Markov chains are a popular way to model sequential data. They form the basis of more complex ideas, such as Hidden Markov Models, which are used for speech recognition and have applications in bioinformatics.
Today I want to run through an implementation of Markov chains for generating text based on an existing corpus. First of course we need to understand what Markov chains are before we can implement one.
The Intuition
I've talked about intuition-first machine learning before, so I'll start with the intuition. The first thing we need to know about is what a Markov chain consists of, then we need to define the Markov assumption.
States and Transitions
In a Markov chain we are assuming our sequence is made up of discrete states. That is, every item in the sequence is one of a finite set of possible values. In text, the states could be every letter of the alphabet and our punctuation marks.
We assume that for every item in the sequence, there is a probability associated with what the next value will be or, more formally, what state we will transition to. This is called the transition probability.
This is different between pairs of states, so the probability of going from state A to state B might be different than going from B to A. There is also a probability of staying in the same state.
However, it would be too difficult to attempt to model the transition probabilities if we always used the entire sequence. This would be like trying to predict the last word in War and Peace and having to take into account every single word that came before it.
To make this easier, we make what's called the Markov assumption.
The Markov Assumption
The Markov assumption is when we assume the Markov property to be true.
The Markov property (of a sequence) is typically formulated like so:
The future is independent of the past, given the present.
That sounds a bit like an old proverb but it's a simple concept.
If a sequence has the Markov property it means that the value of the sequence at the next step depends only on the value at the past \(n\) steps. The value of \(n\) (how far back we assume we need to go) is called the order of a Markov chain.
So a second-order Markov chain is one where we use the current and the previous value to predict the next value. We are assuming that it does not matter what the rest of the text was before the previous word, we only need the current word and the one before it.
In effect, the Markov chain has no "memory" beyond the last \(n\) steps.
The Markov assumption simply states that we believe this Markov property to hold for a given sequence.
This is a simplifying assumption that means it is much easier to compute these Markov chains at the cost of some complexity and accuracy. However simple this assumption may feel, it performs remarkably well.
The Markov Assumption in Text
Let's look at an example. What does this Markov assumption look like for text? We'll work with second-order Markov chains, as defined above.
Let's imagine that our sequence is this string of animals:
cat cat cat dog cat cat cat fish cat dog fish cat dog dog cat
What we say if we assume the Markov property is that our prediction of the next word in the sequence depends only on the last two words: dog cat.
How do we use that for predicting?
We need to calculate the transition probabilities based on the text that we have, then assume that going forward those probabilities are what decide our next words.
Because we are using second-order Markov chains, we calculate the probabilities of each word following a two-word pair. To do this, we simply count what percentage of the time a word appeared after each possible pair.
If we take a single example, "cat cat", we'll see that this pair was followed by "cat" twice, "dog" once and "fish" also once. That means if we're in the state "cat cat" the next word will be "cat" with a 50% probability, or "dog" or "fish" with 25% probability each.
The Code
That's enough intuition, let's get into the code.
I've scraped the lyrics to all songs written by Muse and we'll use this as our corpus. We'll use it to generate some more text, specifically a new Muse song.
Preparation
The first step is to read the file in and clean it so we have a sequence of words and new lines. I won't go into the details, you can see how the cleaning is done in the accompanying Jupyter notebook. I also included the code I wrote to scrape the lyrics in the first place, but going through that is optional as I'll also include the final source file.
Training
When we do the "learning" for a Markov chain, we're just identifying all unique triplets of words, and creating our transition probabilities using the first two words as the current state and the third as the next state.
As an implementation detail, I've made the transition probabilities a Python dictionary where each key is a unique word pair, and the value is a list of all the words that have ever followed that pair. I haven't even explicitly calculated the probabilities, but instead included each word as many times as it appears.
Our "cat cat" example above would look like an entry in a Python dictionary like this:
{ "cat cat": [ "cat", "cat", "dog", "fish" ] }
To generate our transition probabilities we use a helper function:
def generate_triples(word_list):
d = {}
# loop through the text and generate triples
for i in range(len(word_list)):
if i == 0 or i == len(word_list) - 1:
continue
else:
if (word_list[i-1], word_list[i]) in d:
d[(word_list[i-1], word_list[i])].append(word_list[i+1])
else:
d[(word_list[i-1], word_list[i])] = []
d[(word_list[i-1], word_list[i])].append(word_list[i+1])
return d
So given a list of words (our entire text) we go through it word by word, creating dictionary entries for every pair we encounter, and adding the next word into the list associated with that pair.
To then simulate the probability of choosing a word given a pair of words, we randomly sample from the list associated with that pair.
This function gives us the next word each time. This is part of a Markov class, where self.words is our previously trained dictionary.
def get_random_word(self, phrase=None):
if phrase:
# find a phrase from the list of words associated with the last two words in the supplied phrase
phrase_words = [x.lower() for x in phrase.split(' ')]
if len(phrase_words) > 1:
if (phrase_words[-2], phrase_words[-1]) in self.words:
past = self.words[(phrase_words[-2], phrase_words[-1])]
else:
past = self.words[random.choice(list(self.words.keys()))]
else:
past = self.words[random.choice(list(self.words.keys()))]
else:
# no phrase supplied, return a word from our dict at random
past = self.words[random.choice(list(self.words.keys()))]
return random.choice(past)
The function will look at a phrase, check if it is at least two words long and for the last two words, it finds the appropriate dictionary entry and samples from the list associated with that pair.
So if we gave it "banana cat cat" it would look up "cat cat" in the dictionary and sample from the list.
Remember: the words in the list implicitly represent the transition probabilities. Because we have "cat" twice out of four words in our list we've defined the transition probability as 50%.
To generate the text we can then either give it some text to start it off, or let it start with a random pair.
I've added some structure so that the code creates a song title, a chorus and a couple of verses.
It came up with this:
hear me moan
Verse 1
you are just
too much attention
and its gonna be
show me mercy can someone rescue me
make me agitatedChorus
escaped your world
no one is crying alone
i wont let them hurt
hurting you noVerse 2
you know what youve done
bring me peace and wash away my dirt
spin me round and have me toChorus
escaped your world
no one is crying alone
i wont let them hurt
hurting you no
I dare say Matt Bellamy couldn't have written it better himself.
Here's the associated Jupyter notebook. I've also previously published the code as more of a plug and play library, which you can find here.
Next Steps
We could improve this code by doing any of the following:
- Experiment with different orders for the Markov chain
- Giving it more data is never a bad idea, you could mash Muse's lyrics up with another artist
- You could try a character-level Markov chain to see if it would generate meaningful text that way, although for that you might want a Recurrent Neural Network
- Adding support for punctuation
Further Reading
If you want to learn more about Markov chains, and even see them 'in action', this is a great resource to start with.
I glossed over a lot of the maths behind the algorithm, but if you're interested you could start here.
About David
I'm a freelance data scientist consultant and educator with an MSc. in Data Science and a background in software and web development. My previous roles have been a range of data science, software development, team management and software architecting jobs.
