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Can conditional distributions with respect to a sufficient sub-$\sigma$-algebra be represented by a single Markov kernel?

Let $(\Omega, \mathcal{F})$ be a measurable space, and let $\mathcal{P}$ be a collection of probability measures on this space. A sub-$\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ is said to be ...
MrTheOwl's user avatar
  • 111
9 votes
2 answers
429 views

Hermite–Fourier expansion for the median

Let $n$ be an odd positive integer. Let $M : \mathbb{R}^n \to \mathbb{R}$ be the median function: $M(x_1,\dots,x_n)$ is the median of $x_1,\dots,x_n$. What can be said about the Hermite–Fourier ...
Gil Kalai's user avatar
  • 24.7k
3 votes
0 answers
92 views

Tighter Freedman's inequality for a special martingale difference sequence

Let $X_{1}, \ldots, X_{T} \in \{0, 1\}$ be a sequence of Boolean random variables with $$ \mathbb{E}[X_{t} | X_{1}, \dots, X_{t - 1}] = p_{t}. $$ Consider the sequence $Y_{t} := X_{t} - p_{t}$ (which ...
Fellow4's user avatar
  • 41
2 votes
1 answer
208 views

Proving an exponential sum inequality for symmetric Hamming distance sequences in binary vectors

Background: Let $X = \{0,1\}^k$ represent the set of all binary vectors of length $k$. For two binary vectors $x, y \in X$, the Hamming distance $d_H(x, y)$ is defined as the number of positions where ...
tom jerry's user avatar
  • 349
16 votes
1 answer
397 views

Examples of problems in statistics accessible only using information geometry

I am just curious if there are some examples of problems in statistics that are indeed accessible using information geometry while proofs completely avoiding geometry are unknown. In other words, ...
温泽海's user avatar
  • 269
0 votes
0 answers
44 views

Large Deviation Principle for an adaptive sampling rule for Multi Armed Bandits

Consider the following adaptive strategy for sampling from a Multi Armed Bandit with $K$ arms: Split the $T$ rounds into $N (\in \mathbb{N})$ disjoint intervals. Each interval is indexed by $i=1,2,\...
29910622's user avatar
1 vote
0 answers
48 views

Quantile maximization of the difference of random constrained quadratic optimization problems

I am interested in understanding the family of parametrized random variables defined by the pushforward map $$ \lambda_x : \varepsilon \mapsto \underset{z_1 \in \mathbb{R}^n :\, h^T z_1 = 0, \; z_1 \...
NeyPea's user avatar
  • 11
0 votes
1 answer
100 views

Expressing a multivariate normal distribution as a mixture of uniform distributions?

Context: Given a scalar normal distribution $X\sim \mathrm{N}(\mu, \sigma^2)$, it is possible to express $X$ as a mixture of uniform distributions over intervals (compound probability distributions), ...
PiePiePie's user avatar
2 votes
0 answers
77 views

Inequalities concerning cummulative distributions of binomials

For random variable $Z$, let $F_Z$ denote its cdf, i.e., $F_Z(t)=\mathbb{P}(Z\leq t)$. Let $X$ be a binomial distribution with parameters $(n,p)$ and $Y$ a binomial distribution with parameters $(m,p)$...
Marcos Kiwi's user avatar
1 vote
0 answers
61 views

Bound on $\int_0^1\sqrt{\log N_{[]}(\varepsilon,\mathcal{F},d)} \, d\varepsilon$ over the class of half-spaces $\mathcal{F}$ on $\mathbb{R}^d$?

For a class of functions $\mathcal{F}$ and a pair $f,g\in\mathcal{F}$ with $f\leq g$, the interval $[f,g]=\{h:f(x)\leq h(x)\leq g(x),\forall x\in\mathbb{R}^d\}$ is called a bracket for $\mathcal{F}$. ...
Daan's user avatar
  • 141
1 vote
1 answer
56 views

How to study the convergence of the sample mode for arbitrary probability spaces

(This is not the problem I actually care about, but an analogy with similar issues to the problem I'm actually considering.) Consider a probability space with i.i.d. random variables $X_i$ producing ...
cgmil's user avatar
  • 277
6 votes
2 answers
607 views

Whence “uniform distribution”?

The “Earliest Uses” site suggests that the expression “uniform distribution” first appeared in Uspensky (1937), and “uniformly distributed” in Sakamoto (1943). Is that true?
Francois Ziegler's user avatar
1 vote
0 answers
44 views

Constrained random sampling from partitioned sets with quotas

Let $D$ be a finite set, $\mathcal{P} = \{D_{i,j}\}_{(i,j) \in I \times J}$ a partition of $D$, $N: J \to \mathbb{N}$ a quota function, and $k \in \mathbb{N}^+$. A subset $F \subseteq D$ is considered ...
DataGuy553's user avatar
1 vote
0 answers
80 views

Moments from characteristic function for matrices

When $x$ is a random variable with the smooth characteristic function $\phi_x(t) = \mathbb{E}e^{itx}$, we can easily compute the moments as $\mathbb{E}[x^k] = i^{-n}\phi_x^{(n)}(0)$. There is no magic ...
user3826143's user avatar
2 votes
0 answers
56 views

Sum of independent Wisharts

Suppose random vectors $y_1,y_2,\ldots,y_m$ are independent and the distribution of each $y_i$ is a $d$-dimensional complex Gaussian with mean $0$ and covariance $\Gamma_i$, that is $y_i \sim \mathcal{...
user3826143's user avatar
3 votes
0 answers
131 views

Matrix-Gaussian distributions

The point of this question is to ask for references on matrix-variate Gaussian distributions. But I will explain what I mean by a matrix-variate Gaussian with an example (the notion I have in mind is ...
user3826143's user avatar
3 votes
0 answers
353 views

Moments of normalized multivariate Gaussians (and Wick's/Isserlis theorems)

Suppose $x = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}$ is distributed according to the real two-dimensional Gaussian with mean-$0$ and covariance matrix $\Sigma$. I am interested in a closed form for ...
user3826143's user avatar
-2 votes
1 answer
43 views

$E(\mathbf{y}|\mathbf{x}+\mathbf{z})=g(\mathbf{x})$ almost surely, if $\mathbf{z}\perp \!\!\! \perp \{\mathbf{y},\mathbf{x}\}$ jointly?

Let $\mathbf{y},\mathbf{x}$ and $\mathbf{z}$ be real-valued random vectors with possibly different dimensions. If $\mathbf{z} \perp\!\!\!\perp \{\mathbf{y},\mathbf{x}\}$ (i.e., $\mathbf{z}$ is ...
John's user avatar
  • 193
1 vote
1 answer
81 views

Inference for the normal distribution with known variance from multiple clusters

Here's the question: We have: $q \sim N\left(q_p, \frac{1}{\tau}\right), q_i \sim N\left(q, \frac{1}{\zeta}\right), t_n \sim N\left(0, \frac{1}{\eta}\right)$. Let $$ r_n=\sum_{i=1}^{\theta k_{n}} \...
LunaSakura's user avatar
0 votes
0 answers
85 views

When is a family of distributions "closed" with respect to minimal sufficient statistics?

As in the title, I am interested in understanding how to express the idea that a parametric family of distribution is "closed" with respect to minimal sufficient statistics. Before giving ...
Francesco Bilotta's user avatar
3 votes
0 answers
80 views

Seeking strong bounds on KL-divergence and martingales for a hypothesis-testing inequality

Let's say we have a finite set $\mathcal{O}$ of observations, and let $\mathcal{C}(\Delta\mathcal{O})$ denote the space of closed convex sets of probability distributions. We have two hypotheses which ...
Alex Appel's user avatar
2 votes
0 answers
50 views

Weighted squared norm of multivariate truncated normal vector

Let $X \sim \mathcal{N}(0, \Sigma)$ be a multivariate normal vector with zero mean and inverse covariance matrix $$ \Sigma^{-1} = \begin{pmatrix} n & 1 & 1 & \cdots & 1 &...
Jesse van Rhijn's user avatar
1 vote
1 answer
84 views

Limiting value of Stieltjes transform of sum of independent Wishart matrices

Let $n_1$, $n_2$, and $d$ positive integers tending to infinity such that $d/n_k \to \phi_k \in (0,\infty)$ and $n_1/(n_1+n_2) \to p \in (0,1)$. Let $X_k$ be an $n_k \times d$ random matrix with iid ...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
177 views

Optimization over Poisson-binomial distributions

I am studying the problem of how an expected utility maximizer should optimally form a portfolio of uncorrelated Bernoullis. Fix an increasing sequence of $n$ numbers in $(0,1)$, $0<p_1<\dots<...
Francesco Bilotta's user avatar
2 votes
1 answer
170 views

Law of large numbers for a continuum of Bernoullis

Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
Francesco Bilotta's user avatar
0 votes
1 answer
159 views

Theories for "fuzzy" distributions

When calculating the probability density function for the quotients of adjacent values in an empirical time series, the image of the PDF looked like this: It seems to resemble a lognormal ...
Manfred Weis's user avatar
  • 13.2k
0 votes
1 answer
108 views

RMT for modified Wishard matrix $Y'Y$ (where $i$th row of $Y$ is zero if $|x_i^\top u| \le \theta$; else it equals $x_i$)

Let $n$ and $d$ be positive integers tending to infinity such that $d/n \to \phi \in (0,\infty)$. Let $X$ be an $n \times d$ random matrix with iid rows $x_1,\ldots,x_n$ from $N(0, \Sigma)$, where $\...
dohmatob's user avatar
  • 6,853
0 votes
0 answers
89 views

Stein's Lemma for conditional expectation?

Let $X=(X_1,\ldots,X_d)$ be a standard normal random vector in $\mathbb R^d$, let $m:\mathbb R^d \to \mathbb R$ be a function, and let $E=E_m$ denote the expectation operator conditioned on $m(X) > ...
dohmatob's user avatar
  • 6,853
1 vote
1 answer
97 views

Bayes classifiers with cost of misclassification

A minimum ECM classifier disciminate the features $\underline{x}$ to belong to class $t$ ($\delta(\underline{x}) = t$) if $\forall j \ne t$: $$\sum_{k\ne t} c(t|k) f_k(\underline{x})p_k \le \sum_{k\ne ...
BiasedBayes's user avatar
0 votes
0 answers
37 views

Compatibility of 2-copulas

An $n$-copula is the joint distribution function of a distribution on $[0,1]^n$ with uniform marginals. A family of 2-copulas $(C_{i,j})_{i<j\leq n}$ is compatible if there exists an $n$-copula $\...
Stefan Perko's user avatar
4 votes
1 answer
234 views

Maximum entropy probability distribution with fixed interval and variance?

What is the maximum entropy probability distribution if the support is a fixed interval (e.g. $[-1,1]$) with an already known variance? If we know the support is a fixed interval, then the maximum ...
Sarah Rune's user avatar
1 vote
0 answers
53 views

The limit ratio of two Markov Chain Probability

Suppose there are two given SDE in $\mathbb{R}^d$: $$ \begin{align} \left\{ \begin{aligned} dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...
Francis Fan's user avatar
2 votes
0 answers
84 views

Concentration result for self-normalized empirical process

In Theorem 1.1 of this paper by Bercu, Gassiat and Rio, a concentration result is derived for the 'self-normalized' empirical process. Specifically, suppose that $(X,X_n)_{n \ge 1}$ is a sequence of i....
WeakLearner's user avatar
0 votes
0 answers
42 views

Limiting value of trace of resolvent matrix involving two independent Wishart random matrices

Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that $$ d/n_k \to \phi_k \in (0,\infty). $$ Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...
dohmatob's user avatar
  • 6,853
2 votes
1 answer
243 views

Concentration inequalities for heavy-tailed distributions

Suppose $X_1,...,X_N$ are $N$ i.i.d random variables with heavy tailed distributions. For example, $E[X_i^p]\leq 1$ for some $p\geq 1$. Are there some concentration inequalities to bound the tail $$P(\...
jkfds's user avatar
  • 31
1 vote
1 answer
115 views

A property of the distribution related to stochastic ordering

Let $X$ be a random variable with a symmetric support $S\subset[-M,M]$ for some $M>0$. (i.e., if x is a point of increase of CDF $F_X(\cdot)$, so is $-x$.) Has the infimum value of $c$ such that \...
Ben's user avatar
  • 19
0 votes
0 answers
73 views

Asymptotic stochastic ordering for weighted sum of i.i.d. random variables

Are you aware of any literature focusing on the conditions such that for two i.i.d. sequences of discrete r.v.'s $\{X_n\}$ and $\{Y_n\}$, \begin{equation} a_1X_1+a_2X_2+\ldots+a_nX_n\geq_1 a_1Y_1+...
Ben's user avatar
  • 19
1 vote
0 answers
68 views

Gibbs Priors form a Martingale

I am working on adapting variational inference to the recently developed Martingale posterior distributions. The first case, which reduces the VI framework to Gibbs priors, is proving hard to show as ...
BayesRayes's user avatar
1 vote
2 answers
306 views

Joint moments like $\tau(XYXYXY)$ in terms of individual moments of free variables $X,Y$

Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5 $$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$ ...
Yaroslav Bulatov's user avatar
0 votes
0 answers
91 views

Some new questions on Rademacher complexity

For $A\subset R^n$,$A=(a_1,a_2,\dots, a_n)$, $\sigma_i$ are Rademacher random variable. Is $|\mathbb{E}_\sigma \inf_{a\in A}\sum_{i=1}^n\sigma_ia_i| \le |\mathbb{E}_\sigma \sup_{a\in A}\sum_{i=1}^n\...
Hao Yu's user avatar
  • 185
2 votes
2 answers
195 views

Random partition of an interval – Dirichlet distributed?

Let $X_1, \ldots, X_N \sim \operatorname{Unif}[0,1]$ and consider the intervals between successive order statistics: $[0, X_{(1)}], [X_{(1)}, X_{(2)}], \ldots, [X_{(N)}, 1]$. What is the distribution ...
gusl's user avatar
  • 57
1 vote
1 answer
241 views

Expectation of top-K selection of squared Gaussian random variables

Let us have $$ Z = [z_1, z_2, \dots, z_n], $$ where $z_i \sim N(0, \sigma^2)$ and are iid. Additionally, consider $$ X_k := \{ x \in \{0, 1\}^n : e^T x = k \} $$ If $Y = \max_{X \in X_k} |Z^T X|^2,$ ...
Alireza Khayatian's user avatar
1 vote
0 answers
62 views

A small lemma on cache resets (Bloom filters in particular)

Assume a fixed set of message $D$ and an associated distribution for selecting each message $d_i$ such that the total probability $\sum_{i \in D} d_i = 1$. We create a cache with $M$ bits and $k$ ...
Birdy Nam Nam's user avatar
1 vote
0 answers
72 views

Dimension-free sample complexity for the inverse of Gaussian sample covariance?

Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need the inverse of the sample covariance $\Sigma_m^{-1}$ to be $\varepsilon$-close to true inverse covariance $\Sigma^{-1}$ (in ...
axk's user avatar
  • 517
0 votes
0 answers
82 views

High probability bound on number of sparse solutions to Gaussian linear system

Suppose we have a random matrix $A \in \mathbb{R}^{m \times n}$ with all entries i.i.d. from the standard Normal distribution $\mathcal{N}(0, 1)$. Suppose $k$ divides $n$, and let $S \subseteq \mathbb{...
anon's user avatar
  • 43
7 votes
2 answers
235 views

Evolution of the empirical mean of a list as we remove elements proportional to their value

Consider a list of $N$ integers $k_1,k_2,\dots k_N$, drawn independently from some distribution $P(k)$ with $k_i \geq 1$. We denote its mean with $\langle k\rangle=\sum_{k=1}kP(k)$. The first two ...
papad's user avatar
  • 274
3 votes
2 answers
358 views

Minimax optimal multiple hypothesis test

Let us consider the following two-player game between Chooser and Guesser. There is a finite set $\Omega$ and $k$ probability distributions on $\Omega$, denoted by $ \mathcal{P} =\{P_1,\ldots,P_k\} $. ...
Aryeh Kontorovich's user avatar
2 votes
1 answer
245 views

Sum of arrival times of Chinese Restaurant Process (CRP)

Suppose that a random sample $X_1, X_2, \ldots$ is drawn from a continuous spectrum of colors, or species, following a Chinese Restaurant Process distribution with parameter $|\alpha|$ (or ...
Grandes Jorasses's user avatar
3 votes
1 answer
108 views

When does the optimal model exist in learning theory?

In the context of learning theory, we usually have: data $(x,y)\sim P(x,y)$, with $x\in\mathcal{X}\subseteq\mathbb{R}^d$ and $y\in\mathcal{Y}\subseteq\mathbb{R}^k$, a hypothesis class $\mathcal{F}\...
rick's user avatar
  • 199
2 votes
1 answer
138 views

expectation of the product of Gaussian kernels and their input

I was wondering if anybody knows how to solve: $$\mathbb{E}{\mathbf{z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})}\left[ (\mathbf{x}{i} - \mathbf{z})(\mathbf{x}{j} - \mathbf{z})^\top \exp\left( - (\...
patchouli's user avatar
  • 275

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