General description of the Markov Chains
Sources
https://spectrum.ieee.org/andrey-markov-and-claude-shannon-built-the-first-language-generation-models http://langvillea.people.cofc.edu/MCapps7.pdf https://www.irishtimes.com/news/science/that-s-maths-andrey-markov-s-brilliant-ideas-are-still-bearing-fruit-1.3220929 http://www.alpha60.de/research/markov/DavidLink_TracesOfTheMouth_2006.pdf
Histories
Andrey Andreyevich Markov was a Russian mathematician who lived between 1856 and 1922. His most famous studies were with Markov chains, an algorithm that allows to predict future changes once one knows the current state . The first paper on the subject was published in 1906. He was also interested in literature. He tried establishing a linguistic mathematical model using Markov Chains by manually counting letters of Alexander Pusjkins verse novel Eugene Onegin. Next, he applied the method to the novel Childhood Years of Bagrov's Grandson by S.T. Aksakov. This links the Markov Chains directly to the field of literature, text and language. And the link will live firmly throughout the history of this algorithm.
The following text is based on Oscar Schwartz' article for IEEE Spectrum, Andrey Markov & Claude Shannon Counted Letters to Build the First Language-Generation Models.
In 1913, Andrey Markov sat down in his study in St. Petersburg with a copy of Alexander Pushkin’s 19th century verse novel, Eugene Onegin, a literary classic at the time. This work comprises almost 400 stanzas of iambic tetrameter. Markov, however, did not start reading Pushkin’s famous text. Rather, he took a pen and piece of drafting paper, and wrote out the first 20,000 letters of the book in one long string of letters, eliminating all punctuation and spaces. Then he arranged these letters in 200 grids (10-by-10 characters each) and began counting the vowels in every row and column, tallying the results.
In separating the vowels from the consonants, Markov was testing a theory of probability that he had developed in 1906 and that we now call a Markov Process or Markov Chain. Up until that point, the field of probability had been mostly limited to analyzing phenomena like roulette or coin flipping, where the outcome of previous events does not change the probability of current events. But Markov felt that most things happen in chains of causality and are dependent on prior outcomes. He wanted a way of modeling these occurrences through probabilistic analysis.
Language, Markov believed, was an example of a system where past occurrences partly determine present outcomes. To demonstrate this, he wanted to show that in a text like Pushkin’s novel, the chance of a certain letter appearing at some point in the text is dependent, to some extent, on the letter that came before it.
To do so, Markov began counting vowels in Eugene Onegin, and found that 43 percent of letters were vowels and 57 percent were consonants. Then Markov separated the 20,000 letters into pairs of vowels and consonant combinations. He found that there were 1,104 vowel-vowel pairs, 3,827 consonant-consonant pairs, and 15,069 vowel-consonant and consonant-vowel pairs. What this demonstrated, statistically speaking, was that for any given letter in Pushkin’s text, if it was a vowel, odds were that the next letter would be a consonant, and vice versa.
Markov used this analysis to demonstrate that Pushkin’s Eugene Onegin wasn’t just a random distribution of letters but had some underlying statistical qualities that could be modeled. The enigmatic research paper that came out of this study, entitled An Example of Statistical Investigation of the Text Eugene Onegin Concerning the Connection of Samples in Chains was not widely cited in Markov’s lifetime, and not translated to English until 2006. Markov was forced to stop his letter-counting experiments, when he had nearly completely lost his sight due to glaucoma. Even if Markov had had more time and better eyesight to carry his experiments further, extensions would have been very difficult to complete, given the precomputer era he lived in, when computational efforts had to be paid in man-years.
These images show Markov’s original notes in computing the probabilities needed for his Pushkin chain.
Influence
Some of Markov's central concepts around probability and language spread across the globe, eventually finding re-articulation in Claude Shannon’s hugely influential paper, A Mathematical Theory of Communication which came out in 1948.
Shannon’s paper outlined a way to precisely measure the quantity of information in a message, and in doing so, set the foundations for a theory of information that would come to define the digital age. Shannon was fascinated by Markov’s idea that in a given text, the likelihood of some letter or word appearing could be approximated. Like Markov, Shannon demonstrated this by performing some textual experiments that involved making a statistical model of language, then took a step further by trying to use the model to generate text according to those statistical rules.
In an initial control experiment, he started by generating a sentence by picking letters randomly from a 27-symbol alphabet (26 letters, plus a space), and got the following output:
XFOML RXKHRJFFJUJ ZLPWCFWKCYJ FFJEYVKCQSGHYD QPAAMKBZAACIBZLHJQD
The sentence was meaningless noise, Shannon said, because when we communicate we don’t choose letters with equal probability. As Markov had shown, consonants are more likely than vowels. But at a greater level of granularity, E’s are more common than S’s which are more common than Q’s. To account for this, Shannon amended his original alphabet so that it modeled the probability of English more closely—he was 11 percent more likely to draw an E from the alphabet than a Q. When he again drew letters at random from this recalibrated corpus he got a sentence that came a bit closer to English.
OCRO HLI RGWR NMIELWIS EU LL NBNESEBYA TH EEI ALHENHTTPA OOBTTVA NAH BRL.
In a series of subsequent experiments, Shannon demonstrated that as you make the statistical model even more complex, you get increasingly more comprehensible results. Shannon, via Markov, revealed a statistical framework for the English language, and showed that by modeling this framework—by analyzing the dependent probabilities of letters and words appearing in combination with each other—he could actually generate language.
The more complex the statistical model of a given text, the more accurate the language generation becomes—or as Shannon put it, the greater “resemblance to ordinary English text.” In the final experiment, Shannon drew from a corpus of words instead of letters and achieved the following:
THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH WRITER THAT THE CHARACTER OF THIS POINT IS THEREFORE ANOTHER METHOD FOR THE LETTERS THAT THE TIME OF WHO EVER TOLD THE PROBLEM FOR AN UNEXPECTED.
For both Shannon and Markov, the insight that language’s statistical properties could be modeled offered a way to re-think broader problems that they were working on. For Markov, it extended the study of stochasticity beyond mutually independent events, paving the way for a new era in probability theory. For Shannon, it helped him formulate a precise way of measuring and encoding units of information in a message, which revolutionized telecommunications and, eventually, digital communication. But their statistical approach to language modeling and generation also ushered in a new era for natural language processing, which has ramified through the digital age to this day. As David Link notes in his article, Traces of the Mouth, Markov's efforts in retrospect “represent an early and momentous attempt to understand the phenomenon of language in mathematical terms.” It's not an exaggeration to say that Markov's analysis of text is in principle similar to what Google and other firms now routinely carry out on a massive scale: analyzing words in books and internet documents, the order in which the words occur, analyzing search phrases, detecting spam and so on.
Applications
Since Markov chains can be designed to model many real-world processes, they are used in a wide variety of situations. They appear in physics and chemistry when probabilities are used for unknown quantities. In information processing, they have a role in pattern recognition, automatic speech analysis and synthesis and data compression. They are used by meteorologists, ecologists and biologists. Other applications include the control of driverless cars, machine translation, queuing patterns, and prediction of population growth, asset prices, currency exchange rates and market upheavals. Also artists have used Markov chains, such as musician Iannis Xenakis who developed “Free Stochastic Music” based on Markov chains.
In 2006 – the 100th anniversary of Markov's paper – Philipp Von Hilgers and Amy Langville summarized the five greatest applications of Markov chains. This includes the one that is used by most of us on a daily basis, Google's Page Rank. Every time we search on the internet, the ranking of webpages is based on the solution to massive Markov chain. You can say that all the web pages are states, and the links between them are transitions possessing specific probabilities. In other words, we can say that no matter what you’re searching on Google, there’s a finite probability of you ending up on a particular web page. If you use Gmail, you must’ve noticed their Auto-fill feature. This feature automatically predicts your sentences to help you write emails quickly.
And last but not least, have you ever wondered why spam has all those hilarious nonsensical strings of words in it? They’re pretty odd constructions, not as random as if you picked words randomly out of a hat, almost grammatical much of the time, but still clearly gibberish. Also here the Markov chains have taken on a lot of the work.