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Stochastic functions and the Giry monad
Stochastic functions A function \(f\) between sets \(X\) and \(Y\) is a mapping \[\begin{aligned} f &:& X &\to& Y \cr f &:& x &\mapsto& f(y) \end{aligned}\] and one of the most important properties of functions is that they compose, i.
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Variational autoencoders
The following is a summary of some of the results of KW2013. Evidence lower bound Suppose we have a probability space \(X\) from which we can sample, but whose probability density \(p(x)\) is unknown.
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Euler beta function and Dirichlet distribution
Let’s remind ourselves how about the Euler beta function, defined as \[ B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt. \] First, we will express \(B(x,y)\) in terms of the more familiar Gamma function, defined as
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Introduction to Reinforcement Learning
I would like to record the basic formalism of reinforcement learning. Hopefully, this will lead to a post or series of posts giving a basic tensorflow implementation of a simple deep reinforcement learning model.
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Variance Stabilizing Transformations
I want to record here a very interesting thing which I recently discovered, variance-stabilizing transformations. The idea is very simple: suppose we have a random variable \(x\), which follows a probability distribution which is parametrized solely by its mean \(\mu\), with variance \(var(x) = g(\mu)\) a known function of the mean.
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Santalo Formula
This post has been migrated from my old blog, the math-physics learning seminar. Let \(M\) be a simple Riemannian manifold with boundary \(\partial M\). For \((x,v) \in SM\), let \(\tau(x,v)\) denote the exit time of the geodesic starting at \(x\) with tangent vector \(v\), i.
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The Index Form
This post has been migrated from my old blog, the math-physics learning seminar. Let \(f: [0,T] \times (-\epsilon, \epsilon) \to M\) be a family of parametrized curves in a Riemannian manifold \((M, g)\).
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Boundary Distance
This post has been migrated from my old blog, the math-physics learning seminar. Recently, I’ve been learning some topics related to machine learning, and especially manifold learning. These both fall under the general notion of inverse problems: given some mathematical object \(X\) (it could be a function \(f: A \to B\), or a Riemannian manifold \((M,g)\), or a probability measure \(d\mu\) on a space \(X\), etc.
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Hamilton-Jacobi equation and Riemannian distance
This post has been migrated from my old blog, the math-physics learning seminar. Consider the cotangent bundle \(T^\ast X\) as a symplectic manifold with canonical symplectic form \(\omega\). Consider the Hamilton-Jacobi equation
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Classical partition function
This post has been migrated from my old blog, the math-physics learning seminar. Let \((M, \omega)\) be a symplectic manifold of dimension \(2n\), and let \(H: M \to \mathbf{R}\) be a classical Hamiltonian.
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