All Questions
105 questions
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41
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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
vote
1
answer
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 ...
- 277
2
votes
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 ...
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<...
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 ...
5
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1
answer
2k
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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 ...
- 173
0
votes
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
14
votes
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 ...
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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
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 $L^2(\pi)$ ...
- 111
1
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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
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 ...
- 13
39
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3
answers
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 ...
- 1,127
37
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3
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3k
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On Mathematical Analysis of MathSciNet & MathOverflow
This question has two original motivations: mathematical and social.
The mathematical motivation is mainly based on what I have seen about Zipf's law here and there. The Zipf's law simply states ...
0
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0
answers
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,
$$
X_{t_1, \ldots, t_d}=f\left(t_1, \ldots, t_d\right) d ...
- 73
32
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3
answers
12k
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What is the Katz-Sarnak philosophy?
It has been recently mentioned by a speaker (his talk is completely not relevant to random matrix theory/RMT though) that modern statistics, especially random matrices theory, will help solving some ...
- 8,071
3
votes
1
answer
257
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Triangle equality for cosine similarity in high dimensions
I'm trying to understand whether I can use the following equality in my application -- for $u,v,w \in \mathbb{R}^d$:
$$\cos(u,w)\approx \cos(u,v)\cos(v,w)$$
Where $\cos(x,y)$ gives cosine of the angle ...
- 2,252
3
votes
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 ...
- 1,095
2
votes
1
answer
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$...
- 341
14
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1
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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 ...
- 8,071
3
votes
1
answer
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 ...
- 2,247
32
votes
4
answers
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. ...
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
17
votes
4
answers
2k
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Good introduction to statistics from a algebraic point of view?
There are already lots of questions on this subject like
Is there an introduction to probability theory from a structuralist/categorical perspective?
Is there a combinatorial/topological treatment ...
- 283
1
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0
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350
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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-...
- 380
3
votes
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 ...
1
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3
answers
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 &...
- 791
1
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1
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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 ...
- 708
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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}...
user274127
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 $(\...
- 935
2
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1
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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 $(...
4
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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:
...
- 203
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
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 ...
- 2,680
4
votes
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 ...
6
votes
2
answers
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 ...
- 61
1
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1
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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
11
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2
answers
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 ...
- 4,952
9
votes
2
answers
879
views
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 ...
- 2,680
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 $U_{n:1}\le\cdots\le U_{n:n}$ ...
- 128k
0
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1
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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 $v$ ...
- 2,252
0
votes
1
answer
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
$...
- 1,467
2
votes
1
answer
235
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Kolmogoroff condition for truncated random variables
Question summary. Does the Kolmogoroff condition $\sum_{n=1}^\infty\frac{\mathbb V Y_n}{n^2} < \infty$ hold for truncated random variables $Y_n := X_n \cdot 1_{\{X_n \le n\}}$ (see below for a more ...
- 1,326
2
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1
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247
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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 ...
- 10.5k
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
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76
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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 ...
- 103
11
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1
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1k
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What are some of the surprising results of finite sample statistical estimation?
I'm trying to familiarize myself with the latest results in finite sample statistics. It seems to me that these results can be classified into two categories:
Unsurprising results confirm that the ...
- 395
7
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4
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4k
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Estimating the probability that one Poisson RV is larger than another
Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function
$$\mathbb P(X - Y ...
- 8,512
6
votes
1
answer
2k
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Kullback Leibler "variance": does that divergence have a name?
If you consider two probability distributions $p$ and $q$, one way to measure the distance between the two is the Kullback-Leibler divergence:
$$KL(p,q)=\int p \log (p/q) = E_p(\log p/q)$$
and this ...
1
vote
0
answers
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