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Maxwell's demon can (hypothetically) reduce the thermodynamic entropy of a system by using information about the states of individual molecules; but, as Landauer (from 1961) and co-workers [20] have shown, to function the demon himself must increase thermodynamic entropy in the process, by at least the amount of Shannon information he proposes ...
Cross-entropy can be used to define a loss function in machine learning and optimization. Mao, Mohri, and Zhong (2023) give an extensive analysis of the properties of the family of cross-entropy loss functions in machine learning, including theoretical learning guarantees and extensions to adversarial learning. [3]
It's easy to check that the logistic loss and binary cross-entropy loss (Log loss) are in fact the same (up to a multiplicative constant ()). The cross-entropy loss is closely related to the Kullback–Leibler divergence between the empirical distribution and the predicted distribution.
Since an entropy is a state function, the entropy change of the system for an irreversible path is the same as for a reversible path between the same two states. [23] However, the heat transferred to or from the surroundings is different as well as its entropy change. We can calculate the change of entropy only by integrating the above formula.
Multinomial logistic regression is known by a variety of other names, including polytomous LR, [2] [3] multiclass LR, softmax regression, multinomial logit (mlogit), the maximum entropy (MaxEnt) classifier, and the conditional maximum entropy model.
Relative entropy relates to "rate function" in the theory of large deviations. [24] [25] Arthur Hobson proved that relative entropy is the only measure of difference between probability distributions that satisfies some desired properties, which are the canonical extension to those appearing in a commonly used characterization of entropy. [26]
Intuitively, the entropy H X of a discrete random variable X is a measure of the amount of uncertainty associated with the value of X when only its distribution is known. The entropy of a source that emits a sequence of N symbols that are independent and identically distributed (iid) is N ⋅ H bits (per message of N symbols).
Equivalently, the min-entropy () is the largest real number b such that all events occur with probability at most . The name min-entropy stems from the fact that it is the smallest entropy measure in the family of Rényi entropies. In this sense, it is the strongest way to measure the information content of a discrete random variable.