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

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18 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 $Q$ ...
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0answers
36 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) ...
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0answers
31 views

How to put together a set of modified conditional distributions to a joint distribution? [on hold]

I am abstracting my original problem to a simple scenario.Consider a bi-variate multi-modal mixture of gaussian distribution, P(x,y). When we slice through x or y we get a univariate multi-modal ...
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0answers
18 views

Finding the bound of a mixture model percentile [on hold]

I could do with some help on the following issue. I'm trying to obtain $\alpha$ from: $\beta = \int_{\alpha}^{\infty} \sum_{i=1}^{n} p_i f_i(x) dx$ I have that: $0 \leq p_i \leq 1$, $\sum_{i=1}^{n} ...
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1answer
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 ...
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40 views

formula for turning star reviews into upvotes [on hold]

I want to turn reviews of up to 5 stars and the number of reviews into upvotes. Whats a good algorithm for doing this? A venue with 10 reviews total with a 5 star average rating should obviously get ...
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0answers
11 views

How to test the significance of covariance [closed]

I'm using the Mutual Informacion covariance in RNA sequences and I want to know if there exits a way to test if some covariance is significant, let's say, an associated p-value. Thanks to all for ...
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33 views

Backpropagation KL-divergence for training “one level neural network” [closed]

Hi I hope that some one could help me. I have a L matrix (25x1000) (I have 10000 works each work is represent by 25 bits) I map each word to one of 5 classes {vary negative,negative, ...
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0answers
30 views

The computation of a maximazation [closed]

I really need a help to compute the following maximization problem. $\max _{p(x), E(x) \leq \alpha} \int_x \int_y (\frac{p(x,y)^2}{p(x)p(y)})dx dy$. Suppose that : $p(y|x)=\frac{1}{\sqrt{2 \pi ...
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0answers
18 views

Simulate correlated random field of probabilities [closed]

Hi I am trying to simulate a spatial latent random field that are probabilities not correlated binary data as specified in other posts. The goal is to have the probabilities correlated by distance. ...
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2answers
154 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 ...
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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
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0answers
117 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
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1answer
387 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 ...
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2answers
38 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 ...
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2answers
312 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 ...
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30 views

Implementing the Bivariate survival function (Dabrowska estimator etc) [migrated]

I am trying to implement the Dabrowska estimator to apply to some data that has been collected and I am wondering if anyone can help. The original paper is Kaplan-Meier Estimate on the Plane by Dorota ...
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2answers
91 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 ...
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0answers
34 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 ...
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5answers
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 ...
2
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1answer
104 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 ...
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0answers
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 ...
3
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1answer
207 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 ...
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22 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
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2answers
113 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 ...
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3answers
515 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 ...
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1answer
385 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 ...
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0answers
100 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 ...
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2answers
927 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
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0answers
219 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 ...
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55 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, ...
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1answer
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.
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66 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 ...
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1answer
91 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 ...
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1answer
136 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 ...
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0answers
40 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); ...
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2answers
383 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
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1answer
113 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: ...
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0answers
53 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 ...
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1answer
96 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 $\mathbf{S}$ $=$ ...
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2answers
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
1answer
922 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
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4answers
705 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 ...
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0answers
57 views

limit distribution of multinomial distribution with increasing categories

If $\bf{X} \sim \text{multi}(n,p)$ with $k$ categories, we know $$ \sqrt{n}\left( \frac{\bf{X}}{n} - \bf{p} \right) \rightarrow^D N(0,\Sigma),$$ where $\bf{X}=(X_1,\ldots,X_k)^T$ and ...
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0answers
29 views

Inverse of the covariance of the estimate of a covariance

I have a covariance matrix, $V_{ij}$, which (for reasons that aren't important) I'm going to call the visibilities. I have an estimator for the visibilities $\hat V_{ij}$, and I've derived that the ...
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0answers
39 views

Kaplan-Meier estimator for mixtures

Let $\mathbf F=(F_1,\ldots ,F_m)^\top$ be a vector of continuous CDFs and $W$ is a matrix of weight coefficients, such that: $W\in \operatorname{Mat}_{n\times m}([0,1])$ $\forall i\in [n]:\sum ...
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0answers
25 views

Coordinates Poisson Cluster parent point

Is there any method to know the position of parent point in 'Poisson Cluster Process'? For information I use data with poisson distribution. data consist of (longitude, latitude, date). I want ...
2
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0answers
70 views

Sum of the entries of the inverse covariance matrix

Let $T \in\left(0,1\right)$, $n\in\mathbb{N}$ and $e_n = [1,\ldots,1]\in\mathbb{R}^n$. Consider the covariance matrix $\mathfrak{A}_n = ...
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0answers
27 views

How to sample from the ratio between two distributions?

I want to sample a lot of $\theta$s from the density function below: $$ r(\theta) = \frac{prior(\theta)}{Z}\frac{\int p(\theta,z_1)dz_1}{\int q(\theta,z_2)dz_2} $$ where $Z$ is the constant for ...
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1answer
215 views

Continuity of a Functional

A certain functional $T$ is defined as: $$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$ where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$. The result that above functional is ...