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If one considers the text of every book ever published as a sequence, with each symbol being the text of a complete book, and if there are N published books, and each book is only published once, the estimate of the probability of each book is 1/N, and the entropy (in bits) is −log 2 (1/N) = log 2 (N).
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.
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]
Onsager (1931, I) [1] wrote: "Thus the vector field J of the heat flow is described by the condition that the rate of increase of entropy, less the dissipation function, be a maximum." Careful note needs to be taken of the opposite signs of the rate of entropy production and of the dissipation function, appearing in the left-hand side of ...
Despite the foregoing, there is a difference between the two quantities. The information entropy Η can be calculated for any probability distribution (if the "message" is taken to be that the event i which had probability p i occurred, out of the space of the events possible), while the thermodynamic entropy S refers to thermodynamic probabilities p i specifically.
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]
The BSC has a capacity of 1 − H b (p) bits per channel use, where H b is the binary entropy function to the base-2 logarithm: A binary erasure channel (BEC) with erasure probability p is a binary input, ternary output channel. The possible channel outputs are 0, 1, and a third symbol 'e' called an erasure.
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