Questions tagged [pr.probability]

Questions in probability theory

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18 views

Finding a game where the set of Correlated equilibria is different from the set of Coarse correlated equilibria

For the recent exercise of my Game Theory lecture I am asked to find a game where the set of Correlated equilibria (CE) is not equal to the set of Coarse Correlated equilibria (CCE). Because we know ...
3
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1answer
146 views

Proof and interpretation of the No Free Lunch theorem in data privacy

This question relates to a supposed counterexample to the No Free Lunch theorem governing data privacy mechanisms, as stated by Kifer et al (Section 2.1). Colloquially, the theorem states that no ...
3
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99 views

From coin flips to algebraic functions via pushdown automata

Background We're given a coin that shows heads with an unknown probability, $\lambda$. The goal is to use that coin (and possibly also a fair coin) to build a "new" coin that shows heads ...
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131 views

Probability distributions generated by pushdown automata

Background This question is about generating random variates, in the form of their binary expansions, on restricted computing models. Specifically, the computing model is based on pushdown automata (...
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1answer
73 views

Differential privacy definition: subset of range of values vs. equals a value in the range

Consider only $\epsilon$-differential privacy. The textbook definition for this is: Definition 1: "A randomized algorithm $\mathcal{M}$ with domain $\mathbb{N}^{|\chi|}$ is $\epsilon$-...
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143 views

What is the least compressible probability distribution? (under entropy constraint, for an expected squared error metric)

Consider a distribution $\mathcal D$ over the reals, a real parameter $H\in\mathbb R^+$, and an integer parameter $k\in\mathbb N$. The Entropy-Constrained Quantization problem asks what is the best ...
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151 views

"Looking for help understanding a proof by Gossner (1998)."

Although there is no use of cryptographic protocols in Gossner (1998), the author refers to protocols of communication and he has a main result that I struggle to prove, because he does not use a ...
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1answer
176 views

Young Diagrams and distinguishing between two distributions

Introduction: The reference for everything is this paper. The Robinson–Schensted–Knuth (RSK) algorithm is a well-known combinatorial algorithm with diverse applications throughout mathematics, ...
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98 views

converse form of chernoff bound

Suppoese $X_1,\ldots,X_n$ are i.i.d. random variables, all with mean $p$, and we have the statement that $Pr[\sum_i X_i\geq 0.8n]\geq 0.9$, can we use this to conclude anything about $p$ (in ...
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97 views

Using the probabilistic method to fill the gaps in a proof of set disjointness

In the 2-party $k$-sparse set disjointness problem, we have a set $U$ of size $n$ and there are two parties: Alice, who gets a set $X \subseteq U$ and Bob who gets a string $Y \subseteq U$, and it ...
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104 views

Does this sum-product algorithm make more sense?

Recall the belief propagation "sum-product" algorithm (from wikipedia): $\forall x_v\in Dom(v),\;$ $\mu_{v \to a} (x_v) = \prod_{a^* \in N(v)\setminus\{a\} } \mu_{a^* \to v} (x_v)$ $ \mu_{...
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5answers
322 views

Relationship between Random Graph Theory and TCS

Sorry for this large and vague question. I am a new grad probability student recently interested in random graph theory(RG). I heard from someone in math department that RG has close relationship to ...
4
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1answer
117 views

What is tightest known (VC-style) sample complexity bound for uniform convergence of empirical means?

The following result is adapted from Anthony and Bartlett, 1999 (Theorem 4.9). Theorem There exist positive constants $m_0 \le 400$, $c_1 \le 8$, $c_2 \le 41$, $c_3 \ge 1/576$ such that, if $(\Omega,\...
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44 views

Prune length distribution of random binary tree

Consider a random binary tree with $N$ leaves. Each node (except the root node) has a degree of exactly three (two children and one parent). No further restriction is placed on the structure of the ...
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147 views

Theorem on non-decreasing probability of success of an algorithm

Question: What's a standard name/framework for the following, or some variant? Stated briefly for a probabilistic algorithm: define $p_n$ as the conditional probability of success of a fork of the ...
3
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1answer
97 views

Conditioning Probability on a Language With Measure 0

Let $\Sigma = \{ 1, 2, \ldots, n\}$ be some alphabet. Assume that you have a coin with n-sides (each side corresponds to a letter in $\Sigma$), and we get each letter with equal probability. Now you ...
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110 views

Are there any known Chernoff/Hoeffding bounds for the case of "almost independence"?

The usual statement of a Hoeffding bound (e.g. https://sites.math.washington.edu/~morrow/335_17/ineq.pdf) requires independent random variables. My question is: Do there exist bounds similar to ...
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1answer
157 views

Effect of self loops on mixing time?

Consider 2 graphs G1 and G2. G1: Any non-regular graph. G2: Same graph but with added self-loops such that degree of each node is the same (either some $\Delta$, or maximum '$n$', where $n$ is the ...
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41 views

Rademacher complexity of k-fold maxima of hyperplanes

Aryeh Kontorovich wrote a technical report on 'Rademacher complexity of k-fold maxima of hyperplanes.' But its link (https://www.cs.bgu.ac.il/~karyeh/rademacher-max-hyperplane.pdf) is not accessible ...
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0answers
199 views

Power law for degree distribution of random KNN graphs?

Setup Sample random points from say standard Gaussian (or uniform) distribution in $R^{d}$ for some "d" and consider a KNN (K-nearest neigbour) graph for some K. Look at the degree ...
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133 views

Can a tractable sequence distribution prefer satisfying to unsatisfying assignments?

Let $p(x)$ be a Boolean circuit on $n$ bits $x \in \{0,1\}^n$. Consider a program that computes a probability distribution over all sequences $x$, autoregressively factored as $$\pi(x | p) = \prod_{0 ...
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Is there a known notion of "stochastic dependent pair"?

I came upon this when thinking about the semantics of probabilistic programs. Say you have a generative model N ~ Poisson() for n = 1:N X[i] ~ Normal() Then the ...
5
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1answer
179 views

Complexity of finding the most likely edge

Consider a connected, unweighted, undirected graph $G$. Let $m$ be the number of edges and $n$ be the number of nodes. Now consider the following random process. First sample a uniformly random ...
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1answer
105 views

Required sample size to hit certain subset of a ground set

Suppose $X$ is a set of $n$ points in $\mathbb{R}^d$ and $N_1,\cdots,N_k$ are k disjoint (unknown)subsets of $X$. There is a probability distribution $\phi$ on $X$ defined as $\phi(p) = \frac{\lvert\...
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43 views

understanding generalized coupon collector for distributions or learning mixture of distribution

Lets suppose we have a set $S=\{1,\ldots,n\}$ and $P$ is the uniform distribution over two subsets $T_1,T_2\subseteq S$, each of size $m\leq n/100$. Now, suppose somehow is given uniform samples from ...
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1answer
55 views

Notation in proof for Asymptotic Equipartition Property

In the following lecture notes chapter 3, page 12-13, they state the following We begin by introducting some important notation: - For a set $\mathcal{S},|\mathcal{S}|$ denotes its cardinality (...
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1answer
152 views

Binary search on coin heads probability

Let $f:[0,1] \to [0,1]$ be a smooth, monotonically increasing function. I want to find the smallest $x$ such that $f(x) \ge 1/2$. If I had a way to compute $f(x)$ given $x$, I could simply use ...
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3answers
2k views

"Almost all objects have property P" vs. "It is easy to test whether an object has property P"

I am interested in any relation between "almost all objects(from a universe) possessing a particular property P" versus "testing whether an object has property P being poly. time decidable". My guess ...
4
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0answers
84 views

Strong data-processing inequality: bound $TV(T_{\#}P_0,T_{\#}P_1)$ if $\|T(x)-x\|_\infty \le \varepsilon;\forall x \in \mathbb R^p$

Disclaimer. I've moved this question from MO hoping that here is the right venue. Also, this is my first post on this channel, so please have some patience. So, Iet $X = (X,d)$ be a Polish space, ...
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98 views

Weighted circular balls into bins

I would like to ask you for a help about modified balls into bins problem. Consider $n$ weighted balls such that each ball has a weight $w_i$ that is at most $W$. Furthermore, each of $m$ bins has a ...
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116 views

Majority function stability under deletion and addition of entries

It is well known that the majority function is stable under random flipping of bits. That is, if $v$ is a random binary vector, and then we re-sample each bit of $v$ with probability $\delta$ and get $...
4
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1answer
406 views

Maximization of Mutual Information

Let $X\in\{0,1\}^d$ be a Boolean vector and $Y, Z\in\{0,1\}$ are Boolean variables. Assume that there is a joint distribution $\mathcal{D}$ over $Y, Z$ and we'd like to find a joint distribution $\...
4
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1answer
254 views

How tight is the XOR lemma?

The XOR lemma states that if you have a distribution $D$ on $\{0,1\}^n$, and all the Fourier coefficients of $2^n D$ are small, then it is close in $L_1$ to the uniform distribution. Specifically, ...
3
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1answer
104 views

Reconstruction of a sequence generated by a Markov chain - reference request

Let S be a finite sequence of symbols from a finite alphabet, with gaps - that is on some known locations an unknown number of symbols are missing. Assuming that the sequence , including the symbols ...
4
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1answer
192 views

Why is differential privacy defined over the exponential function?

For adjacent database $D,D'$, a randomized algorithm $A$ is $\varepsilon$-differential private when the following satisfies $$\frac{\Pr(A(D) \in S)}{\Pr(A(D') \in S)} \leq e^\varepsilon,$$ where $S$ ...
6
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1answer
283 views

Big-O bounds on the k-th largest element of iid Gaussians

I'm interested in the following problem. Let $X_1, \dots, X_n$ be iid samples with a $N(0,1)$ distribution. Let $X_{[k]}$ be the $k$-th largest element of $\{X_1, \dots, X_n\}$, so e.g. $X_{[1]} = \...
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1answer
93 views

How to play the following game? (placing balls into bins)

Let $n,\ell\in\mathbb N$ for some $n\gg \ell\gg 1$. The goal is to pick two sequences of numbers, $x_1,\ldots,x_\ell$ and $y_1,\ldots,y_\ell$ such that $$\Sigma_{i=1}^\ell x_i = n\quad{}\mbox{and}\...
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0answers
65 views

Mapping of entire balls using Locality Sensitive Hashing (LSH)

LSH functions are useful for approximate nearest neighbor search. They are usually defined, for distance metric $d$ and $c>1$ as follows: A family of hash functions is $(r, cr, p_1, p_2)$-LSH ...
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2answers
175 views

If I know pretty well '(a,b)', I know pretty well 'a', or 'b', or 'a xor b'

I've a quite simple problem: let's imagine I have a couple of bits $(a,b) \in \{0,1\}^2$ sampled uniformly at random. Then, I give a function of these bits $f(a,b)$ (it can be any function, including ...
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0answers
58 views

Probability of detecting small bias of a die in the low confidence regime / balls and bins

We are given a biased $m$-sided die: one of the sides has probability $\frac{1}{m} + \gamma$ and all the rest have probability $\frac{1}{m} - \frac{\gamma}{m-1}$ each. The goal is to figure out which ...
5
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2answers
139 views

Statistical Distance Growth Given K Independent Copies

Let $X$ and $Y$ be distributions with statistical distance (total variation distance) at most $d$. What is the best upper bound you can give on the statistical distance between $k$ independent copies ...
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2answers
329 views

Infinite process balls in bins problem

Given $n$ balls and $m$ bins, let us consider an infinite process, where in each time slot we throw a ball at a random bin. When all $n$ balls are thrown, we take the balls from the bin with the ...
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45 views

Are there any continuous-time stochastic processes in which transition probabilities are discontinuous functions over time? [closed]

In stochastic processes, like homogeneous Markov processes, Poisson processes, Queueing systems etc., the functions that represent (transition) probabilities are continuous over time. This is also ...
8
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2answers
284 views

Is uniform convergence faster for low-entropy distributions?

Let $\mathcal D$ be a probability distribution on $\{0,1\}^d$. Let $X_1, \cdots, X_n \in \{0,1\}^d$ be i.i.d. samples from $\mathcal D$. Let $\mu \in [0,1]^d$ be the mean of $\mathcal D$ and let $\...
2
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1answer
115 views

Empirical Rademacher averages versus Hoeffdings bound

Let $M$ be finite set with $n$ distinct elements. I want to probalistically approximate the relative counts $\frac{|P(Q)|}{|M|}$ of $Q \subseteq M$, where $P(Q) = |P \cap M|$. An upper-bound for ...
2
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1answer
120 views

Sample Complexity for Order Statistics

I have a sample complexity question which seems fairly basic, but for which I'm having trouble finding a reference. Let $F$ be an unknown distribution over $[0,1]$. Denote by $X_{k:n}$ the $k$th of $...
2
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1answer
109 views

A coupon collector type problem with changing probabilities

Suppose we are flipping coins starting at some time $t$. At time $t$ the probability we obtain heads is $\frac{1}{\sqrt{t}}$. If the coin lands tails, at time $t+1$ the probability of heads is now $\...
2
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1answer
141 views

Recovering a rank-one matrix from its eigendecomposition after randomized rounding

Let $A = xy^T$ be a rank-$1$ matrix, and suppose every entry of $A$ is in $[0,1]$. We can create a binary matrix $A_{\rm rounded}$ by setting $$ [A_{\rm rounded}]_{ij} = \begin{cases} 1 & \mbox{ ...
4
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1answer
131 views

Probability of a $k$-path in a random graph

Assume that $G\in G(n,p)$; if $p=\frac{\ln n +\ln \ln n +c(n)}{n}$, the following fact is well known: \begin{eqnarray} Pr [G\mbox{ has a Hamiltonian cycle}]= \begin{cases} 1 & (c(n)\...
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51 views

Expected size of the min-cut, under edge perturbations

Suppose we have a graph $G(V, E)$. Assume that the min-cut of this graph is given $C=(A/B)$ and denote the size of the cut with $|C|$. Create a random modification of the graph: Drop each ...

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