# Questions tagged [sampling]

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### Sampling i.i.d. variables with restrictions

General Problem: Suppose $X_1,\ldots,X_n \sim \mathbb{P}_X^{\otimes n}$ is a finite sequence of i.i.d. (real- or integer-valued) random variables. Suppose $A\subseteq \mathbb{R}^n$ is a set of "...
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### Probability that a random multigraph is simple

Question. Consider a given sequence of $n$ integers $d_1$, $d_2$, $\cdots$, $d_n$ with $\sum_i d_i$ even and $d_i\le n$ for all $i$. One may sample a random multi-graph having this degree sequence ...
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### Algorithm for sampling stratification-constrained permutations

Let $\mathcal{X}$ be some base space, and let $\mathbf{S} =S_1, \ldots, S_N$ be a cover (not a partition, i.e. they can overlap) of $\mathcal{X}$. I will refer to the $S_i$ as strata, and to the cover ...
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### Interior point of a convex polytope

Suppose the convex polytope is the set of feasible solutions $\mathbf{x}\in\mathbb{R}^n$ for the linear system $\mathbf{A}\mathbf{x}=\mathbf{b}\,,\; \mathbf{A}\in\mathbb{R}^{m\times n}$ subject to ...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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} ...
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### 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 ...
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### 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 ...
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(...