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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]
Given the binary nature of classification, a natural selection for a loss function (assuming equal cost for false positives and false negatives) would be the 0-1 loss function (0–1 indicator function), which takes the value of 0 if the predicted classification equals that of the true class or a 1 if the predicted classification does not match ...
In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. [4]
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.
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]
More precisely, the source coding theorem states that for any source distribution, the expected code length satisfies [(())] [ (())], where is the number of symbols in a code word, is the coding function, is the number of symbols used to make output codes and is the probability of the source symbol. An entropy coding attempts to ...
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