All Questions
105 questions
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40
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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 ...
- 111
1
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1
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56
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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 ...
- 128k
2
votes
1
answer
177
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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<...
- 266
0
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0
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85
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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 ...
2
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1
answer
170
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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 ...
- 63.9k
0
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0
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89
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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) > ...
- 6,853
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37
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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 $\...
- 677
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1
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Mathematics research relating to machine learning
What branch/branches of math are most relevant in enhancing machine learning (mostly in terms of practical use as opposed to theoretical/possible use)? Specifically, I want to know about math research ...
- 30.3k
4
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5
answers
2k
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Martingales and Betting Strategies
Does anyone know of a good introduction to the theory of martingales and betting strategies from the point of view of statistics and/or probability theory? I'm looking for something basic, with lots ...
0
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55
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Modeling player interactions in multi-dimensional rating systems
In traditional rating systems (such as Elo), a player's strength is represented by a single scalar value, which is assumed to be consistent across different opponents. However, in some games, the ...
- 1
2
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1
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110
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Lower bound on likelihood of binary outcomes
I am wondering about the following: does there exist a stochastic process $(X_n)_{n \ge 1}$ with values in $\{0,1\}$ on a probability space $(\Omega, \mathcal F, \mathbb P)$ such that for all $n \ge 1$...
CommunityBot
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6
votes
2
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2k
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Interesting thesis topic on statistical inference that is sufficiently mathematical
Hello , I am a student who's gonna start honours in mathematics . Currently , I am at the stage of finding a suitable honours thesis topic . I've chosen my supervisor , who's research interest is on ...
- 2,821
1
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0
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144
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Estimator for the conditional expectation operator with convergence rate in operator norm
Let $X$ and $Z$ be two random variables defined on the same probability space, taking values in euclidian spaces $E_X$ and $E_Z$, with distributions $\pi$ and $\nu$, respectively.
Let $L^2(\pi)$ ...
0
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0
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52
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Classifier-specific lower bounds on the misclassification rate in binary classification
Consider a binary classification problem for $(X,Y)$, and let $\hat{f}$ be a proposed classifier. We wish to bound the misclassification rate $P(\hat{f}(X)\ne Y)$. There are many known lower bounds on ...
CommunityBot
- 1
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0
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49
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Gaussian white noise model in application
I am interested in applications (to data) of non-parametric statistics, and my question concerned the Gaussian white noise model defined by,
$$
X_{t_1, \ldots, t_d}=f\left(t_1, \ldots, t_d\right) d ...
- 63.9k
1
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0
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68
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Limiting value of expectation of $\operatorname{tr}(BR(z))$, where $R(z) := (X^\top X - z I_d)^{-1}$ and $X \sim N_{n,d}(0,A)$
Let $A=A(d)$, and $B=B(d)$ be (sequences of) deterministic positive-definite $d \times d$ matrices and let $X$ be an $n \times d$ random matrix with iid rows from $N(0,A)$. Let $R$ be the resolvent of ...
- 6,853
32
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4
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7k
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Bayesian statistics for pure mathematicians
Could someone please recommend reading on Bayesian statistics presented from a pure mathematical point of view? That is, works that start assuming a good knowledge of measure theoretic probability. ...
- 113
14
votes
1
answer
3k
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How is the "conformal prediction" conformal?
The question is clarified by Prof.V.Vovk. See his answer below for discussion.
Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a ...
- 2,600
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1k
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Multidimensional Berry–Esseen for probability density functions
This is a follow up to this recent question: Berry Esseen type result for probability density functions
There exists a multidimensional version of the usual Berry–Esseen theorem (for cumulative ...
- 1,446
3
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2
answers
348
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General version of $d$-separation
I find the $d$-separation criterion (see, e.g., Theorem 2 here; note however the preceding definition, which basically means we are treating discrete random variables) a really useful sufficient ...
- 6,406
3
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1
answer
257
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Triangle equality for cosine similarity in high dimensions
I'm trying to understand whether I can use the following equality in my application -- for $u,v,w \in \mathbb{R}^d$:
$$\cos(u,w)\approx \cos(u,v)\cos(v,w)$$
Where $\cos(x,y)$ gives cosine of the angle ...
- 128k
14
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8
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3k
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Relevant mathematics to the recent coronavirus outbreak
I would like to ask about (old* and new) reliable mathematical literature relevant to various mathematical aspects of the recent coronavirus outbreak: In particular, standard statistical/mathematical ...
- 82.7k
39
votes
3
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4k
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Manifold of probability measures: connections between two types of metrics
The space of probability measures could be viewed as an infinite-dimensional manifold, equipped with two possible types of metrics — (1) Wasserstein and (2) Fisher-Rao. Metric (1) is connected with ...
- 6,001
4
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0
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164
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Convergence rates for kernel empirical risk minimization, i.e empirical risk minimization (ERM) with kernel density estimation (KDE)
Let $\Theta$ be an open subset of some $\mathbb R^m$ and let $P$ be a probability distribution on $\mathbb R^d$ with density $f$ in a Sobolev space $W_p^s(\mathbb R^d)$, i.e all derivatives of $f$ ...
- 6,853
3
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1
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355
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Is there a complete countable axiomatization of conditional independence? (Graphoids)
Note: A pointer to a reference, or a yes/no answer with a 1-2 sentence incomplete/non-rigorous justification would suffice for answers. I am just curious about whether the result is true; it is fairly ...
- 136
1
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1
answer
106
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What is the maximum possible coefficient of variation for data taking values within a specified range?
I have a question that seems very basic, and yet I have not managed to find an answer after probably several hours of Google-searching.
Fix $0<a<b<\infty$, and let $\mathcal{P}_{[a,b]}$ be ...
- 128k
1
vote
0
answers
349
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Tail bounds for random Gaussian chaos?
Let $g = (g_1, \dots, g_d)$ be a sequence of independent standard Normal random variables, and suppose $\Sigma$ is a $d \times d$ (deterministic), real, symmetric, positive definite matrix. The Hanson-...
- 420
3
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1
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416
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Well-definedness of maximum likelihood estimation
Consider a family $\{\mu_\theta:\theta\in\Theta\}$ of probability measures on a measurable space $X$. Given $x\in X$, the maximum likelihood estimate is the value of $\theta$ which maximizes the ...
- 128k
5
votes
1
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363
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Inverse marginal property of a collection of $\sigma$-algebras
In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space"
I introduced the Inverse marginal property (IMP) for a collection of $\sigma$-algebras.
Let $(\...
CommunityBot
- 1
2
votes
1
answer
268
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Monotonicity of the Hellinger integral/distance
Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are
$H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and $\...
- 61.9k
2
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1
answer
199
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Do enough permutations of an initial set probably cover most permutations?
Fix $\alpha, \epsilon \in(0,1)$. Take $(S_n)_n$ to be any sequence of sets with each $S_n$ containing $ \lceil (n!)^\alpha\rceil$ permutations of $n$ elements. Also build another sequence of sets $(...
0
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1
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209
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Factorisation of Gaussian random matrix into random Hermitian and correction factor
By the Bartlett decomposition, one has that for $k \leq n$ and $\mathbf{\Gamma}_{n\times k} \in \mathbb{R}^{n\times k}$ a standard Gaussian matrix with independent entries
$$\mathbf{\Gamma}_{n\times k}...
CommunityBot
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1
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3
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269
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Practical pseudorandom generators
It is known that existence of pseudorandom generators (PRGs) is equivalent to the existence of one-way functions. In turn, the latter is an open problem.
I am curious if someone developed kind of &...
- 10.4k
0
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0
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171
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A basic property of maximal correlation
Let $𝑋$ and $𝑌$ be random variables. Then the maximal correlation $\rho_{m}(X;Y)$ is defined as:
$$\rho_{m}(X;Y):=\max_{f,g}\mathbb{E}[f(X)g(Y)],$$
where the maximization is taken over real-valued ...
- 11
1
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0
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438
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Chain rule for maximal correlation
Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...
- 11
3
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1
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511
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Complete statistical manifolds
Here, by a statistical manifold I mean a $d$-dimensional Riemannian manifold whose points are probability measures on $\mathbb{R}^n$. What are some well-studied/interesting examples of statistical ...
- 6,001
4
votes
1
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362
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Information monotonicity of divergence => function of $f$-divergence
It is well-known that $f$-divergences defined on $\mathcal P(\mathcal X)$ where $\mathcal X$ is a measure space with $\sigma$-algebra $\mathcal B$ satisfy the property of information monotonicity:
...
- 695
2
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0
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49
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What are some beginner's references on algebraically structured (statistical) models, and their connection with group actions and Fourier transform?
I asked this question on Cross Validated a few days ago, but didn't really get a favorable response, so asking here to see if I get any.
I'm looking at the description of a short-term position in ...
- 223
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1
answer
77
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Fourth moment of a random-variable with block-tridiagonal structure
Let x be a random variable in $\mathbb{R}^d$, $J$ a block tridiagonal $d\times d$ matrix, and probability of $x$ is defined as follows
$$p(x)\propto \exp(-x'Jx)$$
For a fixed $d\times d$ matrix $v$ ...
- 128k
1
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1
answer
149
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Asymptotics of $\chi_m$-distribution where the degree of freedom $m \to \infty?$
I'm interested to see a result where for large degree of freedom $m,$ the chi distribution $\chi_m$ is increasingly well approximated by a family of normal distributions with parameters depending on $...
- 1,467
0
votes
1
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428
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First and last order statistics and their ratio for $\chi^2_{m}$ random samples
Let $X_1, \dots, X_n \sim_{iid} \chi^2_{m}$ be a random sample from a chi-squared distribution with $m$ degrees of freedom (d.f.). I was wondering if there's any known result for the order statistics
$...
- 5,429
11
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2
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78k
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Coin pusher game
While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins).
Essentially, one has a distribution of ...
- 63.9k
0
votes
1
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75
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Density function approximation with respect to $L^1$ distance
Given iid samples $X_1,...,X_N$ drawn from some unknown distribution with not necessarily continuous density function $f(x)$ are there any theorems/papers where based on the data $X_1,...,X_N$ an ...
- 4,729
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0
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141
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What is the distribution of the norm of the multivariate $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d?$
Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. Then what's the distribution (PDF, CDF etc.) of $X?$ When $\mu = 0, \Sigma = I_d,$ we know that $||X||...
- 1,467
1
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0
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64
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Dependence rank: what is the size of the largest subcollection of random variables which is statistically independent?
Let $X_1,\ldots,X_p$ be random variables on the same space. Define their dependence rank, denoted $rank(X_1,\ldots,X_p)$ as the largest nonnegative integer $k$ such that there is a subcollection of $k$...
- 6,853
9
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2
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879
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Is there a combinatorial/topological treatment of statistical independence?
Is there any reference which studies sets of random variables as independence systems, a type of combinatorial object (see below)?
Motivation:
In particular, since independence systems are abstract ...
- 24.2k
2
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1
answer
1k
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Order statistics on the spacings between order statistics for the uniform distribution
For any natural $n$, let $U_1,\dots,U_n$ be independent identically distributed
random variables each uniformly distributed on the interval $[0,1]$. As usual, let $U_{n:1}\le\cdots\le U_{n:n}$ ...
CommunityBot
- 1
1
vote
1
answer
155
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Reference request concerning order statistics from the uniform distribution
Let $U_1,\dots,U_n$ be iid random variables uniformly distributed on the interval $[0,1]$, with the corresponding order statistics $U_{(1)}\le\dots\le U_{(n)}$. Let $G_i:=U_{(i+1)}-U_{(i)}$ for $i=0,\...
- 10.4k
2
votes
1
answer
242
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Overview of interpretations of classical probability
The Stanford Encyclopedia of Philosophy has a nice overview of numerous different interpretations of probability (classical as opposed to quantum) with an extensive bibliography.
What books would ...
- 39.9k
3
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0
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95
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Empirically random, quickly multiplicable matrices
I have encountered a need for fast computation of a transformation $Ax$ where $A\in \mathbb{C}^{K\times N},\ K\sim 10^7,\ N\sim 10^3$ is designed, and $x\in \mathbb{C}^N$ has iid $\mathcal{CN}(0,1)$ ...