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Two bits of entropy: In the case of two fair coin tosses, the information entropy in bits is the base-2 logarithm of the number of possible outcomes — with two coins there are four possible outcomes, and two bits of entropy. Generally, information entropy is the average amount of information conveyed by an event, when considering all ...
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 ...
The algorithm starts with an initial estimate of the optimal value, , and proceeds iteratively to refine that estimate with a sequence of better estimates ,, ….The derivatives of the function := are used as a key driver of the algorithm to identify the direction of steepest descent, and also to form an estimate of the Hessian matrix (second derivative) of ().
In information theory, the Rényi entropy is a quantity that generalizes various notions of entropy, including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. The Rényi entropy is named after Alfréd Rényi , who looked for the most general way to quantify information while preserving additivity for independent events.
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.
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.
Entropy of a Bernoulli trial (in shannons) as a function of binary outcome probability, called the binary entropy function.. In information theory, the binary entropy function, denoted or (), is defined as the entropy of a Bernoulli process (i.i.d. binary variable) with probability of one of two values, and is given by the formula: