14
votes
3answers
557 views

“Entropy” proof of Brunn-Minkowski Inequality?

I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality. The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then $$ ...
7
votes
0answers
186 views

Hasse-Weil Bound and Chebyshev Inequality

I was reading about the Hasse-Weil bound for the number of points in on a curve over the finite field $\mathbb{F}_q$. $$ \big| |C(\mathbb{F}_q)| - (q+1) \big| \leq 2g \sqrt{q} $$ However, this ...
2
votes
0answers
130 views

An optimization in Markov Chain

We are given two correlated random variables $V$ and $X$ supported over a finite alphabets $\mathcal{V}$ and $\mathcal{X}$. Suppose the marginal $P_V$ and conditional distribution $P_{X|V}$ are ...
0
votes
1answer
98 views

Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...
1
vote
1answer
94 views

Kalman filter with long term bias

I was reading about the Kalman filter and I do not understand how it should be used when our measurements have a long term offset like GPS location updates do. As I understand, the Kalman filter ...
0
votes
1answer
240 views

Anyone has Kushner's book “Introduction to stochastic control” 1971? I need a theorem from it

In a paper I'm reading, it refers to Theorem 8, Page 217 of the book "Introduction to Stochastic Control" H. J. Kushner, New York: Holt, Reinhart, and Winston 1971. Unfortunately I don't have it and ...
2
votes
1answer
134 views

Upper bound concerning Snell envelope

Consider, on a filtred probability space $ \left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $ \mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the usuual ...
7
votes
1answer
299 views

Probability density that minimizes the sample range

Let $\mathcal{F}$ denote the set of all "concave probability distributions" on the unit interval, that is, all functions $f:[0,1]\to \mathbb{R}$ such that $f$ is concave, $f(x)\geq 0$ for all $x\in ...
2
votes
1answer
193 views

An optimization problem, non complete bipartite graph and hungarian algorithm

I have two tables at my disposal, one work dataset and one reference dataset. Each dataset has got two columns, lets say these are fields A and B. I would like the rows in reference dataset with the ...
1
vote
2answers
694 views

When do maximum and expectation commute?

Hi, I'm looking for conditions on $G(t,x)$ such that $$ \sup\limits_{t\in [0,1]}E[G(t,X)]=E[\sup\limits_{t\in [0,1]}G(t,X)] $$ where $X$ is a random variable (it's easy to see that $\sup\limits_{t\in ...
0
votes
1answer
241 views

Is it known that every PDF continuous in all $R^n$ has a maximum? [closed]

I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around ...
1
vote
2answers
165 views

Bound on expression from probability distributions

I came across this issue while trying to combine multiple probability distributions into a single distribution which approximates them all simultaneously. This boils down to maximizing this expression ...
4
votes
1answer
433 views

An optimization problem involving sum of binomial coefficients upto some value

I would like to minimize $f(s, n, \epsilon)$ with respect to $s$ where $$f(s,n,\epsilon) = \left( 1 + \frac{n}{2^s} \right)\frac{1}{s} \sum_{k=0}^{\lfloor s\epsilon \rfloor} {s \choose k}~.$$ Note ...
1
vote
1answer
154 views

Max Absolute Difference of Expectations under Change of Measure

Given a sample space $\Omega=\{ 1,\cdots,N \}$, a random variable $x$ defined on $\Omega$ that takes value $x_1,\cdots,x_N$, and a set of strictly positive real numbers $w_1,\cdots,w_N$. Define for ...
1
vote
3answers
431 views

convergence in distribution of stochastic gradient descent.

The stochastic gradient descent algorithm where only a noisy gradient (zero mean noise) is used to update current estimate is known to converge almost surely to the minimizer. However, if one is ...
2
votes
1answer
243 views

Concentration of measure and bounds on variance

I am trying to characterize the sensitivity of a function $f: R^N\to{}R$ to the perturbations in the input vector $\mathbb{x}=\left[x_1,\dots{}x_N\right]$. For that purpose, I evaluate Cramer-Rao ...
3
votes
0answers
320 views

maximum variance unfolding

Consider positive weights $\pi_1, \ldots, \pi_n$ (one can suppose that they add up to $1$) and $n-1$ lengths $d_1, \ldots, d_{n-1}$. Is there an analytical solution to the following problem: find the ...
4
votes
2answers
1k views

Commuting supremum and expectation

Given a one-parametric random function on a probability space $(\Omega,\mathcal F,\mathbb P)$: $X:U\times\Omega\to \mathbb R \text{ and } (a,w)\mapsto X(a,w), \text{ with } \sigma(X(a))\subseteq ...
8
votes
2answers
641 views

Covering a random graph with spanning trees.

Let $G=(V,E)$ be a connected graph, say $V=\{1,\ldots,n\}$. Let $F=(V,E')$ be a uniformly random forest in $G$. (In other words, $E'$ is a subset of edges $E$ not containing a cycle, and it is ...
6
votes
1answer
685 views

Random, Linear, Homogeneous Difference Equations and Time Integration Methods for ODEs

Most methods (that I know of) of numerically approximating the solution of ODEs are "general linear methods". For this type of method, the so-called 'linear stability' is examined by applying the ...
-2
votes
1answer
695 views

Determine noise distribution [closed]

I'm trying to solve the following least squares problem: $\underset{x}{\text{min}} ||Ax - \tilde{b}||_2$ where $Ax = b$ and $\tilde{b} = b + w$ Question: How do I determine which probability ...
4
votes
0answers
457 views

Dynamic programming principle (DPP)

In stochastic control problem, one shall use the measurable selection theorem to prove DPP. It was discussed in discrete time case in [Bertsekas and Shreve 1978]. Is there unified framework in ...
0
votes
2answers
529 views

univariate prior corresponding to weighted sum of L1 and L2 penalties?

Is there a univariate probability distribution $p_{\lambda,\alpha}(\beta)$ over the reals, parameterized by $\lambda > 0$ and $1 >= \alpha >= 0$, such that $p_{\lambda,\alpha} \propto ...