# Questions tagged [it.information-theory]

Questions in Information Theory

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### Given a sequence, how can we define its information content? [closed]

For example, if we take the following sequence: 1113333442 what is its information content?
48 views

### HYPOTHESIS TESTING DIVEGENCE (CHANNEL CODING THEOREM)

Using Hypothesis testing divergence, Show that if we transmit at rate \ $R\leq D^\frac{\epsilon}{2}_{H}(P_{XY}||P_X P_Y) + \log(\frac{\epsilon}{2})$ then $Pr(M\neq\widehat{M})\leq\epsilon$. Where M is ...
52 views

### Interesting statistical experiment concerning data compression

I want to present the following statistical experiment concerning data compression, on which I will ask you to predict the result obviously justifying the choice made. The statistical experiment is ...
138 views

### Maximal uniquely decodable codes

This question is about the Kraft-McMillan inequality: If $w_1,\ldots,w_n$ are words of lengths $l_1,\ldots,l_n$ from an alphabet with $r$ letters, which form a uniquely decodable code, then  \sum_{i=...
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### meaning of "algorithms that do not resample points" in the algorithms that do not resample points theorem

The No Free Lunch theorems for search and optimization demonstrate that for search/optimization problems in a limited search space, where the points being searched through are not resampled, the ...
106 views

### Can theoretical computer science be combined with mechanism and information design and applications in financial markets

I am considering to take a position as a phd student in a computer science department. I am a mathematician with a master degree in finance and my research interests are mainly focused in game theory. ...
201 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 ...
81 views

### Approximate (in hamming distance) subset representation

Let us have a set $S$ and a subset $T \subseteq S$. I want to find an approximate representation of $T$, i.e. I want to represent (exactly) a set $T'$ that is close to $T$. That is, I want the ...
157 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 ...
1 vote
107 views

### Does any physical process constitute a "computation"? [closed]

I am trying to sharpen the convex hull of what seems like a (surprisingly) stubborn concept to enclose based on answers here, as well as conversations with others, around the nature of what actually ...
204 views

### Upper bound on the expected number of correct bits via a "lossy compression"

Consider the following "compression problem" for a pair $(C,D)$ of algorithms: $C$ receives a uniformly random $x \in \{0,1\}^n$ and outputs a smaller bit string $y \in \{0,1\}^s$. Algorithm ...
135 views

### Information and Coding Theory Texts

I am coming from a pure mathematics (in analysis) background and am curious to learn some information and coding theory. I am after some recommendations on texts. Due to my personal background I am ...
200 views

### Does this notion of entropy have a name?

Recently I stumbled upon the following notion of entropy which seems quite natural to me. I am looking for its "real" name and/or any references where it might come up. I tried searching ...
1 vote
52 views

### sophistication or logical depth to detect intelligent extra-terrestrial species

From my understanding, Algorithmic information theory (AIT) gives some ways to define the amount of « structure » in a string: for example sophistication or logical depth (see for instance ), can ...
1 vote
113 views

### Deterministic one way communication complexity for message with arbitrary length

Let Alice have a binary string of length $n$ that it wants to send to Bob along a one-bit communication channel. However, Bob does not know the length of the message. I have been looking into ...
165 views

### Error in entropy properties in Mathematical Theory of Cryptography by Claude E. Shannon

I am reading this classic paper by Claude E. Shannon and I think there may be a couple of errors in his description of the properties of Entropy/Uncertainty. The screenshot shown at the bottom of this ...
414 views

1 vote
189 views

### Explicit Bits-back Coding (a.k.a. Free Energy Coding) applied to Gaussian mixtures

I've been trying to understand Bits-back coding (Frey, B. J., and G. E. Hinton. 1997.) a bit more (pun intended), which can be used to encode data with latent variable models. This tutorial by Pieter ...
1 vote
114 views

107 views

### Notation of sequences in rate distortion theory

I have been reading whatever sources I could get my hands on today, regarding this problem. Most notes online about rate distortion theory come from the book Elements of Information Theory by Thomas ...
213 views

### Expected vs worst-case communication complexity

In the set disjointness problem of 2-party communication complexity, Alice and Bob are both given an $n$-bit string as input; denoted by $X$ for Alice's input, and $Y$ for Bob's input. They need to ...
89 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, ...
105 views

### Representing data with Shannon entropy predicted bits

Let us assume a file based on a character set where each character has equal probability of occurance. This will result in the maximum entropy for that character set. On calculating the entropy, let ...
411 views

1 vote
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### Uniqueness of the distribution maximizing the channel capacity

Setting: We look at a discrete memoryless channel which takes an input probability distribution acting over symbols in $\mathcal{X}$ to an output probability distribution over symbols in $\mathcal{Y}$....
340 views

### Is a binary sequence computable iff the Kolmogorov complexity of its initial segments is bounded?

Disclaimer: I am mostly unfamiliar with theoretical computer science, making it hard for me to navigate literature in the field. I ask the following out of curiosity. Background/Motivation: Coming ...
188 views

### Minimal information needed for determine some function

From calculus, we know that if someone has a continuous function $f$, it is enough to know $f$'s values on the rationals in order to know $f$ on the entire line. In some sense, a "countable amount of ...
359 views

### Is algorithmic information theory still evolving?

I am currently looking for a subject for a thesis and encountered the field of algorithmic information theory. The field seems very interesting for me, but it seems everything is the field was done ...
269 views

### How can AIC converge in the limit when even 2 parameter models can have infinite VC dimension?

AIC-based model-selection converges to zero error in the limit, and also has finite-sample convergence that is rate-optimal with respect to worst case minimax error . (Note that AIC refers to ...
1 vote
65 views

### Is there a theoretical guarantee that an autoencoder $g$ has $I(x;g(x)) \approx H(x)$?

I know that in general, a function $g$ can be a good auto-encoder (i.e., $g(x) \approx x$ for $x \sim D$) and on the same time $I(g(x);x)$ is small. This is the case when $g$ forms a good correlation ...