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almostsuremath.com | ||
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jmanton.wordpress.com
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| | | | | If $latex Y$ is a $latex \sigma(X)$-measurable random variable then there exists a Borel-measurable function $latex f \colon \mathbb{R} \rightarrow \mathbb{R}$ such that $latex Y = f(X)$. The standard proof of this fact leaves several questions unanswered. This note explains what goes wrong when attempting a "direct" proof. It also explains how the standard proof... | |
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mkatkov.wordpress.com
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| | | | | For probability space $latex (\Omega, \mathcal{F}, \mathbb{P})$ with $latex A \in \mathcal{F}$ the indicator random variable $latex {\bf 1}_A : \Omega \rightarrow \mathbb{R} = \left\{ \begin{array}{cc} 1, & \omega \in A \\ 0, & \omega \notin A \end{array} \right.$ Than expected value of the indicator variable is the probability of the event $latex \omega \in... | |
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thatsmaths.com
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| | | | | We consider a simple problem in probability: A thin rod is broken at random into three pieces. What is the probability that these three pieces can be used to form a triangle? This problem is solved without difficulty. If the rod is of unit length, aligned along the interval $latex {0\le x \le 1}&fg=000000$, we... | |
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iclr-blogposts.github.io
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| | | This blog post explores the interplay between the Data Processing Inequality (DPI), a cornerstone concept in information theory, and Function-Space Variational Inference (FSVI) within the context of Bayesian deep learning. The DPI governs the transformation and flow of information through stochastic processes, and its unique connection to FSVI is employed to highlight FSVI's focus on Bayesian predictive posteriors over parameter space. The post examines various forms of the DPI, including the KL divergence based DPI, and provides intuitive examples and detailed proofs. It also explores the equality case of the DPI to gain a deeper understanding. The connection between DPI and FSVI is then established, showing how FSVI can measure a predictive divergence inde... | ||
