![]() ![]() When x > y, the term is already positive. For example, if you're using several such distance metrics and want to compute the average as an overall metric, all terms need to be positive.Īdding y - x to (x log(x/y)) make it positive, when y > x (read final paragraph below). Particularly, it is negative if y > x.Īs KL divergence is used as a distance metric, it may be convenient to make it non-negative. The expression (x log(x/y)) can be positive or negative, depending on the values of x and y. While Mario Boley's accepted response partly answers the question with a good example, the reason why the term -x + y is added is not explained. I am not familiar with the rationale behind the element-wise extra-terms of _div but the documentation points to a reference that might explain more. Print(rel_entr(p, q), sum(rel_entr(p, q))) While both result in the same sum (when used with proper probability vectors the elements of which sum to 1), the second variant ( _div) is different element-wise in that it adds -x +y terms, i.e.,įrom scipy.special import rel_entr, kl_div In contrast, both _entr and _div are "element-wise functions" that can be used in conjunction with the usual array operations, and have to be summed before they yield the aggregate relative entropy value. On the basis of such results, it is indicated that, as was found in earlier work, a thermodynamic theory of irreversible processes can be erected on the compensation function since it is an integral of a one‐form in the thermodynamic space whereas the Boltzmann entropy is not.The default option for computing KL-divergence between discrete probability vectors would be. Such a limit represents a contraction of information as the description of irreversible processes is made in the thermodynamic space contracted from the phase space of 10 23 particles. The time derivative of the relative entropy does not vanish in the limit, but tends to a limit associated with energy dissipation. By using the balance equations for the compensation function and the relative entropy, we investigate the limiting behavior of the rate of relative entropy as the thermodynamic branch of the distribution function becomes convergent in the sense of means (i.e., weakly converges) to the phase‐space distribution function. By using the concept of relative entropy which is the difference between the two quantities, we examine their relations and significance for the mathematical structure of thermodynamics of irreversible processes. The two quantities are not generally the same. The Boltzmann entropy is an information entropy enumerated with the phase‐space distribution function in the phase space, whereas the compensation function introduced in the previous work is a representation of the former in thermodynamic space. Such a distribution function is different in nature from the phase‐space distribution function obtained by directly solving the Boltzmann kinetic equation subject to initial and boundary conditions in the phase space without the functional hypothesis. The conventional solution methods for the Boltzmann kinetic equation such as the Chapman–Enskog method or the moment method provide a thermodynamic branch of the distribution function evolving through macroscopic variables under the functional hypothesis. ![]()
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