All Questions
Tagged with pr.probability st.statistics
1,135 questions
1
vote
1
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341
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Form of minimax estimator
Let $\Delta$ be the set of all probability distributions over $\mathbb{N}=\{1,2,\ldots\}$ and fix some $\mathcal{P}\subseteq\Delta$.
Suppose additionally that $\Delta$ is endowed with some norm $||\...
- 6,541
2
votes
0
answers
84
views
Concentration result for self-normalized empirical process
In Theorem 1.1 of this paper by Bercu, Gassiat and Rio, a concentration result is derived for the 'self-normalized' empirical process. Specifically, suppose that $(X,X_n)_{n \ge 1}$ is a sequence of i....
- 206
1
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2
answers
221
views
Beating the $1/\sqrt n$ rate of uniform-convergence over a linear function class
Let $P$ be a probability distribution on $\mathbb R^d \times \mathbb R$, and let $(x_1,y_1), \ldots, (x_n,y_n)$ be an iid sample of size $n$ from $P$. Fix $\epsilon,t\gt 0$. For any unit-vector $w \in ...
- 6,853
4
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1
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287
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Local maxima of the sum of Gaussian functions in *multiple dimensions* are always strict local maxima - prove/disprove/prove conditionally?
This is a follow up of the question in one dimension, that asked to show that the all the maxima of the sum of Gaussian
$$f_n(x):= \sum_{i=1}^{n}e^{-(x-x_i)^2}, x_1 < x_2 < \dots < x_n$$
are ...
- 1,467
2
votes
1
answer
122
views
Analytical solution for a double integral involving logistic functions and Gaussian distributions
I am working on a mathematical problem involving the evaluation of a double integral, and I am seeking an analytical solution or techniques to solve it. The integral I'm dealing with is as follows:
$$...
- 31
0
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1
answer
231
views
Concentration inequalities for random sampling without replacement
Let a population $C$ consist of $N$ values $c_1, c_2, \cdots, c_N$, with $c_i\in \{0,1\}$. Let $X_1, X_2, \cdots, X_n$ denote a random sample without replacement from $C$ and let $Y_1, Y_2, \cdots, ...
- 105
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156
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Known relations between mutual information and covering number?
This is a question about statistical learning theory. Consider a hypothesis class $\mathcal{F}$, parameterized by real vectors $w \in \mathbb{R}^p$. Suppose I have a data distribution $D \sim \mu$ and ...
- 141
6
votes
2
answers
775
views
Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped
I have a weird question which probably seems out of place here but it has proven more difficult than anticipated. I am going to describe the game without showing work toward a solution. Numerically, ...
user504691
0
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0
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82
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High probability bound on number of sparse solutions to Gaussian linear system
Suppose we have a random matrix $A \in \mathbb{R}^{m \times n}$ with all entries i.i.d. from the standard Normal distribution $\mathcal{N}(0, 1)$. Suppose $k$ divides $n$, and let $S \subseteq \mathbb{...
- 43
1
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1
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69
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Expected value of MGIG distribution
I'm currently dealing with a Gibbs sampler of the multivariate generalized inverse Gaussian distribution (MGIG). In order to check the correctness of the sampler, I'd like to know the expected value ...
- 2,560
0
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0
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42
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Limiting value of trace of resolvent matrix involving two independent Wishart random matrices
Let $n_1$, $n_2$, and $d$ be positive integers tending to infinity such that
$$
d/n_k \to \phi_k \in (0,\infty).
$$
Let $X_1 \in \mathbb R^{n_1 \times d}$ and $X_2^{n_2 \times d}$ be independent ...
- 6,853
4
votes
1
answer
223
views
Existence of disintegrations for improper priors on locally-compact groups
In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...
- 8,512
2
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1
answer
187
views
Law of iterated logarithm for quadratic variation of Brownian motion
Let $(\Omega, \mathcal{F}, \mathbb{P})$ denote a probability space supporting a standard Brownian motion $B$. Let $\Pi=\{\pi_n : n \ge 0\}$ denote the sequence of dyadic uniform partitions of the ...
- 41
23
votes
7
answers
5k
views
What makes Gaussian distributions special?
I'm looking for as many different arguments or derivations as possible that support the informal claim that Gaussian/Normal distributions are "the most fundamental" among all distributions.
...
1
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0
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148
views
conjecture for general form of minimax estimator
I had previously posed an overly ambitious version of this conjecture here,
Form of minimax estimator,
which was quickly shot down by Václav Voráček (on twitter) and Iosif Pinelis (MO answer in the ...
- 6,541
0
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0
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89
views
Stein's Lemma for conditional expectation?
Let $X=(X_1,\ldots,X_d)$ be a standard normal random vector in $\mathbb R^d$, let $m:\mathbb R^d \to \mathbb R$ be a function, and let $E=E_m$ denote the expectation operator conditioned on $m(X) > ...
- 6,853
0
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1
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116
views
Existence and uniqueness of a posterior distribution
I am wondering about the existence and uniqueness of a posterior distribution.
While Bayes' theorem gives the form of the posterior, perhaps there are pathological cases (over some weird probability ...
6
votes
1
answer
275
views
Spectrum asymptotics for a product of $k$ random matrices?
How does the spectrum of a product of $k$ random matrices behave around 0?
In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k=...
- 2,252
1
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0
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72
views
Dimension-free sample complexity for the inverse of Gaussian sample covariance?
Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need the inverse of the sample covariance $\Sigma_m^{-1}$ to be $\varepsilon$-close to true inverse covariance $\Sigma^{-1}$ (in ...
- 517
1
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0
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62
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A small lemma on cache resets (Bloom filters in particular)
Assume a fixed set of message $D$ and an associated distribution for selecting each message $d_i$ such that the total probability $\sum_{i \in D} d_i = 1$. We create a cache with $M$ bits and $k$ ...
0
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1
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107
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Bounding $\|X_1/(X_1+X_2) - Y_1/(Y_1+Y_2)\|_p$ by the closeness of $X$ and $Y$
This question is inspired by the answer to this other question, but I have tried to make it self-contained and to zoom in on the counter-example from this answer.
Suppose $\{(X_n, Y_n)\}_{n=1}^2$ are ...
- 15
14
votes
8
answers
3k
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Relevant mathematics to the recent coronavirus outbreak
I would like to ask about (old* and new) reliable mathematical literature relevant to various mathematical aspects of the recent coronavirus outbreak: In particular, standard statistical/mathematical ...
1
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1
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146
views
Upper bound on difference of correlated ratios
Suppose $(x_n, y_n)$ are i.i.d. samples (that is, $x_n$ and $y_n$ are not independent, but $(x_n, y_n)$ is i.i.d. with regards to $(x_m, y_m)$ if $n\ne m$) from a joint distribution, with $0 < x_n, ...
- 15
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0
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50
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Approximate CDF of integral using the Berry-Esseen theorem
I'm trying to approximate CDF of the integral $$\frac{1}{T}\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt,$$
where $W_t$ is the Wiener process, i.e. $W_t\sim N(0,t)$.
For this I use ...
- 49
3
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2
answers
226
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Recovering measure support from the sequence of I.I.D random variables
Assume we have a Borel probability measure $\mu$ in $\mathbb{R}^n$ and a sequence of $\mu$ distributed I.I.D. random variables $x_n$. Is there a limit formula for $supp(\mu)$, something like closure ...
- 728
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1
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166
views
Does this inequality hold for the cumulant generating function?
Suppose a random variable $X$ is zero-mean and the cumulant generating function is
$$
K\left( t \right) =\log \mathbb{E}[e^{tX}].
$$
Given any positive constant $\tau > 0$, does this inequality
$$
\...
- 49
1
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1
answer
191
views
Concentration inequality for square roots
Given a sequence of (not-necessarily-iid) real-valued random variables $X_n$ that converge to $a\in\mathbb{R}$ in probability, suppose we have an exponential concentration inequality of the form
$$
P(|...
- 13
5
votes
1
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516
views
Bounding the variance of a truncated Gaussian random variable
Suppose $X_1, X_2, X_3 \sim N(0, 1)$ are three independent standard normal random variables. I am interested in showing that:
$$\text{Var}[X_2\mid X_2 \geq X_1 - a, X_1 \leq X_3 + b] < 1,$$
where ...
- 269
3
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1
answer
165
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Why is the logistic regression model good? (and its relation with maximizing entropy)
Suppose we're trying to train a classifier $\pi$ for $k$ classes that takes as input a feature vector $x\in\mathbb{R}^n$ and outputs a probability vector $\pi(x)\in\mathbb{R}^k$ such that $\sum_{v=1}^...
- 7,527
7
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1
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569
views
Singular Fisher information matrix and existence of unbiased estimators
I'm doing some research into the Cramer-Rao bound for time of arrival localization and have come across a rather strange result: the FIM is singular, but there exists an unbiased estimator. My ...
- 75
0
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1
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107
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What's the lower bound for this quantity?
Suppose $p$ is a discrete distribution with $n$ values and the random variable $x$ satisfies $\mathbb{E}_p[x] = 0$ and $|x| < \infty$. Given $\alpha \in (0,1)$, does there exist a lower bound for ...
- 49
0
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0
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140
views
Bounding the difference of weighted-mean estimations
I recently asked a question related to a proof I'm working on as well as a follow-up to that. The detailed counter-examples I received elucidated a lot, but it quickly transpired (i) that I was ...
- 15
2
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0
answers
92
views
Construct a Bregman divergence from Wasserstein distance
I was wondering whether one has studied the Bregman divergence arising from a squared Wasserstein distance.
More precisely, let $\Omega\subset \mathbb{R}^d$ be a compact set and $c\in \Omega\times \...
- 503
49
votes
13
answers
24k
views
Why is it so cool to square numbers (in terms of finding the standard deviation)?
When we want to find the standard deviation of $\{1,2,2,3,5\}$ we do
$$\sigma = \sqrt{ {1 \over 5-1} \left( (1-2.6)^2 + (2-2.6)^2 + (2-2.6)^2 + (3-2.6)^2 + (5 - 2.6)^2 \right) } \approx 1.52$$.
Why ...
- 673
91
votes
8
answers
16k
views
Is there a natural random process that is rigorously known to produce Zipf's law?
Zipf's law is the empirical observation that in many real-life populations of $n$ objects, the $k^\text{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $...
- 114k
0
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2
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261
views
What mathematical formalism might be used to disprove natural selection, on the basis that there are too many independent genetic parameters? [closed]
I have nagging doubts that the random genetic mutation process of natural selection is sufficient to explain evolution, even when coupled with sexual selection (Darwin proposed that evolution is ...
- 41
1
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2
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2k
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Upper bound about Gaussian tail bound
From the definition of sub-Gaussian distribution $X$ w.r.t. $\sigma$ i.e.
$$\mathbb{P}(|X-\mathbb{E}(X)|\geq t) \leq 2 \exp(-\frac{t^2}{2\sigma^2}).$$
It's natural that when $X \sim \mathcal{N}(\mu, \...
- 81
1
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1
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153
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Minimax estimation rate of sparse vector $w_\star$, w.r.t to mixed norm $\|\hat w_n-w_\star\| := \|\hat w_n - w_\star\|_2 + \|\hat w_n-w_\star\|_q$
Let $n,d,s$ be positive integers with $s \le d$, and let $B_0(d,s)$ be the set of all (real) $d$-dimensional vectors with at most $s$ nonzero components. Given an $n \times d$ matrix $X$ with rows $...
- 6,853
1
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2
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336
views
Anti-concentration of gaussian variable
Let $X$ be $\mathcal{N}(\mu,\sigma^2)$ gaussian. Its expectation $\mu$ is positive. Can we derive a lower bound on
$$\mathbb{P}(X\geq\epsilon)\geq g(\epsilon,\mu,\sigma) \text{ where } \epsilon\leq\mu$...
- 405
0
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0
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55
views
Modeling player interactions in multi-dimensional rating systems
In traditional rating systems (such as Elo), a player's strength is represented by a single scalar value, which is assumed to be consistent across different opponents. However, in some games, the ...
- 1
1
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0
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144
views
Estimator for the conditional expectation operator with convergence rate in operator norm
Let $X$ and $Z$ be two random variables defined on the same probability space, taking values in euclidian spaces $E_X$ and $E_Z$, with distributions $\pi$ and $\nu$, respectively.
Let $L^2(\pi)$ ...
- 111
9
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1
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725
views
Popular mistakes in probability
$\DeclareMathOperator\Var{Var}\DeclareMathOperator\Bern{Bern}\DeclareMathOperator\Pois{Pois}$Question: What not-trivial mistakes do students often make when solving problems in probability theory, ...
1
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0
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57
views
Limiting value of expectation of trace of truncated Gram matrix
Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, ...
- 6,853
1
vote
0
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67
views
Random matrix theory: accounting for mean
Assume a random matrix, denoted as $X$, which is an $n$ by $T$ matrix, $T\geq n$. While I understand the typical scenario where the random variables $X_{ij}$ are sampled from a $\mathcal{N}(0,\sigma_{...
3
votes
1
answer
355
views
Quantitative results (with formal proof) on the median approximation of Chi-squared distribution
It is well-known that Chi-squared distribution $X_n$ with degree-$n$ freedom has an approximate formula for its median as $n\left(1-\frac{2}{9n}\right)^3$. Or $(X_n/n)^{\frac{1}{3}}$ is approximately ...
- 457
1
vote
0
answers
112
views
A high probability bound for a Rademacher process
Let $\{x_i(t)\}_{i=1}^n$ be i.i.d. Gaussian processes for $t \in [0, T]$ with
\begin{align*}
\mathbb{E}[x_i(t)] & = 0, \quad \forall i \in [1 : n], \ t \in [0, T] \\
\mathbb{E}[x_i(s) x_i(t)] &...
- 81
1
vote
1
answer
415
views
Approximate the singular values of a certain random dot-product kernel matrix (in the sense of El Karoui, Cheng-Singer, etc.)
Let $g:\mathbb R \to \mathbb R $ be a continuous function which is
"sufficiently smooth" (e.g $\mathcal C^3$) around $0$, and
"sufficiently integrable" (e.g integrable w.r.t $N(0,...
- 6,853
4
votes
1
answer
258
views
When is $\prod_{i=0}^\infty (I-x_i x_i^T)=0$ for isotropic Gaussian $x_i$?
Suppose $x_i$ is sampled IID from isotropic zero-centered Gaussian random variable in $d$ dimensions with covariance $\Sigma=c*I$. When is the following true with probability 1?
$$\prod_{i=0}^\infty (...
- 2,252
2
votes
1
answer
416
views
Bounding Kullback-Leibler
Suppose we have a probability distribution $P$ on a finite set $S$. We draw $N$ i.i.d. samples according to $P$ and use these samples to define an empirical distribution $R$. We measure the Kullback-...
- 3,979
2
votes
2
answers
323
views
Continuity of Nash equilibrium for a family of games
The question may be too vague, but ultimately in search of various (counter)examples or theorems to exhibit the following:
Do continuous families $t\mapsto G_t$ of "games" (say each $G_t$ is ...
- 17.5k