Generative Creativity - lecture 8:
Markov models

Introduction

Generative outputs from Cope's EMI system are modulated by models (partly) derived from examples.

But how are the models obtained?

This lecture will look at the method of Markov modeling/chaining, which plays a key role in a variety of sample-driven GC approaches (including Cope).

In its simplest form, the method offers a way of building a probabilistic model of a sequence.

N-grams

An n-gram is a continuous string of elements taken from a sequence

This may be a sentence, a melody or a linguistic corpus.

For example, from the sentence `give me a break' we can extract various n-grams.

Calculating n-gram probabilities

One way to obtain the n-gram probabilities is to identify each individual n-gram and then compute its relative frequency in the sequence.

But this is exponentially costly.

Instead, we can use the chain rule to build up the probabilities incrementally.

N-gram hierarchy

To obtain n-gram probabilities in a way which exploits the chain rule, work through the sequence building up an n-gram tree structure.
When this is done, convert the observed frequencies into the appropriate conditional probabilities.

Markov models

A set of n-gram probabilities forms a Markov model of a sequence.

The `order' of the model is fixed by the n-gram size.

Markov assumption

The Markov Assumption is that conditional probabilities spanning more than n+1 elements `make no difference'.

If we have a 1st order Markov model (a model based on bigram probabilities), then the Markov assumption means that

In other words, effects outside a certain `window' make no difference.

Markov sequence models

  p(a) = 0.3
  p(b) = 0.4
  p(c) = 0.1
  p(d) = 0.2

  p(a|a) = 0.0           p(c|a) = 0.0
  p(a|b) = 0.0           p(c|b) = 1.0
  p(a|c) = 1.0           p(c|c) = 0.0
  p(a|d) = 0.5           p(c|d) = 0.0
  p(b|a) = 0.5           p(d|a) = 0.3
  p(b|b) = 0.5           p(d|b) = 0.0
  p(b|c) = 0.0           p(d|c) = 0.0
  p(b|d) = 0.0           p(d|d) = 0.5

Generating a Markov chain

To generate a variation of the sequence (from which n-gram probabilities have been sampled) we output n-grams according to their sampled probabilities.

This is the process of generating a `Markov approximation' or `Markov chain'.

The order of the chain is the number of previous elements taken into account in generating each new element.

Markov approximations from Shakespeare

Samples generated from the Complete Works of Shakespeare (939,067 words)

i is that othello with me and a is had no of of have a a the my you i have and make and he sir, and thy of my so me and go the of a an and the and him my to for to the first on in

i know my lords at this is my good angels sing and so is my lord of his head of all the purpose to your grace! by thy sweet self too late to think you shall i beseech you, sir, i see you shall we shall be this to my

i pray thee, when thou hadst struck so to me as well as i do not say there's grain of it shall be made a world of tyre act i scene ii the palace. [enter a servant] servant o my most of all things are like to see a mess of such a great deal with my lord, you well, my lord. king henry vi the bird the lie, and lie open to give her to the duke of all the duke is like a man of the house of lancaster; and i have of it. pistol 'tis 'semper

3rd-order approximation

i would speak with those that have a sword, and so, i pray to thee, thou shalt be my lord of westmoreland, and attendants] king edward iv now let the general wrong of rome-- as fire drives out fire, so noble and so noble and so am i for i have heard him say, brutus i do not what 'twas to be revenged on itself. cleopatra so am i for measure act i pray thee, breathe my soul into the chantry by: and still as well as i come to speak truest not truer office of mine at once. no, my good lord; and in this is the man is the lord polonius my lord, i am a room in the benefit of his power unto octavia. cleopatra o, that which i would i might never in my life before this ancient to the general. second senator howsoever

Markov approximations of Beatles' lyrics

you the to you don't my i a of could and i that to you the should me come you i you i the been to you the she and to you i the you it but you it you i she's the i the i i you please my

you can see the girl when you can show but it's getting so many years yes, wait till tomorrow way get back in my apart but i know we will love you know my life i've never be mine, i want that's why i know i know i held each

you i want me to dance with you and i feel as though you ought to do what he left it won't be my baby, now went wrong i've got a boat on me and so my name you know my baby everybody's trying to be a boat on the hill sees the sheik of love chains of love. come and you know my name you know my name well don't get me mine

3rd-order

you want to dance with me i'm in love with another oh, when i kiss her majesty's a pretty nice girl but i'm miles above you tell me, i'm so alone don't bother me i'm a loser i'm a loser and i'm not a second time that was so long bye, bye, bye. lady madonna children at your man i wanna be your man i wanna be let it be, you know she thinks of him steal your heart away i got nothing to say but it's okay when i'm home we're on our way back home we're going home we're on our love to fast but i'm miles away and after dark people think for yourself 'cause it's going fast but i'm miles and my feet are hurting all the lonely people where they all the lonely people where they all my heart love i'll give you

Markov chain applet

Using Markov chains to generate music

Markov chaining is particularly useful for generative music and is very widely used.

In an early example, Harry F. Olson at Bell Labs used Markov chains in the 1950s to analyse the music of American composer Stephen Foster, and generate scores based on the analyses of 11 of Foster's songs.

Lejaren Hiller used a computer at Princeton in 1955 to generate the Illiac Suite (the first genuine case of music GC, according to Geraint Wiggins). Combined Markov chaining and application of rules of 16th century counterpoint. There are many good websites on this including

http://www.music.psu.edu/Faculty%20Pages/Ballora/INART55/illiac_suite.html and http://www.lim.dico.unimi.it/eventi/ctama/baggi.htm

Lejaren Hiller and Robert Baker also worked with Markov processes to produce their `Computer Cantata' in 1963.

Summary

Resources

Exercises

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