1
vote
1answer
143 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 ...
2
votes
0answers
33 views

Where to read about this kind of “measure of irredundancy” of a set from a family of sets?

Studying a very practical problem from psychometrics, I encountered the following construction. Let $(X,\mu)$ be a measure space; if preferred, you can presume $\mu$ is a probability measure. In any ...
0
votes
0answers
48 views

Linear Bounds on estimation error

Consider a markov chain on discrete state space $\mathbb{S} = \left\{1,2,..,S \right\}$, with transition probability matrix defined as $A = [a_{ij}]_{S \times S}$ where $a_{ij} = ...
5
votes
3answers
225 views

How do you call the problem of approximating a continuous distribution with a simple discrete distribution?

The following problem came up on the Mathematica forum as "Generating a list of integers that roughly satisfy a distribution": Given $n$, find $n$ integers (possibly with duplicates) whose ...
1
vote
0answers
57 views

Small ball probabilities for functions of correlated normals

Let $f : \mathbb{R}^k \rightarrow \mathbb{R}$ and let $X$ be distributed k-dimensional normal with mean $0$ (with "arbitrary" covariance matrix). I am looking for references with bounds of the form: ...
3
votes
1answer
126 views

Concentration rates for the posterior distribution

Sanov's theorem and Dvoretzky–Kiefer–Wolfowitz's inequality tell us how fast the empirical distribution concentrates around the true underlying probabilty distribution. What is known about the ...
9
votes
2answers
267 views

“Fractional sampling” from a probability distribution

My question concerns an operation on probability distributions which has arisen in some applied research. It is well-defined mathematically (at least in a limited context), but I don't know how to ...
3
votes
0answers
93 views

A simplified MCMC / MH algorithm. Are there known convergence results?

Hi, I hope this isn't too basic. We were working on a simulation using a Monte Carlo Within Metropolis algorithm and noticed that the whole thing could be expressed in the form below and simplified ...
0
votes
0answers
90 views

Two Different Representations of Multivariate Bernstein Polynomials

In the literature the multivariate Bernstein polynomial of a function $f:[0,1]^m\rightarrow\mathbb{R}$ is often defined as the following: $$B_{f,n}(x_1,\dots,x_m)=\sum_{\mathbf{k}\in ...
8
votes
0answers
254 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 ...
4
votes
2answers
693 views

Interesting thesis topic on statistical inference that is sufficiently mathematical

Hello , I am a student who's gonna start honours in mathematics . Currently , I am at the stage of finding a suitable honours thesis topic . I've chosen my supervisor , who's research interest is on ...
11
votes
1answer
403 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 ...
2
votes
0answers
105 views

Quantifying the amount of structure in a data set via random matrix theory

Given a data matrix, $M \in \mathbb{R}^{n \times p}$, I am interested in methods quantifying the amount of structure in present in $M$. I've found a few approaches, but I would like to learn more ...
1
vote
0answers
105 views

time derivative of the median of a stochastic process

Suppose you have a cumulative distribution that is changing with time, namely $ P_t(x) $. Assume $ P_t $ is monotone increasing and smooth enough so that we can define $ x_t(P) = P_t^{-1} $. We want ...
1
vote
1answer
252 views

Stieltjes convolution with white noise

I'm looking for a reference that would discuss a Stieltjes convolution between a wiener process and a function of bounded variation. Additionally, I had a question about this sort of convolution. Is ...
8
votes
3answers
1k 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. ...
4
votes
2answers
282 views

Convolutive noise removal

I have the time domain signal $$ u_o(t) = u(t)e^{-t/\tau}\eta(t) + \sigma(t) $$ where $\tau$ is known, $\eta$ is non-Gaussian noise, and $\sigma$ is Gaussian noise. The distribution of $\eta(t)$ is ...
4
votes
1answer
263 views

For an approach to the Hadamard-matrix-problem: is there a proof, that the iterative plane-wise orthogonal rotations (Quartimax/Varimax) converge to global maximum?

I've asked this question at stat-exchange and at the "Semnet"-mailing list of professionals in statistics. The reference to some articles in Psychometrica (for instance ten Berge 1995, Jennrich 2001) ...
4
votes
1answer
128 views

Mean occurrences of letters in complete strings given by a Bernoulli scheme

Suppose one has an alphabet of $K$ letters, from which we draw sequentially letters; assume that the $n$-th letter occurs with a fixed probability $p_n$ independently of the others and of the previous ...
0
votes
0answers
100 views

A name for this distributional condition?

I have come across a needed condition on a continuous probability distribution defined over $[0, \infty]$ and wonder whether it has a name. For a distribution with CDF $F$ and pdf $f$ I need that the ...
5
votes
0answers
483 views

Multidimensional Berry-Esseen for probability density functions

This is a follow up to this recent question: Berry Esseen type result for probability density functions There exists a multidimensional version of the usual Berry-Esseen theorem (for cumulative ...
0
votes
1answer
124 views

How are epidemic models simulated in case of mobility?

I am not a mathematician but out of curiosity I am trying to implement the SIS epidemic model when the nodes have mobility to understand how it will change the results. I understand how to perform ...
9
votes
1answer
3k 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 ...
1
vote
0answers
173 views

What is the MP pseudoinverse's role in statistical learning and Self-Organizing Maps?

During a discussion in our lab last month, a professor mentioned to me that the behavior of Self-Organizing Maps can be described in terms of repeated applications of the Moore-Penrose psuedoinverse, ...
7
votes
4answers
1k views

Estimating the probability that one Poisson RV is larger than another

Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function $$\mathbb P(X - Y ...
1
vote
1answer
76 views

Sampling with non-uniform costs

Suppose that I have a population, each represented by a bit $b_i$ for $i \in \{1,\ldots, n\}$. I would like to compute an estimate $\hat{B}$ to the statistic $B = \sum_{i=1}^nb_i$ so that with high ...
1
vote
3answers
3k views

Combining variances

I have a set of N bodies. The size of each body is being measured $m_i$ times ($m_i>1$ and different for each body). I would like to describe the resulting measurement. Particularly I'm interested ...
58
votes
13answers
11k 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 ...
4
votes
2answers
223 views

near independence of markov chain observations at high lags

I have to simulate independent draws from a very complicated distribution. They only feasible way appears to be using MCMC. I was considering running thousands of chains in parallel, but that would ...
4
votes
4answers
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 ...