# Tagged Questions

**2**

votes

**1**answer

59 views

### Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law)

I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular Teh's hierarchical LM based on PY processes. A motivation frequently stated in research ...

**5**

votes

**3**answers

258 views

### A. Markov's papers?

A. Markov published several papers on his chains, starting in 1906, so it is written, in the journal:
(1) Извѣстія Физико-математического общества при Казанском университете
I am surprised by the ...

**2**

votes

**2**answers

235 views

### Distribution of a random walk on a directed line

Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line?
$x_0 = n$
$x_t$ is a uniformly random integer between 1 and ...

**2**

votes

**1**answer

164 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

**0**answers

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

**0**answers

50 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

**3**answers

231 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 ...

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vote

**0**answers

59 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

**1**answer

134 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

**2**answers

270 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**

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**0**answers

95 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

**0**answers

93 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

**0**answers

260 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

**2**answers

728 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

**1**answer

421 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

**0**answers

107 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

**0**answers

108 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

**1**answer

266 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

**3**answers

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

**2**answers

283 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

**1**answer

264 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

**1**answer

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 ...

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**0**answers

101 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

**0**answers

494 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

**1**answer

127 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

**1**answer

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

**0**answers

174 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

**4**answers

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

**1**answer

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 ...

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vote

**3**answers

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 ...

**59**

votes

**13**answers

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

**2**answers

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

**4**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 ...