**1**

vote

**1**answer

47 views

### Convexity of truncated expectation

Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$.
Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...

**7**

votes

**2**answers

888 views

### The James–Stein estimator - counterintuitive estimation of the mean. What means it is better than least squares ? (Understanding Wikipedia)

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 ...

**-3**

votes

**0**answers

28 views

### Probability problem [on hold]

In an urn there are 8 blue and 5 red balls. Consequently, we are taking 1 ball without returning it, until all of one of the colors are taken out. Let X is the amount of balls. Find the type of ...

**1**

vote

**0**answers

31 views

### Covariance of order statistics [on hold]

I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Note that I have also posted on math.stackoverflow, but given that this seems to be ...

**0**

votes

**0**answers

24 views

### How to incorporate novel observation into covariance matrix [closed]

I have a 3D covariance matrix with known values (but not the data it was calculated from).
edit: I do have the means.
|a 0 0|
|0 b 0|
|0 0 c|
I receive a novel ...

**6**

votes

**1**answer

4k views

### Conjugate prior of the Dirichlet distribution?

What is the conjugate prior distribution of the Dirichlet distribution?

**5**

votes

**3**answers

13k views

### Distance metric between two sample distributions (histograms)

Context: I want to compare the sample probability distributions (PDFs) of two datasets (generated from a dynamical system). These datasets depend on a set of parameters, and I want a concise way to ...

**-3**

votes

**0**answers

29 views

### Compute score for a set of data [closed]

So let's explain my problem. I have a set of items which have a score from 0 to 100. This set is dynamic which means that several values are expected to be added from moment to moment. In each item is ...

**1**

vote

**0**answers

45 views

### Show this function is strictly concave

Please help me show that $f(w)$ is strictly concave in $w\in[0,\infty)$:
$f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $
where
$P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq ...

**14**

votes

**1**answer

638 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 ...

**1**

vote

**1**answer

180 views

### connection between the statistical properties of a scalar field and its columns

Consider a scalar field $s:[0,1]^3 \to \mathbb{R}$ and its "column" field
\begin{equation}
c: [0,1]^2 \to \mathbb{R}: (x,y) \mapsto \int_0^1 s(x,y,z) \,\mathrm{d}z.
\end{equation}.
What can be said ...

**0**

votes

**0**answers

15 views

### Product of lognormal random variables

Let $X_1$ and $X_2$ be two normal random variables. Write $X_1\sim N(\mu_1, \sigma^2_1)$ and $X_2\sim N(\mu_2, \sigma^2_2)$, to fix ideas.
Consider the corresponding log-normal random variables: ...

**2**

votes

**1**answer

74 views

### Do product distributions (or graph products) eventually cluster as more products are taken?

Say we have a joint distribution on a finite alphabet $\mathcal{X}\times \mathcal{Y}$. It could be a communication link where we want to send a random message $X$ over a channel, but it gets garbled ...

**2**

votes

**2**answers

142 views

### The necessary sufficient condition for recurrence of a Markovian random walk

Suppose $\sigma_{1},\sigma_{2},...$are i.i.d random variables.$S_{0}=0$. Define $S_{n}=S_{0}+\sum_{i=1}^{n}\sigma_{i}$, then ${S_{n}}$ is a Markovian random walk.
I want to figure out the necessary ...

**4**

votes

**1**answer

53 views

### Concentration of U-statistics for exchangable distributions (and the unbounded case)

Consider the following so-called $U$-statistic of order 2: $$U = \frac1{\binom{m}{2}} \sum_{i < j} h(w_i,w_j)$$ where $w_1,\dots,w_m$ are IID from some distribution and $h$ is symmetric. If ...

**0**

votes

**0**answers

52 views

### Unsupervised classification [migrated]

In a typical supervised learning problem, one observe $(X,Y)$ where $Y$ is a categorical variable. We confront the such a problem that $Y$ is hidden and instead $(X,Z)$ is observed where both ...

**4**

votes

**1**answer

115 views

### Can samples be compressed?

The Fisher information of a random variable $Y$ about a parameter $\theta$ upon which the probability of $Y$ depends is:
$\mathcal{I}_Y(\theta)= -E\left[\left.\strut \frac{\partial^2}{\partial ...

**0**

votes

**0**answers

19 views

### Feature selection: wrapper or embedded? [migrated]

I have searched for information on what could be the reasons for preferring wrapper feature selection to embedded feature selection. I found a comparative study which tells when filters are better, ...

**9**

votes

**2**answers

9k views

### 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 ...

**5**

votes

**1**answer

133 views

### Stochastic Covering Number of a Convex Set

Consider a convex set, say $S = [0,1]^d$. Let $X_1, X_2,\ldots,X_n, \ldots$ be i.i.d. random variables that are uniformly distributed on $S$. Denote the Euclidean ball centered at $x \in \mathbb{R}^2$ ...

**0**

votes

**0**answers

31 views

### Experimental Investigations on the Statistics of Infinite, Discrete, Evenly Distributed Pointsets in the Euclidean Plane

I am trying to estimate the distribution of certain planar polygons in the Euclidean plane; to accomplish that, I generate finite set of points, that are evenly distributed in w.l.o.g. the ...

**7**

votes

**1**answer

196 views

### In what sense is the Bayesian posterior mean a “convex combination”?

I asked this on math.stackexchange with no response, I'm hoping someone here might have something.
Suppose I want to estimate $x \in \mathbb{R}^n$ from two signals with zero mean, normally ...

**2**

votes

**0**answers

848 views

### Moments of function of Poisson process

(I'm new to Poisson processes, so please edit if my terminology is incorrect.)
Edit: per comments, here is a (more) general version of the originally posted problem (which is now at the bottom, below ...

**0**

votes

**0**answers

6 views

### What's the definition of multivariate mode? [migrated]

In the case of grouped data where a frequency curve have been constructed to fit the data, the mode will be the value (or values) of x corresponding to the maximum point (or points) on the curve. From ...

**1**

vote

**0**answers

73 views

### Convergence of an rcll process along a random subsequence

I have a process $X_s$, for $s \ge 0$, taking values in a Polish space $T$ with an rcll version where I have shown, for every nonrandom increasing sequence $s_n$, that $X_{s_n} \to c$ in probability, ...

**5**

votes

**1**answer

434 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 ...

**2**

votes

**1**answer

72 views

### What is the order of the constant $K$ in the multidimensional Dvoretzky-Kiefer-Wolfowitz inequality($Ke^{-c z}$)?

Let $F_n$ be the empirical distribution obtained from an i.i.d. sample
of the distribution $F:R ^d \to [0, 1]$.
Kiefer (1961) shows that the convergence of the empirical distribution is like
$$
...

**1**

vote

**2**answers

5k views

### Why 1.5IQR whiskers in boxplot? [closed]

Hi math people.
I'm in the process of analyzing some data that I collected through an experiment. The data are (somewhat) normally distributed and I represent the different data-sets using boxplot, ...

**1**

vote

**0**answers

101 views

### Limit theorem : reproduce a proof with an adaption from discrete to continuous time

Im considering Theorem 5.2.2 in M. Sørensen "Exponential Families of stochastic processes".
The setup is as follows:
We have a Levy-Process $X_t$ fullfilling the CLT
\begin{align}
...

**0**

votes

**0**answers

59 views

### Basic Monte Carlo Integral Approximation

On the very first page of a well-known book on Monte Carlo techniques, there is the following statement. Let
\begin{equation}
I = \int_D g(\textbf{x})d\textbf{x},
\end{equation}
where $D \subset ...

**2**

votes

**0**answers

45 views

### Literature on transformed Gaussian matrices

I am considering real $n$-by-$m$ matrices of the following type:
$$
M=SM^\prime,\\
M^\prime_{ij}\sim^{iid}N(0,1).
$$
Here, $S$ is a fixed $n$-by-$n$ matrix and the entries of $M^\prime$ (same size ...

**7**

votes

**1**answer

101 views

### Choosing a sample based on where the density function is highest

Is there a name for the following process?
Say I have an absolutely continuous probability density function $f$ with compact support, and I take $k$ independent samples $x_1,\dots,x_k$ from $f$. ...

**3**

votes

**1**answer

109 views

### Is there a closed form expression for $E(X e^{-\mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$?

Is there any closed form expression for $E(X e^{- \mu \sqrt{X}})$, where $X\sim Poisson(\lambda)$ and $\mu >0$? If not, is there any tight upper bound for this quantity? Any idea how to proceed?

**2**

votes

**0**answers

40 views

### A question about probabilistic graphical models

Say one is given a probabilistic graphical model and a cut of the underlying graph. Do we know any statements about when and how can one or many of the marginals (of the sources) or the conditionals ...

**1**

vote

**0**answers

37 views

### 2-step sampling from a conditional density

The setting is as follows:
We are given two random variables $X : \Omega \to \mathbb{R}$ and $\Theta : \Omega \to T$ for some 'parameter space' $T \subset \mathbb{R}$, and
1) we know the density of ...

**5**

votes

**1**answer

173 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?

**0**

votes

**0**answers

24 views

### Adaptive refinement of integral domain

In electromagnetics we need to calculate the radiated power which is defined as something like
$P_r=\int_0^{2\pi}\int_0^{\pi}R(\theta,\phi)\sin{\theta}d{\theta}d\phi$
We already have ...

**1**

vote

**0**answers

19 views

### Robust weighted estimator of location

Let $X = (x_1, \ldots, x_n)$ be a sample of i.i.d values. There are several robust estimators of sample location, most notably sample median and Hodges-Lehmann estimator.
Now let $W = (w_1, \ldots, ...

**3**

votes

**1**answer

108 views

### Two minimization problems using singular value decomposition

Posted here too: http://math.stackexchange.com/questions/1711026/two-minimization-problems-using-singular-value-decomposition
Let $q_0, q_1:[0,1]\to \mathbb{R}^n$ be two maps whose components are ...

**0**

votes

**0**answers

18 views

### Error propagation with black boxes: add uncertainty in quadrature, or use a weighted standard deviation?

I have a measurement $x$ with a known uncertainty $\sigma_m$.
I have a black box that can take an error-free measurement $x$ and produce a value $y$ with a known uncertainty $\sigma_{b}$ (which is ...

**18**

votes

**4**answers

824 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 ...

**1**

vote

**0**answers

57 views

### Norm-averaging reference request

(Apology in advance for the broadness of this question) I recently came across a relatively simple application where I needed to "balance" the "spreaded-out-ness" of a function with the "peaked-ness" ...

**3**

votes

**3**answers

2k views

### Integral over error function and normal distribution

Help me understand why
$\int_{-\infty}^{\infty}\frac{1}{2}[1+\operatorname{erf}(\frac{\theta-x}{\sqrt{2q^2}})]\frac{1}{\sqrt{2\pi\sigma^2}}{\exp(-\frac{(x-\mu)^2}{2\sigma^2})}dx \approx ...

**3**

votes

**1**answer

132 views

### Moment matching on the standard simplex

Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j ...

**3**

votes

**0**answers

75 views

### An inequality involving conditional variance and its connection to information theory

Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$ ...

**3**

votes

**1**answer

105 views

### Learn a distribution from distributions on samples

There's many good ways to learn a distribution $p_X$ of an r.v. $X$ over $k$ symbols given many i.i.d. samples $X_1,\ldots, X_n$. The simplest is to use the sample relative frequencies $\hat{f}_X$ as ...

**0**

votes

**0**answers

34 views

### A functional's expectation using both known and unknown pdf

Suppose we have a random variable $X$ with a known distribution $f$ over an interval $[a,b]$ and another r.v $Y$ over the same interval but with an unknown distribution $g$. We also have a functional ...

**2**

votes

**1**answer

96 views

### An Inequality Regarding the Squared Conditional Variance

Given absolutely continuous random variables $(X, Y)$ with joint distribution $P_{XY}$, we construct $Z:=\sqrt{\gamma} Y+N_\mathsf{G}$ where $N_\mathsf{G}\sim N(0, 1)$ and is independent of $(X,Y)$. ...

**3**

votes

**0**answers

78 views

### How does Jensen Shannon divergence and KL divergence correlate?

I am wondering if there is way to derive the correlation between Jensen Shannon divergence and KL divergence for two distributions: P and Q, in order to show that if JSD(P,Q) decreases, KLD(P,Q) ...

**5**

votes

**3**answers

262 views

### The mean of points on a unit n-sphere $S^n$

A unit n-sphere is defined as $$\mathcal{S}^n = \{\mathbf{p} \in \mathbb{R}^{n+1}: \|\mathbf{p}\| = 1\}$$
The distance between two points $\mathbf{p}$, $\mathbf{q}$ on $\mathcal{S}^n$ is the ...