Markov Models

Markov Models

• The future is independent of the past given the present
• if know state of the world right now, then knowing state of the world in the past is not going to help you predict the future
• uses: temporal data, or some sequence of data
  • weather
  • economic
  • language
    • speech recognition
    • Mark V. Shaney
  • music
    • automatically generated music
  • etc.

example data

ex CO2 levels in atmosphere

• x axis is time
• y axis time sequence data with periodicity and some randomness

position of robot GPS

• x & y-axes are lat long
• noisy measurements of position
• what is the actual position in time t?

fill in the blank language ex

• what is the word at the end of this ___________?

formal def'n

• sequential data D=(x1, ..., xn)
• take random variables X1, .. ,Xn
• there's clearly some dependency on the points that are nearby in time, can't call them iid's.
• if everything is dependent, totally intractable problem
• for the most accurate prediction of what's gonna happen in the near future is what's happening right now.
• if looking xn + 1, just look at more recent data, don't look at data from distant past
• recent past tells you more than distant past.
• Xt depends on Xt-1, Xt-2, ... , Xt-m (fixed m)
• Simplest case: m = 1
• simplifying assumptions: discrete time (duh) discrete space, i.e., xi is discrete variable that happens at discrete times
• discrete r.v.s X1, ..., xn form a discrete time Markov Chain
• i.e., joint distribution p(xt|x1, ..., xt-1) = p(xt|xt-1)
• AND p(x1,...,xn) = p(x1)*p(x2|x1)*p(x3|x2)*...*p(xn|xn-1)

different types (generalizations)

• can also have second order markov chain where m = 2
• can also have continuous time markov chain
  • poisson process
  • brownian motion continuous time
    • e.g., modelling stock prices, brownian motion in 2-D to model a particle of pollen in a glass of water
• e.g. taking a random walk along the integer - discrete time, discrete space, p(moving up) = 1/2, p(movingdown) = 1/2
• e.g., four states of weather. transition probabilities between any two states
• discrete-time, continuous space = "state-space"
• BUT: Can't expect to observe the perfect information about true state of the world (system) ... noisy observations, measurements
• Answer - to acknowledge fact that there's hidden information that we're not seeing
• can break up system into observed and hidden parts of the state
• can model with hidden (latent) variables

Hidden Markov Model

• A state machine, but you don't know the state
• But there's a probability
• NLP
  • Only information is the order of hte sentence
  • Capures probabl
• can estimate parameters
• goto baseline model for sequential data
• some rand vars z1, ... , zn in discrete integers in some finite set (like 26 letters)
• hidden vars x1, ..., xn in some set capital X
• the trelis diagram: x's are observed data D = (x1, ..., xn), and z's are hidden /latent vars
• joint distribution of all random variables p(x1,..,xn,z1,...,zn) factors = p(z1)*p(x1|z1)* PRODUCT( from k = 2 to n: p(zk|zk-1)*p(xk|zk) )

NLP considerations

• Only information is the order of hte sentence
• Captures probability of moving to part of speech
• Transition probaility matrix - for that particular row, the sum of probabilities has to be 1 because that covers all the possibilities of transition
• Incoming probabilities does not have to sum to 1
• observation is a word emission
• observation liklihood = how many times you see that word, divided by how many words in the corpus
• E.g., when I start the sentence, what is the probabiltity that the part of speech will be an article.

uses

• handwriting recognition - x is observed scribble (the strokes of the letter), z is the actual letter.
• parameters:

1. transition probabilities - T(ij)= p(zk+1= j| zk = i)
2. emission probabilities ei(x)= p(x|Zk = i) for i in set and x in its set .. e sub i is a prob dist. on space capitol X
3. initial distribution