Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [st.statistics]

Applied and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments.

100
votes
14answers
22k views

Statistics for mathematicians

I'm looking for an overview of statistics suitable for the mathematically mature reader: someone familiar with measure theoretic probability at say Billingsley level, but almost completely ignorant of ...
78
votes
8answers
13k views

Is there a natural random process that is rigorously known to produce Zipf's law?

Zipf's law is the empirical observation that in many real-life populations of n objects, the $k^{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $n$ (...
35
votes
12answers
16k views

Why is it so cool to square numbers (in terms of finding the standard deviation)?

When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do $$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$. Why ...
33
votes
3answers
2k views

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 ...
33
votes
5answers
6k views

Inference using Topological Data Analysis: Is it worth it for a regular statistician to learn TDA?

After having read Gunnar Carlsson's Topology and Data I feel enthusiastic to use some topological data analysis (TDA) methods in my current research, mostly in social sciences. We often handle huge ...
32
votes
4answers
3k views

“Entropy” proof of Brunn-Minkowski Inequality?

I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality. The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then $$ m(...
29
votes
4answers
6k views

Applications of algebraic geometry to machine learning

I am interested in applications of algebraic geometry to machine learning. I have found some papers and books, mainly by Bernd Sturmfels on algebraic statistics and machine learning. However, all this ...
29
votes
5answers
2k views

You pass X people and Y people pass you: how relatively fast are you?

This question occurs to me every time I go jogging. I suspect every runner probabilist in the world must have thought of it (though I'm no probabilist), but I could not specifically find it online. I ...
28
votes
2answers
1k views

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 ...
27
votes
3answers
10k views

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 ...
27
votes
0answers
9k views

Research situation in the field of Information Geometry

I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field. I have read (4) and parts of (3). ...
25
votes
3answers
2k views

Why do statistical randomness tests seem so ad hoc?

Wikipedia describes Kendall and Smith's 1938 statistical randomness tests like this: The frequency test, was very basic: checking to make sure that there were roughly the same number of 0s, 1s, ...
22
votes
2answers
6k views

L1 distance between gaussian measures

L1 distance between gaussian measures: Definition Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...
22
votes
2answers
1k views

Drawing natural numbers without replacement.

Suppose we start with an initial probability distribution on $\mathbb{N}$ that gives positive probability to each $n$. Let's call this random variable $X_1$ so we have $P(X_1=n)=p_{1,n}>0$ for all $...
21
votes
3answers
2k views

Persistent homology of Gaussian Fields in Euclidean space

If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be ...
21
votes
1answer
2k views

What kind of random matrices have rapidly decaying singular values?

I've been told that in machine learning it's common to compute the singular value decomposition of matrices in order to throw out all information in the matrix except that corresponding to, say, the $...
20
votes
4answers
4k views

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. ...
20
votes
0answers
797 views

Random Distance Matrices

My question is motivated by the following recent paper: http://arxiv.org/abs/1110.6333 Assume you have a metric space $(X,d)$ equipped with a Borel probability measure $\mu$. We can further assume ...
19
votes
3answers
3k views

What is quantum Brownian motion?

It seems that the current state of quantum Brownian motion is ill-defined. The best survey I can find is this one by László Erdös, but the closest the quantum Brownian motion comes to appearing is in ...
17
votes
3answers
821 views

Deceptively simple inequality involving expectations of products of functions of just one variable

For a proof to go through in a paper I am writing, I need to prove the following deceptively simple inequality: $$(*)\qquad E(X^a) E(X^{a+1}\log X) > E(X^{a+1})E(X^a\log X) $$ where $X>e$ has ...
17
votes
2answers
2k views

Calculating the “Most Helpful” review

How would you calculate the order of a list of reviews sorted by "Most Helpful" to "Least Helpful"? Here's an example inspired by product reviews on Amazon: Say a product has 8 total reviews and ...
17
votes
1answer
666 views

How fast can extreme eigenvalues of the average of random matrices converge to their expectation?

Suppose that $X_1,X_2,\ldots,X_m$ are independent $d\times d$ random matrices and let $\overline{X} := \frac{1}{m}\sum_{i=1}^m X_i$. One of the questions studied under the theory of random matrices is ...
17
votes
1answer
2k views

Intuitive Proof of Cramer's Decomposition Theorem

Cramer's decomposition theorem states that if $X$ and $Y$ are independent real random variables and $X+Y$ has normal distribution, then both $X$ and $Y$ are normally distributed. I've seen a few ...
16
votes
7answers
8k views

Lower bound for sum of binomial coefficients?

Hi! I'm new here. It would be awesome if someone knows a good answer. Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case $\...
16
votes
4answers
2k views

Are gaussians with different moments far in total variation distance?

If two Gaussians disagree on one moment, it seems like this should imply that they have a large variation distance--equivalently, if two Gaussians are close in variation distance it's hard for their ...
16
votes
2answers
2k views

When is the function of a median closer to the median of the function than the mean of the function is to the function of the mean?

Background notation: RV= random variable, $\mu=$ mean $m=$ median Jensen's Inequality considers the relationship between the mean of a function of an RV and the function of the mean of an RV. If $f(...
16
votes
1answer
1k views

Gini Coefficient and Renyi Entropy

Gini coefficient (aka Gini Index) is a quantity used in economics to describe income inequality. It is 0 for uniformly distributed income, and approaches 1 when all income is in hands of one ...
16
votes
1answer
1k views

Applications of the Giry monad in probability and statistics

In another thread, I asked about the $M$ endofunctor on the category $\operatorname{Meas}$ of measurable spaces, which sends a space $X$ to its space of measures $M(X)$. Will Sawin described the ...
16
votes
2answers
782 views

The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game

This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...
15
votes
4answers
2k views

Easier reference for material like Diaconis's “Group representations in probability and statistics”

I'm teaching a class on the representation theory of finite groups at the advanced undergrad level. One of the things I'd like to talk about, or possibly have a student do any independent project on ...
15
votes
5answers
2k views

Is a fair lottery possible?

I'm trying to come up with a scheme for a lottery where each individual has roughly the same chance of becoming the winner, regardless of the number of tickets one holds. So no individual should have ...
15
votes
2answers
463 views

How to sample uniformly from singular matrices

I would like to uniformly sample from all singular $n$ by $n$ Bernoulli matrices (that is each entry is $1$ or $0$ with probability $1/2$). I could of course just sample from all $n$ by $n$ Bernoulli ...
15
votes
2answers
2k views

Bounding sum of multinomial coefficients by highest entropy one

When does the following hold? $$\sum_{(i_1,\ldots,i_k)\in E} \frac{n!}{i_1! \ldots i_k!} \le \exp(n H^*)$$ where $H^*=\max_{(i_1,\ldots,i_k)\in E} -(\frac{i_1}{n}\log \frac{i_1}{n}+\ldots +\frac{...
15
votes
1answer
2k views

Distribution of maximum of random walk conditioned to stay positive

I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it ...
14
votes
7answers
3k views

Correlation and Causation. When can we believe correlation (reasonably, at least) imply causation

We always hear, when reading on correlation, that "correlation does not imply causation." Still, I have never seen any source that tries to answer the question of when can we reasonably conclude a ...
14
votes
3answers
1k views

entropy and flatness of densities

I was reading C.R Rao's Linear Statistical inference. Rao presents the entropy of a continuous distribution (expectation of -log density) as a measure of closeness to the uniform distribution, and ...
14
votes
3answers
2k views

James-Stein phenomenon: What does it mean that a James-Stein estimator beats least squares estimator?

Background James-Stein estimator and Stein's phenomenon, as described in Wikipedia are rather counterintuitive and amazing. It is claimed that if one wants to estimate the mean $\Theta$ of Gaussian ...
14
votes
1answer
2k views

Using Fisher Information to bound KL divergence

Is it possible to use Fisher Information at p to get a useful upper bound on KL(q,p)? KL(q,p) is known as Kullback-Liebler divergence and is defined for discrete distributions over k outcomes as ...
14
votes
1answer
1k views

Table with the most seated customers in Chinese restaurant process

Suppose we have some initial configuration of people seated at some tables. We start taking new customers and seat them following Chinese restaurant process. Is there some known work on finding the ...
13
votes
7answers
4k views

Uniformly Sampling from Convex Polytopes

How to choose a point uniformly from a convex polytope $P \subset [0,1]^n$ defined by some inequalities, $Ax < b$? (Here $A$ is an $m \times n$ matrix, $x \in \mathbb{R}^n$, and $b \in \mathbb{R}^...
13
votes
1answer
878 views

Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...
13
votes
2answers
611 views

Archaeogenetics

This question is meant to be applied to recover historic information from genetic data. The following model, is (probably) the simplest possible which takes recombinations into account. First, let ...
12
votes
7answers
15k views

Why isn't Likelihood a Probability Density Function?

I've been trying to get my head around why a likelihood isn't a probability density function. My understanding says that for an event $X$ and a model parameter $m$: $P(X|m)$ is a probability density ...
12
votes
7answers
1k views

Probabilistic (and other mathematical) methods of physics without the physics?

Many of the methods of physics are vastly more general than their use in that discipline. For example, information theory overlaps with a lot of statistical mechanics, and the latter actually ...
12
votes
4answers
695 views

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 ...
12
votes
4answers
1k views

How long for a simple random walk to exceed $\sqrt{T}$?

Let $R_n$ be a simple random walk with $R_0 = 0$, and let $T$ be the smallest index such that $k\sqrt{T} < |R_T|$ for some positive $k$. What is an expression for the probability distribution of $...
12
votes
1answer
3k views

What's the maximum entropy probability distribution given bounds [a,b] and mean?

What is the continuous probability distribution that maximizes entropy, given only the bounds of the random variable [a,b] and the mean mu of the probability distribution? For example: if a=0, b=1, ...
11
votes
5answers
1k views

What is hidden in Hidden Markov Models? [closed]

Why the word "hidden" present in hidden markov model? What exactly is hidden. Whatever is hidden in HMM isn't it hidden in normal Markov Models?
11
votes
1answer
1k views

What areas of algebra could be interesting to probability theorists?

I would like to find some topic of algebra (beyond linear algebra; algebraic number theory is fine) that would be interesting both to a student that wants to specialize in probability theory and to me ...
11
votes
1answer
220 views

Probability distribution derived from gamma function - does it have a name?

Consider the complex gamma function, denoted by $\Gamma(\sigma+it)$. Now, let's fix $\sigma$ and let t vary. Then consider the following expression: $$|\Gamma(\sigma+it)|^2$$ For any choice of $\...