P(Y=1|R=0) = 0.3. What is KL(P(Y|R=1) || P(Y|R=0)) in bits?数学中等derivation未尝试面试订阅3455Posterior-vs-Prior KL After No Alert ArrivesA regime prior is 0.25. An alert has hit rate 0.85 and false-alarm rate 0.15. If no alert arrives, what is the KL divergence in bits between the posterior on the regime and the prior?数学困难derivation未尝试面试订阅3456Why KL Is the Right Tax for MiscalibrationIn probability forecasting, why does KL divergence naturally show up as the penalty for using the wrong predictive distribution?数学中等essay未尝试面试订阅3457Why Mutual Information Can Be Small Even for a Useful SignalGive a clean reason why a signal can be economically useful in a rare-event problem while still carrying only a small number of bits of mutual information.数学中等essay未尝试面试订阅3458Data Processing as an Information BudgetWhy is coarsening a predictive score into a few buckets usually best understood as spending down an information budget?数学中等essay未尝试面试订阅3459Why Two Highly Correlated Signals Add Little Incremental MIWhy does a second forecast feed that mostly repeats the first usually contribute very little extra mutual information about the target?数学中等essay未尝试面试订阅3460Why KL Is Asymmetric but Still UsefulKL divergence is not symmetric. Why does that asymmetry make sense rather than being a defect in forecasting problems?数学中等essay未尝试面试订阅3461Why Log-Loss Improvement Can Be Tiny Yet RealA new model improves average log-loss only slightly. Why can that still correspond to a genuine improvement in information quality?数学中等essay未尝试面试订阅3463Why a Gaussian MI Formula Is a Reasonable ApproximationWhy do quants often use the Gaussian mutual-information formula as a back-of-the-envelope approximation even when the true signal is not exactly Gaussian?数学中等essay未尝试面试订阅3464Why Label Leakage Makes Mutual Information Look Too GoodWhy can label leakage or look-ahead information make estimated mutual information appear enormous in-sample but vanish out of sample?数学中等essay未尝试面试订阅3465Why Conditional KL and Unconditional MI Are Different ObjectsExplain in one clean paragraph why the KL divergence between posterior and prior after one realized observation is not the same object as unconditional mutual information.数学中等essay未尝试面试订阅