Questions tagged [pr.probability]

Questions in probability theory

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7
votes
1answer
136 views

Random Deterministic Automata

I am familiar with the term of random graphs, such as $G(n,p)$- a distribution over simple undirected graphs with $n$ vertices, where each edge appears in a graph w.p. $p$. That is, each graph $G=(V,E)...
1
vote
0answers
50 views

Problem dependent lower bound for stochastic bandits with full information

Suppose you have a $K$ armed stochastic bandit problem but with full information. There are $K$ arms with mean rewards $\mu_1,...,\mu_K$. At each step we have to select an arm, collect the reward from ...
3
votes
2answers
210 views

Hardness of exact binomial tail bounds

Chernoff bounds, in their various forms, bound the tails of a Binomial$(n,p)$ random variable $B$. Define the function $F(n,p,t):=P(B>t)$. Naively, computing $F$ requires exponential (in $n$) time. ...
6
votes
1answer
217 views

A random walk that moves to less-visited nodes

Consider a random walk on an undirected graph that keeps track of how many times it has visited every node. At each step, it moves to the node among its neighbors which has been visited the least ...
4
votes
1answer
451 views

Relation between variance and mutual information

Given two discrete random variables $X,Y$ such that $X,Y \in \mathbb{R}$ and $0 \leq X,Y \leq 1$, is it true that $$|\text{Cov}[X,Y] \leq \sqrt{\frac{1}{2} \text{I}[X,Y]}|. $$ This bound may be ...
6
votes
1answer
127 views

Directed graph with bounded in-deg can be partitioned in a balanced way

I want to prove that for all $n$, there exists a constant $c(n)$ such that if $G=(V,E)$ is a directed graph with in-degree bounded by $n$, it is possible to partition the set of vertices $V$ into two ...
9
votes
0answers
162 views

Fourth(?) moment method for minimum value

I would like to lower bound the quantity $\Pr[X\ge t, Y\ge t]=\Pr[\min(X,Y)\ge t]$ using the small moments of $X$, $Y$ and $XY$. In particular I am interested in the case where $E[X]=E[Y]=0$, but $E[X ...
6
votes
2answers
481 views

Reference for the number of samples needed to distinguish two probability distributions

I am looking for a reference (and/or a full proof) for this statement: $O(1/\epsilon^2)$ samples suffice to distinguish any two probability distributions with variation distance $\epsilon$. I ...
3
votes
1answer
164 views

Lower bound on probability of getting two close points in a sample of $n$ points

Let $D\subseteq \mathbb{R}^k$. Assume that $Pr_{u,v\leftarrow U_D}[|u-v| <B]>p$ for some $B,p$, where $|u-v|$ is the L1 distance of the vectors. $S\subseteq D$ is obtained by sampling $n$ ...
7
votes
2answers
268 views

What is the connection between moments of Gaussians and perfect matchings of graphs?

Today, I heard the following statement in a talk: The 4th moment of a $1$-dimensional Gaussian distribution with mean $0$ and variance $1$ is the same as the number of perfect matchings of a ...
2
votes
1answer
90 views

randomness extraction of real valued sequences of numbers

I have a sequence of numbers $x_1, x_2, \dots, x_n, \dots \in \mathbb{R}$ I would like to extract fair bits from that sequence. My first thought was to use the Von Neumann extractor. For a ...
3
votes
3answers
171 views

Approaches for Theoretical Analysis of Estimates of Probability Distributions

Consider that you have a probability distribution p of some quantity X and you have obtained (via some algorithm) an estimate q of p according to some definition of closeness. Are there methods/papers ...
6
votes
1answer
342 views

Understanding proof of Theorem 3.3 in Karp's “Probabilistic Recurrence Relations”

Background: In Karp's paper on Probabilistic Recurrence Relations, he develops tail-bounds for random variables satisfying the following recurrence: $$ T(x) = a(x) + T(h(x)) $$ where $T(x)$ is a ...
6
votes
1answer
260 views

Expected Kolmogorov complexity under Kolmogorov complexity distribution

If $K(w)$ is the Kolmogorov complexity of a string $w$, where programs are prefix-encoded so $\sum_{w} 2^{-K(w)} \leq 1$, what is $$\lim_{n\to\infty} \frac{\sum_{|w|=n}2^{-K(w)} K(w)}{\sum_{|w|=n} 2^{...
2
votes
1answer
160 views

Probability of random variable $X$ less than $max(Y_i)$

For $\lambda > \mu$, consider independent random variables (i) $X \sim Pois(\mu)$ and (ii) $Y_i \sim Pois(\lambda),\;i\in{1,2}$. Question : Can we upper-bound the following? $$\mathbb{P}\big(X\...
7
votes
1answer
316 views

Minimum Spanning tree on a complete “random” graph

Consider a complete undirected graph with $n$ vertices, $K_n$. Let weight of an edge between vertices $i\; \& \;j$ be a random variable $E_{ij}$. Let $E_{ij} \sim exp(\lambda)$, where $exp(\lambda)...
9
votes
2answers
679 views

Statistical distance between uniform and biased coin

Let $U$ be the uniform distribution over $n$ bits, and let $D$ be the distribution over $n$ bits where the bits are independent and each bit is $1$ with probability $1/2-\epsilon$. Is it true that ...
1
vote
1answer
81 views

Issue with inequalities involving probablities

I'm reading this paper http://eccc.hpi-web.de/report/2016/044/download/, at page 7 you can see this equation: $$ \sum_t \mathbb{E}_{w \in W} [|p_t - p_{t|w,w}|] = 2 \sum_t \mathbb{E}_{w \in W} [\max(...
9
votes
2answers
260 views

Can the “mutual independence” condition in the Lovász local lemma be weakened?

The Lovász local lemma, as stated in Corollary 5.1.2 here, is given as follows. Lemma. Let $A_1, \ldots, A_k$ be events such that each $A_i$ has probability at most $p$ and such that each $A_i$ is ...
22
votes
2answers
2k views

Bounds on $E[f(x)]$ in terms of $f(E[x])$ other than Jensen's inequality?

If $f$ is a convex function then Jensen's inequality states that $f(\textbf{E}[x]) \le \textbf{E}[f(x)]$, and mutatis mutandis when $f$ is concave. Clearly in the worst case you cannot upper bound $\...
2
votes
1answer
89 views

Subspace-evasive set performance in the random case

A subspace evasive set is defined as a large subset of a vector space which has small intersection with any $k$ dimensional affine space. That is, it "evades" all affine subspaces of small enough ...
1
vote
1answer
106 views

The noise distribution on $F_2^n$: probability of landing in a subspace versus a coset

Let $V = F_2^n$ be the $n$-dimensional vector space over the field of two elements. The $\epsilon$-noise distribution on $V$, denoted $\mu_\epsilon$, is a probability distribution on $V$ for which ...
1
vote
0answers
98 views

Percolation probabilities

I have a finite undirected graph with a probability $p_e$ given for each edge $e$. This gives a random graph by removing each edge e with probability $1-p_e$ independently of the others. I'm ...
12
votes
6answers
749 views

Computing the approximate population of a bloom filter

Given a bloom filter of size N-bits and K hash functions, of which M-bits (where M <= N) of the filter are set. Is it possible to approximate the number of elements inserted into the bloom filter? ...
14
votes
1answer
418 views

Expected minimum influence of a random Boolean function $f\colon\{-1,1\}^n \to \{-1,1\}$

For a Boolean function $f\colon\{-1,1\}^n \to \{-1,1\}$, the influence of the $i$th variable is defined as $$ \operatorname{Inf}_i[f] \stackrel{\rm def}{=} \Pr_{x\sim\{-1,1\}^n}[ f(x) \neq f(x^{\oplus ...
1
vote
1answer
157 views

Policy Adjustment in Markov Decision Process

I was using MDP on my work to make optimal decision. I used discrete time, finite state MDP. I assumed that I will have an initial parameters, like the Reward/Cost, state transition probabilities and ...
1
vote
0answers
139 views

Maximal correlation vs correlation coefficient when one RV is Gaussian

Last week I asked a question on MOF (see here), but I got no reply. So I am asking my question here. Let a pair of random variables $(X,Y)$ be continuous random variables (i.e., they both have ...
31
votes
6answers
4k views

Reverse Chernoff bound

Is there an reverse Chernoff bound which bounds that the tail probability is at least so much. i.e if $X_1,X_2,\ldots,X_n$ are independent binomial random variables and $\mu=\mathbb{E}[\sum_{i=1}^n ...
16
votes
0answers
256 views

When does adding edges decrease the cover time of a graph?

When first learning about random walks on a graph $G$, one may have an intuitive feeling that adding edges to $G$ will decrease its cover time $C(G)$. However, this is not the case. The path graph $...
1
vote
1answer
121 views

Average Regret Bounds for Linear Stochastic Bandits

I am reading this paper on linear stochastic bandits : http://papers.nips.cc/paper/4417-improved-algorithms-for-linear-stochastic-bandits.pdf All the results are stated in a high-probability ...
8
votes
1answer
653 views

Exponential Concentration Inequality for Higher-order moments of Gaussian Random Variables

Let $X_1,\ldots, X_n$ be $n$ i.i.d. copies of Gaussian random variable $X \sim N(0, \sigma^2)$. It is known that \begin{align} \mathbb{P}\Bigl( \Bigl|\frac{1}{n}\sum_{j=1}^n X_j \Bigl| >t\Bigr) &...
6
votes
1answer
229 views

Graph that maximizes minimum hitting time?

Let $G$ be some connected bidirectional (or undirected) graph. We define a random walk as a walk that begins at a vertex chosen uniformly at random, and at each step proceeds to one of its current ...
4
votes
1answer
1k views

Paths and Probabilities for a Random Walk on a Graph

I'm working on a problem about $N$ nodes that are randomly positioned on a rectangular grid. I want to take a sample of $n\leq N$ nodes by randomly selecting the first node then visiting the nearest ...
4
votes
1answer
663 views

Orlicz norm of random variable and variance

In probability and statistics Orlicz norms are frequently used in concentration inequalities. For example, for Bernstein's inequality, we have versions for sub-exponential random variables using $\...
2
votes
1answer
261 views

Squared Hellinger distance between Binomial(n,p) and Binomial(n+1,p)

I came across a claim in a paper Trace reconstruction revisited (in Lemma 11), which is basically that the squared Hellinger distance between Binomial$(n,p)$ and Binomial$(n+1,p)$ grows as $O(1/n)$. $...
5
votes
1answer
253 views

About a random walk with constant spectral gap

The existence of a graph with certain strange-looking properties will imply a counterexample to something I am playing with. I'm stuck on figuring out whether such a graph can exist and so I thought I'...
6
votes
1answer
156 views

Are bins with more black than white balls negatively 1-correlated?

Suppose we throw $m_b$ black balls and $m_w$ white balls independently and uniformly at random in $n$ bins. Let $\left\{ B_i \right\}_{i \in \left\{ 1,\ldots,n \right\}}$, $\left\{ W_i \right\}_{i \...
8
votes
1answer
157 views

A bounded-independence variant of the Berry-Esseen theorem

I came across a presentation by Ryan O'Donnell regarding invariance principles. After proving the Berry-Esseen theorem, there is a slide that discusses extensions of the theorem and one that is ...
1
vote
1answer
92 views

Inequality for p-norm of probability distributions

I want to show the following inequality, which seems like it should have an elementary proof (or even a well-known name). Suppose $p, q$ are discrete probability distributions. And suppose that $p_i \...
1
vote
0answers
62 views

Computing the distribution from which this algorithm samples from

Assume we have a set of integers $X_0=\{x_1\ge x_2\ge\ldots\ge x_n\}$. Let $r\in(0,1]$ be a parameter and consider the ranking process: i=0 while ($X_i\ne\emptyset$) let $M = \max \{x\in X_i\}$ (...
21
votes
1answer
531 views

Number of distinct differences of $\omega(\sqrt{n})$ integers chosen from $[n]$

I encountered the following result during my research. $$\lim\limits_{n\to \infty} \mathbb{E}\left[ \frac{\#\{|a_i-a_j|,1\le i,j\le m \}}{n} \right] = 1$$ where $m=\omega(\sqrt n)$ and $a_1,\...
7
votes
1answer
693 views

Reservoir sampling of distinct values

I'm faced with a situation where I need to get m samples from a data stream (without replacement). Only one pass through the data is possible. In my case, the stream contains many duplicate values, ...
-2
votes
2answers
237 views

Lemma needed for my machine learning research [closed]

Say $\sigma_1, \sigma_2, \dots, \sigma_m$ are i.i.d distributed $\pm1$ variables. How do I show that for any choice of $S_1, S_2, \dots, S_d$ subsets of $\{1, 2, \dots, m\}$, the expectation of the ...
2
votes
0answers
68 views

Concentration Bounds for Thompson sampling

This paper gives concentration results around the mean of the regret for variants of UCB algorithm in multi-armed stochastic bandits. However, I could not find any similar results for Thompson ...
19
votes
0answers
496 views

Are monotone Boolean functions in P well-approximated by monotone polynomial-size circuits?

Question 1: Is it true that for every polynomial $p(n)$ and $\epsilon >0$ there is a polynomial $q(n)$ such that every monotone Boolean function on $n$ variables that can be expressed by a Boolean ...
2
votes
1answer
135 views

Turing degree of Solomonoff semi-measure

We define the Solomonoff semi-measure $m$ on finite strings $x$ by $$m(x) = \sum_{p: U(p) = x} 2^{-l(p)},$$ where $U$ is a universal prefix Turing machine, $U(p) = x$ means $U$ outputs $x$ on input $...
2
votes
0answers
146 views

High-Probability bounds for Stochastic multi-armed Bandit Problems

This paper gives some high probability results for UCB algorithm of the form, \begin{align} \mathbb{P}(R_{n} > r_0.x) \leq O(n^{-2\rho x + 1} + x^{-2 \rho + 1} ) \end{align} where $\rho$ is a ...
4
votes
3answers
210 views

Approximating distributions from samples

One claim I find in many papers about identity testing, and closeness testing is that any distribution over $[n]$ can be approximated to within $\ell_1$ distance $\epsilon$ in $O\left(\frac{n}{\...
4
votes
1answer
598 views

Example of pairwise independent random process with expected max load $\sqrt{n}$

This question was previously posted at https://math.stackexchange.com/questions/1220292/example-of-pairwise-independent-random-process-with-expected-max-load-sqrtn where it has no answers. I now ...
10
votes
3answers
533 views

Is there any known CCC closed under a probabilistic powerdomain operation?

Equivalently, is there a known denotational semantics for probabilistic higher-order functional programming languages? Specifically, is there a domain model of pure untyped $\lambda$-calculus extended ...