**1**

vote

**2**answers

127 views

### Average Multivariate Gaussian

Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in [-...

**-2**

votes

**1**answer

83 views

### About the boundary conditions of the Black-Scholes-Merton PDE [closed]

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve.
Let $c(t,x)$ be the value of the ...

**0**

votes

**0**answers

47 views

### convergence of empirical distribution of random vectors

Given
(a) random matrices $A^{n} \in \mathbb R^{n\times n}$ with iid normal
entries $A_{ij}\sim \mathcal N(0, 1/n)$; and
(b) $X^{n} \in \mathbb R^{n}$ with its empirical distributions converging ...

**5**

votes

**0**answers

169 views

### Is there a name for this quantity between two distributions?

Let $f$ be a probability density on a compact domain $D$, and say that $x_1,\dots,x_n$ are samples from $f$. If we wanted to compute the Wasserstein distance between $f$ and the empirical ...

**2**

votes

**1**answer

124 views

### Where can I find a copy of Moussatat's 1976 thesis “On the Asymptotic Theory of Statistical Experiments and Some of Its Applications”?

It was apparently written at Berkeley under the direction of Le Cam, and it is cited in a number of contributions to mathematical statistics, for example in Strasser's (1985) book "Mathematical Theory ...

**0**

votes

**0**answers

45 views

### Mixture model: optimization vs regression

Consider a sample $\mathcal D = \{T_n\}_{n=1}^N$ of independent random variables, s.t.:
$$
p(T_n) = p_n(T) = \sum _{m=1}^Mp_n(\mathcal C_m)p_n(T\mid \mathcal C_m) = \sum _{m=1}^Mw_{nm}q_m(T)
$$
I will ...

**0**

votes

**0**answers

26 views

### K nearest neighbors estimation with a kernel

If I have a bunch of data points $x_1,\dots,x_n$, I can build a density function $f(x)$ based on these data points by defining $f(x) = c/d_k(x)$ for an appropriate constant $c$, where $d_k(x)$ is the ...

**8**

votes

**4**answers

146 views

### Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints
$$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq \vec{...

**5**

votes

**5**answers

350 views

### Generate Bernoulli vector with given covariance matrix

I am from different background, so please forgive me if the answer is so well known.
Let $C=(c_{ij})$ be a given $n\times n$ matrix. Do we have a way to generate samples of random Bernoulli vectors ...

**3**

votes

**0**answers

117 views

### 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 ...

**3**

votes

**1**answer

119 views

### Bounds on the probability of k-of-n events in terms of bounds on single and pairwise probabilities

Let $A_1,\dotsc,A_n$ be events in a probability space, and let $N = \sum_{i=1}^n \mathbf{1}_{A_i}$ be the random number of events that occur. For a fixed value $k \in \{1,\dotsc,n\}$, what can be ...

**2**

votes

**1**answer

213 views

### Expected value (probability) maximization with binomial distribution

I need to solve an optimization problem that involves an expected value like
$$F(n,x) = \sum_{k=0}^n \binom{n}{k} p^k(1 - p)^{n - k} f(k,x).$$
Here $f(k,x)$ is actually a probability coming from a ...

**0**

votes

**1**answer

84 views

### Finding the distribution of a random variable numerically with sample data? [closed]

Just a thought that I had recently. Suppose given discrete data points for a random variable, could one numerically generate the probability function values at these discrete values? I tried looking ...

**7**

votes

**4**answers

179 views

### What can be said about the concentration of measure of product of Gaussian variables?

I have a set of random variables $X_1,\ldots,X_n$, all Gaussian with mean 0 and variance 1, indepedent. Let $p(x_1,\ldots,x_n)$ be some polynomial that takes products and sums of $x_1,\ldots,x_n$.
...

**2**

votes

**0**answers

25 views

### Is there some kind of lower bound for estimation error of the estimation of (near) low-rank matrices in high-dimension?

I'm reading S.Negahban and M.J.Wainright's paper, ESTIMATION OF (NEAR) LOW-RANK MATRICES WITH NOISE AND HIGH-DIMENSIONAL SCALING. In the paper, they give a upper bound for estimation error of the ...

**4**

votes

**1**answer

390 views

### variance of compound binomial distributions

The below is motivated by a problem I'm observing in my experimental data
I have m boxes, where each box is supposed to contain k molecules of mRNA. The measurement process includes labeling all the ...

**0**

votes

**0**answers

113 views

### What is the concentration of measure for Gaussian random variables which are independent, but are transformed?

This might be a too easy question for Mathoverflow, but Googling led to similar questions and answers here (though not the one I was looking for).
The question is split into two:
I have a matrix $X ...

**1**

vote

**0**answers

89 views

### How do you use the bits you get back from Bits Back Coding?

Bits Back coding is a scheme to transmit an observation x.
You can read about it here [1]. To my understanding, it works like this:
The encoder samples a message z from a distribution Q(z|x) that it ...

**0**

votes

**0**answers

53 views

### Recursive parameter estimation for partially observed Ito SDEs

I'm trying to get my head around online (recursive) maximum-likelihood parameter estimation in the language of stochastic processes and in the context of stochastic filtering, i.e. where we have a ...

**0**

votes

**1**answer

214 views

### Are such averages known with representations of $S_n$?

Like is there a sense in which one can quantify that for two group elements (in different conjugacy classes) their characters are "close" for some fixed irreducible representation? (feel free to stick ...

**4**

votes

**2**answers

349 views

### Expectation of Mahalanobis norm

Let $(g_i)_{i=1,...,d}$ sampled i.i.d. from a standard Gaussian, and $(\lambda_i)_{i=1,...,d}$ non-random s.t. $\max_i(\lambda_i)=1$ and $\lambda_i>0, \forall i$.
I am looking for the expectation ...

**7**

votes

**1**answer

254 views

### A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as
$$\rho_2(X;Y):=\sup\frac{\mathbb{E}[f(X)g(Y)]}{||f||_2||...

**4**

votes

**1**answer

146 views

### Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES.
And in the paper, they provide an inequation of the Schatten-p (quasi-)norm, ...

**2**

votes

**0**answers

101 views

### Implication of MGF inequality

Let X and Y be two random variables. Denote by $F_X(x)$ and $F_Y(y)$ their CDFs and by $M_X(t)$ and $M_Y(t)$ their MGFs.
It is known that X and Y have the same CDF iff they have the same MGF.
My ...

**2**

votes

**2**answers

118 views

### Do all positive distributions on $N$ variables factor pairwise?

The Hammersley-Clifford theorem says that any positive probability distribution satisfies one of the Markov properties with respect to an undirected graph G if and only if its density can be ...

**5**

votes

**1**answer

93 views

### Rate-Distortion theory: What is the distribution of distortion on an optimal Gaussian encoder?

If we wish to encode a gaussian source, $X\sim\mathcal{N}(0,\sigma^2)$ at rate $R$, then decode it to create an estimate $\hat{X}$, rate-distortion theory tells us that the lowest mean-squared-error ...

**1**

vote

**0**answers

141 views

### Converse for Levy's continuity theorem

Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function $\varphi(t)...

**1**

vote

**1**answer

197 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

59 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 ...

**3**

votes

**1**answer

116 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 $\boldsymbol{S} = \boldsymbol{X} \boldsymbol{\Lambda} \boldsymbol{X}'$ ($\boldsymbol{\Lambda}$...

**2**

votes

**0**answers

64 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 ...

**0**

votes

**2**answers

319 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 ...

**5**

votes

**1**answer

175 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

1k 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) = \frac{1}{\...

**0**

votes

**0**answers

133 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 E_n(...

**1**

vote

**0**answers

52 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 $(i,j),...

**6**

votes

**2**answers

558 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 > ...

**1**

vote

**0**answers

81 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) ...

**2**

votes

**0**answers

224 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 ...

**6**

votes

**2**answers

552 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 ...

**3**

votes

**2**answers

161 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 $...

**5**

votes

**1**answer

128 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 ...

**5**

votes

**0**answers

111 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 ...

**4**

votes

**0**answers

338 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 ...

**1**

vote

**0**answers

89 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, we'...

**0**

votes

**1**answer

53 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

197 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
$\mathbb{E}|\sum_{...

**2**

votes

**1**answer

285 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

537 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

84 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); \...