**-1**

votes

**0**answers

21 views

### Variance under time-scaling of sample measurements [on hold]

I am struggling with explain something I read in a Whitepaper. The essence is as follows.
Let's begin with a random variable $X$ defined as number of events in an hours. Further, we assume that $X ...

**1**

vote

**1**answer

170 views

### Computing probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s

This question came up in my research: What is the probability that $Ax\geq0$ where $x$ is a vector of iid gaussians and $A$ is matrix of $1$s and $0$s?
So far I only figured out that I can do Monte ...

**1**

vote

**0**answers

46 views

### Subclass of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great.
An Ito semimartingale is a martingale for which the ...

**2**

votes

**1**answer

36 views

### Reducing eigenvalues of symmetric PSD matrix towards 0: effect on ratios of original matrix elements?

Let $\boldsymbol{S}$ be $k \times k$ positive semi-definite real symmetric matrix with eigen decomposition $\boldsymbol{S} = \boldsymbol{X} \boldsymbol{\Lambda} \boldsymbol{X}'$ ...

**22**

votes

**11**answers

8k 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 ...

**0**

votes

**2**answers

190 views

### Generalized expression for balls and bins problem

$n$ number of balls are thrown randomly to $m$ number of bins, standing in a row. The balls are labeled as $1,2,3,....n$ and bins are also labeled as $1,2,3,...,m$. The probability of $i_{th}$ ball ...

**8**

votes

**0**answers

1k views

### Blinding a paper : the acknowledgements section [closed]

I am about to submit a paper (my first!) to a journal in statistics. There is a requirement to submit a "blinded" version. Should I remove the acknowledgements section too for this? If I do, what are ...

**7**

votes

**1**answer

439 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 ...

**0**

votes

**2**answers

41 views

### Is it possible to find an asymptotic distribution for the LRT without the ML estimators being consistent?

I'm reading a comment(last page) to a paper, and the author states that sometimes, even though the estimators (found by ML or maximum quasilikelihood) may not be consistent, the test may be ...

**2**

votes

**0**answers

30 views

### Derivation of gradient of SSE in Geodesic Regression

On page 79 (or page 5) of this this paper the gradient of the SSE of the Geodesic model is described explicitly. My question is how are these equitations derived in detail; where can I find the ...

**4**

votes

**0**answers

76 views

### power laws emerging from the sandpile model

Is there a rigorous proof that the abelian sandpile model generates a power law distribution of avalanche lengths?

**5**

votes

**1**answer

209 views

### Reference on (discrete) log-concave probability distributions

A discrete distribution $p$ over $\mathbb{N}$ is said to be log-concave if it satisfies the following conditions:
The support of $p$ is a contiguous interval, i.e. $\exists a \leq b$ s.t. $p_i > ...

**4**

votes

**1**answer

184 views

### Square root of normal distribution

Let $X$ and $Y$ be independent random variates with the same probability distribution, $P(x)$. Assuming that the product $Z=XY$ is a random variate with normal distribution, say $$f_Z(x) = ...

**1**

vote

**0**answers

26 views

### Is there a way to find an estimator for this problem?

The Hausdorff metric or measures the distance between between sets of points.
Resource 1: Paper formulates a probabilistic approach for Hausdorff distance:
On Probabilistic Hausdorff matching. I ...

**0**

votes

**0**answers

26 views

### How do I compute the variance of expected number of fair coin flips for HTH sequence using linear system of equations? [migrated]

Assuming fair coin flips, I know how to compute the expected number of coin flips to see HTH sequence by writing out the linear system of equations from the state transition diagram below.
...

**2**

votes

**1**answer

108 views

### Maximization of specific Likelihood function

N coins have probability $p_n = e^{-t_n/s}$ of heads, $t_n$ being specific for each coin. Coins 1 to m came up heads and m+1 to N came up tails. Now I'm trying to estimate $s$ using the Maximum ...

**0**

votes

**0**answers

83 views

### How to decide a value of learning rate for Stochastic Gradient Descent?

I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation,
$w_{i+1} = w_i + -\eta \nabla ...

**3**

votes

**1**answer

227 views

### Mutual information decrease with coarse-graining

Let $X,A,Y,B,C,D$ be random binary variables. $D$ is independent from $X,A,C$ and $C$ is independent from $Y,B,D$.
Is it true that:
If $I(Y:B|D=0)\leq \epsilon$ then $I(X\oplus Y:A\oplus ...

**1**

vote

**0**answers

31 views

### Maximum likelihood estimation with several distributions

My question concerns using Maximum likelihood to estimate unknown parameters used by several (poisson) distributions.
The parameters are the pairs $(a_1,b_1),\dots,(a_N,b_N)$, and for each pair ...

**3**

votes

**2**answers

169 views

### PDF of the product of normal and Cauchy distributions

I am having trouble in finding out the resulting PDF of the product of normal and Cauchy distributions. It turns out that we have a general formula for calculating the PDF of product of two random ...

**0**

votes

**0**answers

33 views

### Upper bound for chi-square divergence in terms of KL divergence

In my research I need an upper bound for chi-square divergence in terms KL divergence which works for general alphabets. To make this precise, note that for two probability measures $P$ and $Q$ ...

**1**

vote

**0**answers

57 views

### How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...

**0**

votes

**1**answer

281 views

### Size of KL-divergence neighbourhoods

I am new here. I was reading another
post
here and this got me wondering what can be said about the size of the following kl divergence neighborhoods.
Consider these two kl-divergence neighbourhood ...

**0**

votes

**2**answers

157 views

### Third order central moment of a positive linear combination of log-normal random variables

What is the sign (+tive/-tive) of the third order central moment of a positive linear combination of log-normal random variables?
It seems to be a common notion that the skewness of random variables ...

**0**

votes

**0**answers

41 views

### Taking power of the integrand in a Riemann-Stieltjie Integral

This is a problem I am trying to solve as part of a calculation for Value-at-Risk.
Given that
$P(X<x)=F(x)=\int_{\theta}F(x|\theta)dG(\theta)=1-\alpha$,
where $F$ and $G$ are CDF's, is there a ...

**2**

votes

**0**answers

154 views

### Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go:
Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...

**5**

votes

**2**answers

329 views

### Applications of cohomology to probability and statistics

Are there interesting/useful applications of cohomology (and homological algebra in general) to probability and statistics, or information theory?
By "interesting/useful", I mean "not merely ...

**2**

votes

**2**answers

95 views

### What is the sum capacity of a scalar gaussian broadcast channel?

"On the Achievable Throughput of a Multiantenna Gaussian Broadcast Channel" by Giuseppe Carie and Shlomo Shamai talks, in part, about the following type of link (paraphrasing):
A transmitter with ...

**0**

votes

**0**answers

37 views

### Is a conditional copula invariant under strictly increasing transformations?

currently I am working on conditional copulas and I have a theoretical question. In "An Introduction to Copulas", Nelsen (2006) there is a theorem (2.4.3) which says:
Let $X$ and $Y$ be continuous ...

**4**

votes

**5**answers

2k views

### book recommendation on data analysis and statistics

I am looking for a book on data analysis and statistics.
My objective is to better analyse and understand data over time (like trends or events) and extract useful information from raw statistics. I ...

**1**

vote

**0**answers

31 views

### Is it possible to distinguish between to edge orientation while learning a network structure?

I'm considering the case of learning bayesian network structure using a dataset $\mathcal{D}$ with scoring methods :
$$\mathcal{G}^*=\max_{\mathcal{G}}\text{Score}(\mathcal{G}, \mathcal{D})$$
I'm ...

**1**

vote

**0**answers

23 views

### assumptions on local rademacher complexities

A lot of the work on Local Rademacher complexities of Koltchinskii, and Bartlett for fast rates of convergence is based on Bousquet's version of Talagrand's inequality [1] (Theorem 2.11). However the ...

**2**

votes

**3**answers

516 views

### When do binomial distributions occur?

A binomial distribution is the distribution of the number of successes of n independent, identical Bernoulli trials. What happens when the trials are dependent and the Bernoulli trials are not ...

**0**

votes

**1**answer

391 views

### central limit theorem for binomial random variable

I'm confused about applying central limit theorem to Bernoulli random variables. Let
$X_i=\frac{n}{\sqrt{n-1}}(Z_i - \frac{1}{n})$ where $Z_i$ is iid Bernoulli($\frac{1}{n}$). Then $E[X_i]=0$ and ...

**5**

votes

**0**answers

101 views

### What's the variance in the Six Degrees model?

Recall the six degrees of Kevin Bacon game. You can even play the game at The Oracle of Bacon, and their search works via Breadth First Search.
I interpret the punchline as saying that if I start ...

**1**

vote

**2**answers

942 views

### Derivative of a differentiable stationary Gaussian process

Thanks for your help in advance. I'm interested in understanding the properties of derivatives of a differentiable stationary Gaussian process. Specifically, is the derivative also a Gaussian ...

**4**

votes

**0**answers

227 views

### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate
$$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$
where $dH$ is the unit invariant Haar measure on the group of unitary ...

**0**

votes

**0**answers

56 views

### Continuous self-information

Let $I(X,Y)$ be the mutual information between two continuous random variables $X$ and $Y$.
We have $I(X,Y) = H(X)-H(X|Y)$, and setting $X=Y$ leads to $I(X,X) = H(X)-H(X|X)$. If $X$ was discrete, ...

**0**

votes

**1**answer

32 views

### distances-based dispersion measuring approach

Is there any known approach or method to measure the dispersion of a set depending on the distances between its points (i.e.: without calculating the average or the mean) ?
thanks.

**1**

vote

**0**answers

81 views

### Bound the expectation of trace norm of random Hermitian matrix

Suppose $H_i$ are traceless $d\times d$ Hermitians, $X_i$ are Standard normal distribution for $1\leq i\leq d^2$.
We would like to bound the following expectation on the trace norm
...

**1**

vote

**1**answer

99 views

### 1-wasserstein distance v.s. total variation distance

Suppose that $\mu_1$ and $\mu_2$ are two distributions defined on $\mathbb{R}^n$ and $\gamma$ is a symmetric distribution (around $0$) on $\mathbb{R}^n$ with compact support. Let $\gamma_x$ denote the ...

**2**

votes

**1**answer

170 views

### Statistical distance between discrete and continuous distributions

Are there any statistical distance functions that are capable of comparing a continuous and a discrete distribution? From reading this list
http://en.wikipedia.org/wiki/Statistical_distance
the only ...

**0**

votes

**0**answers

42 views

### Integral over conditioning variable of a Gaussian

The marginal of a multivariate Gaussian can be computed in closed form, i.e.,
$p(x) = \int_y \mathcal{N}((x,y);\mu,\Sigma)\ dy$
is simple. But what I need is
$L(x) = \int_y \mathcal{N}((x\mid y); ...

**8**

votes

**2**answers

385 views

### Inequality in information theory

I am reading the paper "chain independence and common information" (http://ttic.uchicago.edu/~yury/papers/independ.pdf). In this paper, an inequality is used several times (without proof) which looks ...

**3**

votes

**1**answer

120 views

### Does bounding moments make distributions close in total variation distance?

Let $W\sim\mathcal{N}(0,\sigma^2)$ be a "reference" Gaussian random variable.
Suppose I have a set of distributions, $\mathcal{W}$, where $W_a\in\mathcal{W}$ if it satisfies the following criteria:
...

**1**

vote

**0**answers

56 views

### Lower bound on difference between polynomials at moderate distance

Fix $r > 0$ and $k, n \in \mathbb{N}$. Also consider a function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$. Let $x_{1},\ldots, x_{n+1}$ be points chosen uniformly from $[-r,r]^{d}$. For $1 \leq i ...

**0**

votes

**1**answer

97 views

### Independence of Eigenvalues of Wishart

This question regards a previous post, but it is not immediately obvious the two are related, so I ask it anyways: are the eigenvalues of a Wishart matrix $\mathbf{S}$ $=$ ...

**0**

votes

**2**answers

112 views

### What are some examples of isotrophic sets?

What are some examples of isotrophic sets? and is there a "good" way to describe them?
Isotrophic meaning that a random vector X uniformly distributed in the set has the isotrophic property for all ...

**2**

votes

**1**answer

957 views

### Choice of Lipschitz constant for proximal gradient optimization

I'm trying to use proximal gradient methods (Forward-Backwards Splitting and FISTA) to minimize a function
$f(B) = \frac{1}{2}|| XB - Y||_F^2 + \frac{\gamma}{2}||B C^T||_F^2$, where $X \in ...

**8**

votes

**4**answers

711 views

### What does it mean when we say we have computed a number to a certain accuracy using a probabilistic algorithm?

My intention is to ask a general question about probabilistic (Monte Carlo) algorithms. But to keep things simple, I will focus on a few specific examples.
Let me start the discussion with ...