{{ message }}

Instantly share code, notes, and snippets.

# Thomas Bunke thoma5B

• Bad Homburg, Germany
Last active Aug 29, 2015
View file2
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters
 We are interested in a probability distribution $\rho:X \rightarrow \mathbb R_+$, such that 1. (Constraints) $d_i^{-1} (\rho d\mathcal L ) = \mathbb P_i$, i.e. $\forall B \subseteq \mathbb R$ we have $\int 1_{\{d_i(x) \in B \} }\rho(x) d \mathcal L(x) = \mathbb P_i(B)$. 2. (Optimization problem) $\int \rho \log \rho d\mathcal L$ extremal - this expression maximizes entropy under the constraints. This takes account to the fact that less is known about the actual distribution of the probability of presence. To assure positivity we set $\rho = e^\varphi$ and obtain $\rho \log \rho = \varphi e^\varphi$ for the entropy. For simplicity we assume for $\mathbb P_i$ only discrete probabilities $\mathbb P_i = \sum_k p_{i,k} \delta_{x_{i,k}}$ The Gâteaux differential $D^{Gâteaux}$ in direction $\psi$ is given by $\partial_h F(\varphi + h\psi)\mid_{h=0}$
Last active Aug 29, 2015
View discussion
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters
 #A probability distribution (updated) ### suggested by Tilman J. Bohl We are interested in a probability distribution $\rho:X \rightarrow \mathbb R_+$, such that 1. (Constraints) $d_i^{-1} (\rho d\mathcal L ) = \mathbb P_i$, i.e. $\forall B \subseteq \mathbb R$ we have $\int 1_{\{d_i(x) \in B \} }\rho(x) d \mathcal L(x) = \mathbb P_i(B)$. 2. (Optimization problem) $\int \rho \log \rho d\mathcal L$ extremal - this expression maximizes entropy under the constraints. This takes account to the fact that less is known about the actual distribution of the probability of presence. To assure positivity we set $\rho = e^\varphi$ and obtain $\rho \log \rho = \varphi e^\varphi$ for the entropy. For simplicity we assume for $\mathbb P_i$ only discrete probabilities $\mathbb P_i = \sum_k p_{i,k} \delta_{x_{i,k}}$