Questions tagged [pr.probability]

Questions in probability theory

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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 ...
Gil Kalai's user avatar
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16 votes
0 answers
276 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 $...
Tyson Williams's user avatar
11 votes
0 answers
478 views

Cheeger's inequality for directed graphs?

Cheeger's inequality can be used to relate the size of the worst cut in the graph to the eigenvalue gap of a simple random walk on that graph. I am wondering if it possible to extend this result to ...
robinson's user avatar
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11 votes
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369 views

Is Bayesian updating computationally unfeasible?

Bayesian theory is a very popular theory of probabilities based upon a subjective framework of beliefs. However, subjects and beliefs have to be embodied, meaning to be feasible, it ought to be ...
Timmy's user avatar
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9 votes
0 answers
160 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 ...
Nav89's user avatar
  • 209
9 votes
0 answers
202 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 ...
Thomas Ahle's user avatar
9 votes
0 answers
318 views

Beating Nonuniformity by Oracle Access

Informally, we say that a Turing machine $M(\cdot)$ approximates a function $f(\cdot)$ if their outputs on a series of inputs are indistinguishable. More formally, let $L$ be a language, $M(\cdot)$ ...
Sadeq Dousti's user avatar
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8 votes
0 answers
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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 ...
R B's user avatar
  • 9,448
7 votes
0 answers
210 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 ...
Peter O.'s user avatar
  • 191
7 votes
0 answers
150 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 (...
Peter O.'s user avatar
  • 191
7 votes
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Convergence speed in Lévy’s continuity theorem

I suppose the answer to my question follows from the proof of the Lévy’s continuity theorem, but probably one could suggest me a direct reference to the corresponding answer. The question is as ...
Mikhail Berlinkov's user avatar
7 votes
0 answers
190 views

Testing the degree of a vertex

Let's say we have a graph $G$ with $n$ vertices. Given $\epsilon>0$ and a specific vertex $v$, consider the problem of deciding whether $\mathrm{deg}(v) < \frac{\epsilon}{3}n$ or $\mathrm{deg}(v)...
Belle's user avatar
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7 votes
0 answers
200 views

Tree search guided by a probabilistic oracle

I'm trying to find a solution for the following problem. I have a tree $T$ of branching factor $b$ and depth $d$. For the moment, I only care about the case where I restrict $b=2$, but I would be ...
Foo Barrigno's user avatar
6 votes
0 answers
122 views

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 ...
phipsgabler's user avatar
6 votes
0 answers
77 views

Fast convergence of a contagion process in special graphs

The process: Given is a clique $C_n$ of size $n$. Consider the following synchronous process, also known as the (synchronous) voter model (e.g., Even-Dar and Shapira): Define an indicator variable $...
JoelO's user avatar
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6 votes
0 answers
185 views

Squares of random binary matrices

Are there results/techniques pertaining to the analysis of squares of random matrices ? More specifically, let $A$ be an $n\times n$ matrix such that each entry is $1$ or $-1$ independently and with ...
NAg's user avatar
  • 666
6 votes
1 answer
274 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 ...
Yonathan B.'s user avatar
5 votes
0 answers
117 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 $...
Bartolinio's user avatar
5 votes
0 answers
131 views

Spectrum of absorbing random walk for regular graphs

I have a symmetric Markov chain given by the matrix $P$. Let $M$ be a set of special states of the chain (called marked states, they correspond to solutions to some problem). We can write $P$ as \...
costelus's user avatar
5 votes
0 answers
141 views

Sampling small separators

Is there a tractable way to define an approximate distribution over small vertex separators of the graph? I'm looking for something along the lines of [Bayati,2008], a way to turn a single "this is a ...
Yaroslav Bulatov's user avatar
5 votes
0 answers
317 views

Geometric / Visual explanation that the average height of a random binary tree of given size $n$ is asymptotically $2\sqrt{\pi n}$

I just finished reading the proof that the average height of a random binary of given size $n$ is asymptotically $2\sqrt{\pi n}$. I'm now searching for an intuitive, or geometric, or visual proof of ...
Clément's user avatar
  • 281
4 votes
0 answers
249 views

Maximize the mutual information between 2 discrete random variables

I have two random variables $X$ and $Y$. $X$ follows Poisson-Binomial distribution with parameters $\{q_1, \ldots, q_k\}$. Thus, $X$ can take values in the set $\{0,1,\ldots,k\}$. $Y$ is a binary ...
wanderer's user avatar
4 votes
0 answers
152 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 ...
zfkmz's user avatar
  • 187
4 votes
0 answers
107 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, ...
dohmatob's user avatar
  • 291
4 votes
0 answers
250 views

Sums of products of bernoulli random variables

Let $x_1 \ldots x_a,y_1 \ldots y_b$ be independent random variables taking values +1 or -1. Consider the sum $$S = \sum_{i,j} x_iy_j.$$ I wish to upper bound the probability $P(|S| > t)$. The ...
NAg's user avatar
  • 666
4 votes
0 answers
240 views

The balls and bins model: bounding the marginal contributions in the m>>n regime

Consider the standard balls and bins process, where $m$ balls are thrown into $n$ bins, and consider the case where $m >> n$. Denote the load on bin $i$ by the RV $L_i$. Given a set $S \...
JoelO's user avatar
  • 531
4 votes
0 answers
161 views

Concentration Bounds for Dependent Rounding

Consider the following random process which is defined on $n$ numbers $0\leq x_1,\ldots,x_n\leq 1$: At each step, pick an arbitrary number, say $x_i$. Then randomly (and independently) change its ...
afshi7n's user avatar
  • 271
4 votes
0 answers
70 views

Set-systems with some version of independence

Let $S \subset [N]$ be a fixed set of size $n$. Suppose $p$ is the probability that a random set $T$ of size $m$ intersects $S$ in $k$ or more points. That is, $$ \Pr_{\substack{T\subset [N]\\|T| = m}}...
Ramprasad's user avatar
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4 votes
0 answers
157 views

Adding noise to an assignment

Suppose I'm given a CNF formula with $m$ clauses (and $k$ literals in each clause), with a total of $n$ variables in the formula, where each variable is in at most $c$ clauses, along with a satisfying ...
shay's user avatar
  • 41
3 votes
0 answers
46 views

Approximate decomposition of a many-to-one assignment

Suppose we have $n$ items and $n$ agents and we want to assign one item to each agent. We have a probability matrix $P$ such that $p_{i,j}$ is the probability that agent $i$ gets item $j$. If $\sum_j ...
Erel Segal-Halevi's user avatar
3 votes
0 answers
107 views

Outputting true with probabiltiy $P(A|B)$ given $P(B), P(B|A)$, and a function which returns true with probability $P(A)$

I have a black-box function which returns true with probability $ P(A) $, that I don't know how to calculate. I receive evidence B, and I want to create a function which returns true with probability $...
Command Master's user avatar
3 votes
0 answers
259 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 ...
Alexander Chervov's user avatar
3 votes
0 answers
62 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 ...
Vitaly's user avatar
  • 881
3 votes
0 answers
239 views

Probabilistic sorting given pairwise comparison probability

Let $X = \{x_1, \dots, x_n\}$ be a set, and $f:[1..n]^2 \to [0, 1]$ be a function, such that $$f(i, j) \cdot f(j, k) \le f(i, k)$$ For all $1 \le i, j, k \le n$. Does there exist a randomized ...
Federico Lebrón's user avatar
3 votes
0 answers
78 views

One kind of dependence relation between a pair of random variables

I have been working on privacy and come across a neat problem. Suppose two random variables $X$ and $Y$, over finite alphabets $\mathcal{X}$ and $\mathcal{Y}$, are given with joint distribution $P_{...
SAmath's user avatar
  • 425
3 votes
0 answers
279 views

Probability distributions and computational complexity

Some probability distributions are easier to work with than others. Consider the following two problems. Given a number $n$, return $i$ with $0 \leq i < n$ with uniform probability, i.e. $\mathbb{...
Martin Berger's user avatar
3 votes
0 answers
93 views

A sampling and learning question

Suppose there is an oracle that returns a number $b \in \mathbb{Z}_{n}$ whenever I press the button. We have $b = a + e$, where $a \in \mathbb{Z}_n$ is a fixed number and $e$ is sampled according to ...
Richard's user avatar
  • 31
3 votes
0 answers
112 views

Sampling from a distribution with a given covariance matrix

Let $\bf X$ be a binary vector of $n$ (non-independent) random variables $X_1,\ldots, X_n$. Covariance of two random variables is defined as follows: $$\mathrm{cov}(X_i, X_j) = \mathrm{E}(X_i - \mu_i)...
avsmal's user avatar
  • 251
2 votes
0 answers
53 views

Approximately sampling from a discrete unimodal distribution with large support

I have an algorithmic problem and I am curious if a solution is known in the literature, because I cannot find it. I came up with an algorithm of my own, but would be curious if something is known. I ...
user2316602's user avatar
2 votes
0 answers
165 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 ...
fgrieu's user avatar
  • 109
2 votes
0 answers
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 ...
Geoffrey Irving's user avatar
2 votes
0 answers
57 views

Is there a complete and finite axiom scheme for conditional independence? (Graphoids)

Note: This is a better-written version of an unanswered question asked before on MathOverflow. Question: Is there a complete and finite axiom scheme for conditional probability? If so, is there a ...
Chill2Macht's user avatar
2 votes
0 answers
63 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 ...
rajatsen91's user avatar
2 votes
0 answers
73 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 ...
rajatsen91's user avatar
2 votes
0 answers
196 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 ...
rajatsen91's user avatar
2 votes
0 answers
115 views

On a possible probabilistic log-rank conjecture

The communication matrix of a function $f$ is a $2^{n}\times 2^{n}$ matrix $M_f$ where the indices of the rows (columns) correspond to the inputs of Alice (Bob), and each entry $M_f (a,b)$ is the ...
Turbo's user avatar
  • 12.8k
2 votes
0 answers
131 views

Strong Dependence

I asked this question on MO, but no answer. I don't know if this definition has been already given. Suppose $X$ and $Y$ are two random variables over finite alphabets $\mathcal{X}$ and $\mathcal{Y}$...
SAmath's user avatar
  • 425
2 votes
0 answers
228 views

What is the optimal strategy for this 2 player game?

Let some finite array of integers is given initially. There is a number k which is initially '0'. In a move, a player will select a number from the array arr[i] and change k to gcd(k,arr[i]). Also, ...
user2573642's user avatar
2 votes
0 answers
122 views

How to find a merge tree for a set of words?

Consider a set $S \subseteq \Sigma^n$ where $\Sigma$ is a finite alphabet and $p : \Sigma \rightarrow [0,1]$ is a probability function. Let $T$ be a tree leaf-labeled by the elements of $S$. Consider ...
NisaiVloot's user avatar
  • 1,292
2 votes
0 answers
98 views

Convergence of iterated stochastic matrices

(This question has been unsuccessfully asked on mathoverflow: https://mathoverflow.net/questions/161728/convergence-of-iterated-stochastic-matrices) It is well-known that for a stochastic aperiodic ...
Nathanaël Fijalkow's user avatar