All Questions
11 questions
48
votes
2
answers
14k
views
Research situation in the field of Information Geometry
I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field.
I have read (4) and parts of (3). ...
- 8,071
16
votes
1
answer
397
views
Examples of problems in statistics accessible only using information geometry
I am just curious if there are some examples of problems in statistics that are indeed accessible using information geometry while proofs completely avoiding geometry are unknown. In other words, ...
- 269
7
votes
1
answer
1k
views
reverse KL-divergence: Bregman or not?
I am having a little trouble getting my head around the two "directions" of the Kullback-Leibler divergence:
Definition (Kullback-Leibler divergence) For discrete probability distributions $...
- 101
5
votes
0
answers
191
views
Divergence for Bhattacharya Information matrix
The Fisher information matrix (in the scalar parameter case) can be obtained from the Kullback-Leibler divergence by
$$g(\theta) = -\frac{\partial}{\partial \theta}\frac{\partial}{\partial \theta'}D(...
- 779
4
votes
1
answer
362
views
Information monotonicity of divergence => function of $f$-divergence
It is well-known that $f$-divergences defined on $\mathcal P(\mathcal X)$ where $\mathcal X$ is a measure space with $\sigma$-algebra $\mathcal B$ satisfy the property of information monotonicity:
...
- 203
4
votes
0
answers
144
views
Exponential families closed under affine transformations
Let $(\Omega,\Sigma,\mu)$ be a probability space and let $\mathcal{M}$ be an exponential family of probability distributions for $\mu$ of the following form: There are $\varphi_1,\dots,\varphi_n:\...
- 5,405
2
votes
1
answer
282
views
Convexity of exponential family
It is known that (given a $\sigma$-finite Borel reference measure $\nu$ on $\mathbb{R}$) the parameter space of an exponential family is convex in Euclidean space. However, my question is, for an the ...
- 5,405
2
votes
0
answers
217
views
Inequality on the Kullback-Leibler divergence
Let us define the arithmetic, geometric, and harmonic means of $x,y \in \mathbb{R}$ weighted by $\alpha =(\alpha_x,\alpha_y) \in [0,1]$, respectively as
\begin{equation}
a_\alpha(x,y) = \frac{\...
- 245
2
votes
0
answers
396
views
Connecting Wasserstein distance with mutual information?
Suppose I have Markov chains:
$$X \rightarrow f(X) \rightarrow g(X)$$
$$Y \rightarrow f(Y) \rightarrow g(Y)$$
where it is known that minimizing the $\mathbb{E}(g(X)) - \mathbb{E}(g(Y))$ minimizes the ...
- 67
1
vote
0
answers
213
views
Jensen-Shannon Divergence of Sample Distributions
Given normal distributions with a single positional and variation parameter each, $p_1=\big[\mu_1, \sigma_1\big]$, $p_2=\big[\mu_2, \sigma_2\big]$, we define their Jensen-Shannon divergence as:
$$
\...
- 429
0
votes
0
answers
50
views
General formula for Fisher information matrix reparameterization?
Prefacing apology for likely having unclear notation in the question and possible unclear concepts, because I'm not a mathematician.
The Fisher Information Matrix (FIM) for a multivariate normal ...
- 75