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In this form the relative entropy generalizes (up to change in sign) both the discrete entropy, where the measure m is the counting measure, and the differential entropy, where the measure m is the Lebesgue measure. If the measure m is itself a probability distribution, the relative entropy is non-negative, and zero if p = m as measures.
They all have in common a dependent variable to be predicted that comes from one of a limited set of items that cannot be meaningfully ordered, as well as a set of independent variables (also known as features, explanators, etc.), which are used to predict the dependent variable. Multinomial logistic regression is a particular solution to ...
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.
Slow motion video of a glass cup smashing on a concrete floor. In the very short time period of the breaking process, the entropy of the mass making up the glass cup rises sharply, as the matter and energy of the glass disperse. The concept of entropy can be described qualitatively as a measure of energy dispersal at a specific temperature. [65]
In information theory, the cross-entropy between two probability distributions and , over the same underlying set of events, measures the average number of bits needed to identify an event drawn from the set when the coding scheme used for the set is optimized for an estimated probability distribution , rather than the true distribution .
The cosmological evolution of the void regions differs drastically from the evolution of the universe as a whole: there is a long stage when the curvature term dominates, which prevents the formation of galaxy clusters and massive galaxies. Hence, although even the emptiest regions of voids contain more than ~15% of the average matter density ...
The joint information is equal to the mutual information plus the sum of all the marginal information (negative of the marginal entropies) for each particle coordinate. Boltzmann's assumption amounts to ignoring the mutual information in the calculation of entropy, which yields the thermodynamic entropy (divided by the Boltzmann constant).
In general, this entropy will depend on the partition; however, for many dynamical systems, it is the case that the symbolic dynamics is independent of the partition (or rather, there are isomorphisms connecting the symbolic dynamics of different partitions, leaving the measure invariant), and so such systems can have a well-defined entropy ...