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-1
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1answer
41 views

Using common samples to numerically estimate pairwise equality of three random variables

Let $X,Y,Z$ be three discrete random variables which I can numerically sample. I need to numerically estimate the probability that $X=Y$ and the probability that $X=Z$. I would like to know whether ...
2
votes
0answers
96 views

Can computers take uniform samples from a polytope?

For each $r \in \mathbb N $ write $\mathbb Z/ 10^r = \{a/10^r: a \in \mathbb Z\}$ and $P(r)$ for the lattice $(\mathbb Z/10^r)^N \subset \mathbb R^N$. Suppose the plane $P \subset \mathbb R^N$ is ...
0
votes
0answers
25 views

Efficiency of importance sampling in terms of the size of the the support of sampling distribution

In importance sampling, one proposes to compute an integral $I:=\mathbb E_{x \sim P}[h(x)]$ by rewritting it as $$ I=\mathbb E_{x \sim Q}\left[w(x)h(x)\right],\text{ with }w(x):=\frac{p(x)}{q(x)}, $$ ...
0
votes
2answers
54 views

Use Importance sampling for multimodal and multivariate distribution draws, how to choose proposal distribution?

I'm in trouble trying to generate samples following a particular distribution which is not numerically known perfectly. Let us consider a $R^n$ space provided with an orthonormal base $( e_{1},...,e_{...
1
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0answers
25 views

Supremum of expectations over a family distributions close to a base distribution

Let $p$ be a probability distribution on a space $X$ , $f:X \rightarrow \mathbb R_+$ be a measureable function, and $0 < \alpha \le 1 \le \beta < \infty$. Define a sub-collection $\mathcal Q \...
0
votes
0answers
21 views

Approximate in $W_1$ sense, an empirical distribution with restriction of true distribution on a set

Let $\mu$ be a probability distribution on a metric space $X=(X,d)$ (to avoid unnecessary complications, assume the full filtration $2^X$) and let $x_1,x_2,\ldots,x_N$ be a sample of size $N$ drawn i....
2
votes
2answers
72 views

Draw samples from distribitions in the neighborhood of a fixed distribution

Disclaimer Sorry in advance for vagueness. I'm still trying to get my ideas right on this one. Setup So, let $P$ be a distribution on a Euclidean space $X$ with an $\ell_p$ metric, and let $P_\...
8
votes
1answer
217 views

Expectation inequality for sampling without replacement

Is the following proposition correct? $X_1, X_2, X_3$ are uniformly at random sampled from a finite set $\mathcal X$ without replacement. $f : \mathcal X^2 \rightarrow \mathbb R_{\ge0}$ is symmetric:...
3
votes
1answer
297 views

Minimize the variance of a Boltzmann distribution

N.B.: Sorry for cross-posting from https://stats.stackexchange.com/posts/347804/edit (I realized it was the wrong venue for the question, but couldn't find an easy way to transfer the question here). ...
1
vote
1answer
91 views

Uniform sampling of random connected graph with given number of vertices/edges

I am looking for algorithms for the exact uniform sampling of connected labelled graphs with $n$ vertices and $m$ edges. By "exact" I mean that every such graph should be generated with precisely (not ...
2
votes
1answer
61 views

Sampling with non-uniform probabilities

Let $p_1,p_2,...,p_n$ are given probabilities. ($\sum_{i=1}^n p_i =1, p_i \geq 0 $). Is there any distribution, which picks $k\leq n$ distinct elements from $1,2,...,n$ such that $P(i \in S) = k p_i$ ...
1
vote
1answer
99 views

Plancharel-Pólya inequality for functions of exponential type

If $f(z)$ is an entire function of exponential type $\tau$ and $p$ a positive number such that that $$\int_{-\infty}^{+\infty}|f(x)|^pdx<\infty$$ then it can be proven that $$\int_{-\infty}^{+\...
4
votes
1answer
231 views

Combinatorial computational problem about 0-1 vectors and sampling algorithms

Let $M \in \{0,1\}^{m\times n}$, where $n\gg 1$ and $m\le n$. A procedure consisting of the following three steps is repeated $t\gg 1$ times: A row $\require{amsmath} \boldsymbol{r}$ of $M$ picked in ...
1
vote
1answer
103 views

Sampling set: relatively dense and uniformly discrete

The Paley-Wiener space of a domain $\Omega\subset\mathbb{R}^d$ is the set $$PW_\Omega:=\{f\in L^2(\mathbb{R}^d):\text{supp}\widehat{f}\subset\Omega\}.$$ We say that a discrete set $\Lambda\subset\...
5
votes
1answer
409 views

Randomly covering a sphere

Let $S$ be the $n$-dimensional unit sphere in the Euclidean space. Further, let $X_1,\ldots,X_k$ and $Y_1,\ldots,Y_m$ be iid $S$-valued random variables with common (unknown) distribution $\mu$. With $...
3
votes
1answer
293 views

How to measure distribution of high-dimensional data

I have to methods of projecting random samples in $\mathbb{R}^n$ onto a manifold defined by $C(q)=0$, which is a lower-dimensional subset. Now, samples in $\mathbb{R}^n$ are uniformly distributed. ...
1
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0answers
34 views

RMHMC sampling in non-parametric setup

The aim is to sample distributions using Fisher information (as mass matrix in Hamiltonian MCMC sampling). Details can be found in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.190.580&...
0
votes
0answers
74 views

Expected value of parametrized Gibbs distribution w.r.t another probability distribution

Let $\mu$ be a compactly supported probability measure on a finite-dimensional euclidean space (for simplicity) $\mathbb E$, and suppose $\mu$ has density. For a random point $x \sim \mu$, consider ...
1
vote
1answer
157 views

How to sample a path between 2 states in a Markov chain

Given an ergodic Markov chain and 2 states $x$ and $y$, how may one algorithmically sample a path (of finite length) $x =: s_0 \rightarrow s_1 \rightarrow \ldots \rightarrow s_T = y$ between $x$ and $...
1
vote
1answer
97 views

Bounding function by random sampling

Let $f(x): \mathbf{C} \to \mathbf{C}$ be a complex valued polynomial of degree $n$, and suppose that for a random point $z$ sampled uniformly from the unit disk $|z|\leq 1$ we have $$\mathbf{P}_{z}( |...
0
votes
1answer
92 views

Efficiently Sampling of Multivariate Distributions in the Vicinity of a Manifold

I am given a multivariate distribution, that maps each point of $\mathbb{R}^n$ to its probability of being drawn as sample and, the convolution $\mathcal{S_r}\times\mathcal{M}$ of a manifold $\mathcal{...
3
votes
1answer
426 views

Sampling from a particular multivariate probability distribution

Given $3$ real variables $x_1, x_2, x_3 \equiv \bf{x}$, consider their probability density function (PDF) \begin{equation} P({\bf x}) = C \, p(x_1) \cdots p(x_3) \exp[f({\bf x})], \end{equation} where ...
0
votes
0answers
67 views

What distributions fit this strange experimental model?

Suppose you have a set $\mathcal T$ of cardinlity $T$. An experiment is designed in such a way that sampling probabilities are governed by following rules. In each sampling with replacement you can ...
1
vote
0answers
53 views

2-step sampling from a conditional density

The setting is as follows: We are given two random variables $X : \Omega \to \mathbb{R}$ and $\Theta : \Omega \to T$ for some 'parameter space' $T \subset \mathbb{R}$, and 1) we know the density of $...
3
votes
2answers
203 views

Sampling Theorem for non-bandlimited Functions

The classical Shannon sampling theorem states that a bandlimited function with $\mbox{supp } \hat f\subset [-1/2,1/2]$ can be uniquely determined by its samples $(f(i))_{i\in \mathbb{Z}}$ (The symbol $...
1
vote
0answers
61 views

Simulate a graph from a certain distribution

I am wondering if anyone can indicate whether the following is a solved problem. I don't care about time of the algorithm currently. Consider a general probability distribution F on simple graphs ...
6
votes
1answer
134 views

Sample integer points of cross-polytope uniformly

For $r,d\in\mathbb{N}$, let $$C_{r,d}=\{x\in\mathbb{Z}^d: \|x\|_1\le r\}\subset\mathbb{Z}^d$$ be the set of integer points of the $d$-dimensional cross-polytope with radius $r$. What is (...
0
votes
0answers
37 views

Efficient sampling from a polytope with large number of contraints [duplicate]

As far as I know, the most popular way to sample from a polytope (in H-representation) \begin{equation} \mathcal{P} := \{z \in \mathbb{R}^n | (Az)_j \le b_j\; \forall j=1,2,\ldots,m\} \end{equation} ...
8
votes
4answers
191 views

Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints $$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq \vec{...
2
votes
1answer
525 views

Sampling point uniformly at random satisfying equality constraints

First of all, I apologize in advance if the question has already been asked in some way on this site and/or if there is a widely known solution to this problem. The description of my problem is ...
2
votes
0answers
137 views

Probabilities involving Beurling density

I am interested in calculating probabilities involving Beurling densities. Since it's likely probabilists are not familiar with the definitions, I give them below. Definitions. A metric space is ...
2
votes
1answer
82 views

Estimating mean and variance of a distribution based on error-prone estimates of its cdf

Suppose I have some random variable $X$ taking values in $[a, b]$ with unknown distribution (I am happy to assume the distribution is smooth, though it would be nice to not have to). I have a ...
5
votes
1answer
190 views

Exactly sampling from a distribution with access to the probabilities only

There is a discrete distribution where integers, $k$, from $1$ to $n$ occur with probability $p_{k}$, all $p_{k}$ are unknown. Rather than having access to the distribution we have access to $n$ ...
0
votes
1answer
140 views

Monte Carlo estimator with autocorrelated samples

Given an integration problem $I=\int{f(x)dx}$, we can construct an ordinary Monte Carlo estimator as $E[I]=\sum\limits_i\frac{f(x_i)}{p(x_i)}$ where the samples $x_i$ are usually i.i.d. and drawn ...
3
votes
2answers
1k views

Approximating Probability Distribution by Sampling

Consider a discrete probability distribution over $n$ events. Assume that the probabilistic kernel is a black box, that is, we can only sample from it without knowing anything about the type or ...
6
votes
0answers
184 views

Sampling from a Convex Body with Many Extremal Points

Let $p_{1}, \ldots, p_{N}$ be a collection of points in $\mathbb{R}^{n}$. I would like to sample uniformly from the convex hull of these $N$ points in an `efficienct' way. In my setting, I have $n$ ...
1
vote
2answers
175 views

Sampling uniformly from all possible line segments of a given length that fit inside a container

Consider that task of randomly placing a line segment of some length $L$ near a plane s.t. a point $p$ at the center of the line segment is at most a distance $H$ from the plane and intersections ...
1
vote
0answers
72 views

Stochastic process inference from partial observations

Consider a set $U$. My signal is a piece-wise constant "function" $Sig: t \mapsto s$, i.e. the signal at time $t$ equals to some subset $s \subset U$. One can see $Sig(t)$ as a stochastic process. ...
1
vote
1answer
332 views

Discretizing a cosine function?

I'd like to start by noting that for some fixed natural $N$ basis functions for my system will be generated by $f(k,x)$ as defined and explained here or in numerous other sources: $$f(k,x) = \sqrt2 \...
1
vote
3answers
295 views

A sampling and learning question

Suppose there is an oracle that returns a number $b \in \mathbb{Z}_{n}$ whenever I press the button. We have $b = a + e$, where $a \in \mathbb{Z}_n$ is a fixed number and $e$ is sampled according to ...
0
votes
1answer
514 views

transform a polynomial into another one upto a constant

I have a polynomial $p(x)=a_Nx^N+a_{N-1}x^{N-1}+\dots+a_0$. I want to convert this into another polynomial of same order, say $b_Ny^N+b_{N-1}y^{N-1}+\dots+b_0$. Is it possible to find a transformation ...
3
votes
2answers
1k views

Sampling without replacement until hitting a subset

I randomly sample uniformly from $ \{1,..,N \}$ without replacement until drawing a number $ \leq k$. Denote the expected number of draws by $R(N,k)$. I want a good approximation for $\sum_{k=1}^N R(...
3
votes
1answer
418 views

When can you describe a population and its component subpopulations with the same parametric family of distributions?

I believe that it is often the case that you are trying to select the best probability distribution to use to describe some phenomenon you are studying, and you have data not only for a population, ...
0
votes
1answer
3k views

Uniform sampling hemisphere and project in a specific direction [closed]

Hi, I need to generate a 'uniform sample' over an hemisphere and once done project it in a specific vector direction. I have try the following, but it produce some errors... maybe you have an idea ? ...
3
votes
1answer
379 views

Is it possible to sample uniformly on the surface of a high-dimensional polytope?

There are some pretty simple methods to do uniform sampling on the surface of high-dimensional spheres or cubes. Are there any methods that sample uniformly on the surface of a high-dimensional ...
7
votes
2answers
1k views

Sampling uniformly from a sphere

Let $B^{n} _p= ${$ (x_1, \dots, x_n) : |x_1|^p + \dots |x_n|^p = 1 $} be the unit ball in $\mathbb{R}^n$ in the $\ell^p$ norm. If $X_1,\dots,X_n$ are iid $\exp(1)$ -distributed random variables, then ...
0
votes
2answers
444 views

Estimating a sum

Sorry for the vague title but I couldn't find a better one. I want to compute the sum $S = \frac{1}{N}\sum_{i=1}^N c_i x_i$ where $c_i$s are known positive constants. The problem is that computing ...
8
votes
3answers
4k views

Random Sampling a linearly constrained region in n-dimensions…

Hi, So here is my problem: Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$ $x_n \le c_n$ and $\sum_{n=1}^N x_n = 1$ find an ...
3
votes
0answers
285 views

Sampling from a partition of a hypercube by convex polytopes.

I have a binary space partitioning (BSP) tree which recursively partitions a hypercube using linear hyperplanes. That is, a hyperplane splits the hybercube in half, creating two convex polytopes. Each ...
16
votes
2answers
1k views

Sampling from the Birkhoff polytope

The set of $n\times n$ real, nonnegative matrices whose rows and columns sum to one forms the well-known Birkhoff polytope Recently someone asked me if I knew How to sample (in polynomial time) ...