August 21, August 11, August 16, What imagination can Biotechnology as a teacher bring to Artificial Intelligence? How to do some restrictions on Artificial Intelligence in the future?
Some things you should know if you are the Artificial Intelligence startups. Introduction of Computer Vision Machine Learning development. Artificial Intelligence emotion recognition may still be far away. Beginners learning Artificial Intelligence must read mathematics books recommendation with PDF download. The 10 best machine learning websites with reviews.
TensorFlow 2. Driven by community feedback, this release provides a complete set of tools for developers, enterprises, and researchers to easily build ML applications. Federated Learning keeps privacy in mind without centralizing data while allowing edge devices to use machine learning. But should it be up to Facebook to decide what content is acceptable?
Share this: Twitter Facebook. Leave a Reply Cancel reply. Thinking What imagination can Biotechnology as a teacher bring to Artificial Intelligence?
Thinking How to do some restrictions on Artificial Intelligence in the future? Thinking Some things you should know if you are the Artificial Intelligence startups 27 Oct, Thinking Artificial Intelligence emotion recognition may still be far away 8 Aug, Learning The 10 best machine learning websites with reviews 8 Sep, Click here to sign up.
Download Free PDF. Elements of Information Theory. Wojciech Szpankowski. A short summary of this paper. Chapter 6 Elements of Information Theory Entropy and mutual information were introduced by Shannon in , and this began a remarkable development of information theory. Entropy, the Shannon-McMillan- Breiman theorem and random coding technique nowadays are standard tools of analysis of algorithms. In this chapter, we discuss elements of information theory and illustrate its applications to analysis of algorithms.
In particular, we prove three main results of Shannon, that of source coding, channel coding and rate distortion. In the application section, we discuss a variety of data compression schemes based on exact and approximate pattern matching. In this chapter it is convenient to use the logarithm to base 2.
If the base of the logarithm is 2, we measure the entropy in bits; with the natural logarithm we measure the entropy in nats. As we shall see, the entropy represents average uncertainty of X , and in fact it is the number of bits on the average required to describe the random variable cf. Exercise 1. The real virtue of entropy is manifested the best when investigating entropy rates of stochastic processes. Let us start with a simple observation: Consider an i.
We conclude that typical sequences have approximately the same probability equal to 2 nh. We make this statement rigorous in this chapter. He established three main results of infor- mation theory, namely that of source coding, channel coding and rate distortion. It turns out that you cannot beat a certain limit which is equal to entropy times the length of the original message.
In channel coding, the goal is the opposite to source coding, namely, we want to send reliably as much information as possible.
In such cases, we can only describe approximately the source of information. How well we can do? To answer this question we enter the realm of rate distortion theory. We touch in this chapter some of these problems. In fact, a large part of this chapter is based on [59]. In the analysis of algorithms, which is the main topic of this book, entropy appears quite often in disparate problems as we shall see in the application section of this chapter.
We shall follow the presentation of Cover and Thomas [59]. We sometimes write h P or simply h for the entropy. It is also the minimum average number of bits required to describe X , as illustrated in the following example. Example 6. We can use 3 bits to encode 8 messages, however, we can do much better by encoding more likely outcomes with shorter codes and less likely with longer.
For example, if we assign the following codes to eight messages: 0, 10, , , , , , , then we need on the average only 2 bits. Notice that the description length in this case is equal to the entropy.
In Exercise 1 the reader is asked to extend it to general discrete random variable. These properties are easy to prove by elementary algebra, and the reader is asked in Exercise 3 to verify them cf. Exercise 4 for more properties of the entropy. Theorem 6. We prove this fact, and some others, formally below. Then 6. Finally, 6. We shall prove that for stationary processes the limits exist and are equal. First of all, observe that the conditional entropy h Xn j Xn 1 ; Xn 2 ; : : : ; X1 is non- increasing with n and hence has a limit.
Indeed, by 6. By the chain rule 6. Let us consider a stationary Markov source of order one. The main consequence of this theorem is the so called Asymptotic Equipartition Property discussed in the next section. Let us put some rigor into the above rough description. We are aiming now at showing that the above conclusion is true for much more general sources. As the next step, let us consider mixing sources that include as a special case Markovian sources cf.
Chapter 2. We prove it using a Markov approximation discussed in Lemma 4. The main idea is to use the kth order Markov approximation of a stationary distribution. Using 4. In the above, hk is the entropy of the kth order Markov chain, which we know to exist. Now, we consider a lower bound by using 4.
Exercise 6 of Chapter 5, we know that P x0 jX k1 is a martingale. Therefore, by the martingale convergence Theorem 5. P x0 jX 11 a:s: for all x0 2 A. Since x log x is bounded and continuous, the bounded convergence theorem allows to interchange of expectation and limit, yielding 2 3 X lim k!
Section 6. Memoryless Source. Markovian Source. In Section 4. Consider a di- graph on A with weights equal to log pij where i; j P2 A. Exercise 10 of Chapter 4. The Shannon-McMillan-Breiman theorem implies the following property that is at the heart of information theory.
Its probability is smaller than ". The following is a generalization of AEP to jointly typical sequences. Finally, for 6.
This completes the proof. We shall discuss it in depth in Section 6. One suspects that it is a lost manuscript of a famous Polish poet Adam Mickiewicz. In other words, we estimate the probability of a sample xn1 generated by source P with respect to the known distribution Q. We call this problem: AEP for a biased distribution. Below, we restrict our discussion to jointly mixing processes that will guarantee the existence of the limit.
First, however, we prove the existence of the limit in 6. For ex- ample, not all Markov chains belong to JMX , but Markov chains with positive transition probabilities are jointly mixing. By Theorem 5. We should point out that the above AEP for biased distributions cannot be extended to general stationary ergodic processes cf.
However, the following general result is known the reader is asked to prove it in Exercise 9; cf. Lemma 6. Using AEP, we can asymptotically estimate this probability e. But, in some situations e. To answer such a question a lossy or approximate extension of AEP is needed. We discuss it next. Another distortion measure is useful for image compression; e. In the course of proving 6. We now wrestle with 6. Chapter 5. Part iii is a direct consequence of Part ii. For example, to prove 6.
The evaluation of the generalized R'enyi's entropies is harder than expected. The function r1 D is convex with respect to D. In addition, r0 D is convex with respect to D. We only prove part iii. The reader is referred to [] for the whole proof, or even better, is encouraged to provide the missing parts.
Then, the value of the maximum in 6. Indeed, for b! Thus, 6. We must point out, however, that one can envision another AEP for lossy case that is actually quite useful in the rate distortion theory discussed in Section 6.
We brie y discuss it here. We only prove 6. Namely, that of universal source coding theorem, channel coding theorem and rate distortion theorem. In this section, we discuss them and provide simple proofs.
In Shannon's spirit, we shall deal mostly with memoryless sources to present succinctly the main ideas. Throughout this section, we use a new technique, yet not seen on pages of this book, called random coding.
This method selects codes at random to establish the existence of a good code that achieves the best possible code rate. We prove it below for the simplest case. Its extensions can be found in [59]. This is an easy exercise on trees. A path from the root to a node creates a code. The converse part is easy to prove, and we omit it here. In other words, there is a limit how much we can compress on the average. In Theorem 6.
We shall return to the code redundancy in Section 8. To formulate a universal source coding problem, we start by introducing the bit rate R of a code C n. Hereafter, we only deal with extended codes for X1 ; : : : ; Xn generated by a source with the underlying probability measure P. Let M be the number of codewords for C n. Observe that R represents the number of bits per source symbol, and it can be viewed as the reciprocal of the compression ratio.
In other words, on average the best bit rate is equal to the entropy of the source. We assign codes that are reproduced correctly to elements of the set Gn". Clearly, then the error PE is bounded by ".
Let C 0 be the subset of all codewords C that are encoded without an error. This proves the theorem. Such a communication is performed by sending the signal X W that is received as a signal Y depending on X , hence on the message W.
This communication process is illustrated in Figure 6. As before, we shall write P for the probability measure characterizing the source and the channel.
0コメント