Python Programming for Biology: Bioinformatics and Beyond

Markov chains

When  we  have  referred  to  the  rolling  of  dice  or  subsequent  DNA  positions,  we  have

already been considering a chain of events that occur from subsequent probabilistic trials.

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Also, until now we have only dealt with situations where the probabilities of the various

outcomes  are  independent  of  their  position  in  the  chain  (i.e.  that  the  probabilities  don’t

change for the same kind of outcome at different locations in the chain of trials). As we

have  mentioned  before,  this  is  a  simplification  for  many  types  of  random  process.

Naturally there could be all sorts of dependencies, where the probabilities of the outcomes

vary  according  to  context.  In  a  general  sense  we  can  describe  this  context  in  terms  of

conditional probabilities, where a probability is assigned for an event given the case that

another event has occurred.

A  Markov  chain  is  a  model  which  uses  conditional  probabilities  for  a  chain  of  trials

with  the  specific  criterion  that  the  probabilities  of  a  trial  are  conditioned  only  on  the

outcome  of  the  previous  trial.  Phrasing  this  differently  we  could  say  that  the  chain  does

not  have  any  memory  beyond  its  current  state.  This  is  certainly  a  simple  model,  and  in

reality  a  probability  may  actually  be  dependent  on  more  than  just  the  outcome  of  the

previous trial. However, the fact that we have a whole series of conditional probabilities

from the start of the chain means that information is relayed throughout the whole chain.

The fact that Markov chains have proven useful in various areas of biology is undeniable.