All Questions
16 questions
23
votes
1
answer
767
views
The Euler-Mascheroni constant and entropy
I would like to know if I have discovered or merely rediscovered the following pretty fact.
A partition of $[0,1]$ into intervals of lengths $p_{i, i=1\ldots n}$ induces a probability distribution ...
- 17.6k
14
votes
1
answer
3k
views
How is the "conformal prediction" conformal?
The question is clarified by Prof.V.Vovk. See his answer below for discussion.
Recently, early works of Gammerman, Vanpnik and Vovk[4] are rediscovered by Wasserman et.al[1] and proposed it as a ...
- 8,071
10
votes
2
answers
1k
views
Continuity of the mutual information
The mutual information $I(\mathfrak A_1;\mathfrak A_2)$ of two complete $\sigma$-algebras $\mathfrak A_1$ and $\mathfrak A_2$ in a Lebesgue probability space $(X,m)$ is the integral of the logarithm ...
- 17k
9
votes
1
answer
966
views
A necessary condition for differential entropy to be finite
An ensemble corresponding to a probability distribution usually has finite free energy so it is not a great loss of generality to assume that the ensemble also has finite energy in following ...
- 8,071
7
votes
3
answers
790
views
Expected cardinality of a randomly chosen element of the family of subsets of $\{1,\ldots,n\}$ with at most $k$-elements
Assume that $1\le k \le n$ and let $\mathscr{Z}$ be the family of all subsets of $\{1,\ldots,n\}$ with at most $k$ elements. Pick a random element $X$ of $\mathscr{Z}$ (we consider the probablity ...
- 1,658
6
votes
2
answers
813
views
Kolmogorov's approach to probability theory
Question:
Did Kolmogorov develop a set of axioms for probability theory motivated by Algorithmic Information Theory in the 1960s?
Context:
In 1965, Andrey Kolmogorov considered three approaches to ...
- 3,871
4
votes
1
answer
362
views
Information monotonicity of divergence => function of $f$-divergence
It is well-known that $f$-divergences defined on $\mathcal P(\mathcal X)$ where $\mathcal X$ is a measure space with $\sigma$-algebra $\mathcal B$ satisfy the property of information monotonicity:
...
- 203
3
votes
1
answer
152
views
Mutual information in large deviation theory
Many information theoretic quantities such as entropy and relative entropy appear in rate functions in large deviation theory (LDT). Is there any result in LDT that relates mutual information and rate ...
- 1,608
3
votes
1
answer
306
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 B|C=0,D=0)\...
- 585
2
votes
1
answer
222
views
Given iid samples from the joint distribution $P$ of pair of r.v.'s $(X,Y)$, how to get iid samples from independence coupling $P_X \otimes P_Y$?
Let $(X,Y)$ be a pair of random variables on a measure space $\mathcal T \subseteq \text{"subsets of }\mathbb R^2\text{"}$, with joint probability distribution $P$.
We don't assume $X$ and $Y$ are ...
- 6,853
2
votes
1
answer
199
views
Do enough permutations of an initial set probably cover most permutations?
Fix $\alpha, \epsilon \in(0,1)$. Take $(S_n)_n$ to be any sequence of sets with each $S_n$ containing $ \lceil (n!)^\alpha\rceil$ permutations of $n$ elements. Also build another sequence of sets $(...
1
vote
1
answer
124
views
References: error and stability estimates for information projection
$\newcommand\SS{P}\newcommand\TT{Q}$I will call a Gaussian probability measure $\SS$ on $\mathbb{R}^d$ isotropic if its covariance matrix is diagonal with non-vanishing determinant; i.e. $\Sigma_{i,i}&...
- 364
1
vote
0
answers
438
views
Chain rule for maximal correlation
Let a pair of random variables $(X,Y)$ be defined over finite alphabet $\mathcal{X}\times \mathcal{Y}$ with joint distribution $P_{XY}$. The maximal correlation $\rho(X;Y)$ between $X$ and $Y$ is ...
- 1,109
0
votes
0
answers
85
views
When is a family of distributions "closed" with respect to minimal sufficient statistics?
As in the title, I am interested in understanding how to express the idea that a parametric family of distribution is "closed" with respect to minimal sufficient statistics. Before giving ...
0
votes
0
answers
52
views
Classifier-specific lower bounds on the misclassification rate in binary classification
Consider a binary classification problem for $(X,Y)$, and let $\hat{f}$ be a proposed classifier. We wish to bound the misclassification rate $P(\hat{f}(X)\ne Y)$. There are many known lower bounds on ...
- 13
0
votes
0
answers
171
views
A basic property of maximal correlation
Let $𝑋$ and $𝑌$ be random variables. Then the maximal correlation $\rho_{m}(X;Y)$ is defined as:
$$\rho_{m}(X;Y):=\max_{f,g}\mathbb{E}[f(X)g(Y)],$$
where the maximization is taken over real-valued ...
- 11