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3433Entropy of a Dyadic Four-State SourceWhat is the entropy of the distribution (1/2, 1/4, 1/8, 1/8)?数学中等derivation未尝试面试订阅3436Why Uniform Distributions Maximize Entropy Under a Support ConstraintWhy does spreading probability mass more evenly over a fixed finite support raise entropy?数学中等essay未尝试面试订阅3437Why Coarse-Graining Lowers EntropyWhy does merging labels generally reduce entropy rather than increase it?数学中等essay未尝试面试订阅3438Why Side Information Cannot Increase Remaining UncertaintyWhy is it impossible for an informative side signal to make the conditional entropy of the original source larger on average?数学中等essay未尝试面试订阅3439Why Fixed-Length Codes Waste Bits on Non-Power-of-Two AlphabetsWhy does a fixed-length binary code necessarily waste some average code length when the number of equally likely symbols is not a power of two?数学中等essay未尝试面试订阅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5899Betting Kelly on the Wrong ProbabilityAn even-money coin truly wins with probability p=0.55, but you overestimate it as p=0.65 and bet the Kelly fraction implied by your estimate. What is your actual long-run expected log-growth rate per round? Compare it to the growth you would have earned betting the correct Kelly fraction, and state what the sign of your actual growth implies.概率困难数值题未尝试面试订阅5900Higher Expected Return, Lower Compounded GrowthAn even-money coin wins with probability 0.6. Trader A always stakes the fraction f A=0.10 of wealth; Trader B always stakes f B=0.40. (i) Whose stake has the higher one-round expected (arithmetic) profit? (ii) Whose wealth compounds faster over many rounds? Explain the apparent conflict.概率中等数值题未尝试免费