Does anybody know about that the random stuff that we do in
our daily life can be predicted. But how?
It is done by modeling your present state to predict the probability
of your future state. In simple words, it is the Markov chain that tells you
the probability of hoping or transitioning from one state to any other state.
Markov Model;
In probability theory, a Markov
model is a random process used to model randomly changing systems where it is
assumed that future states depends only on the current state not on the events
that occurred before it. This process is named after Russian mathematician
Andrei Andreyevich Markov.
A Markov model is a model where
the next state is only chosen based on the current state. In other words, it is
a process that involves the operation of chance.
Example;
Let, if we made a Markov chain of the
behavior of baby, we might include playing, eating, sleeping and crying as
states, which together with other behaviors could form a ‘state space’; a list
of all possible states. e.g., we could tell that the chance that the baby
currently playing fall fall asleep in the next five minutes without crying
first.
A simple two state Markov chain is given below;
With two states A and B, there are four possible transition
states, not two because a state can transition back in itself. If we are at A,
we could transition to B or stay at A. if we are at B, we could transition to A
or stay at B. in this two state diagram, the probability of transitioning from
one state to another is 0.5.
In real world problems, we rely on transition matrix to
tally the transitioning probabilities. Every state is included once as a row
and once as a column. And each cell in the matrix tells you the probability of
transitioning from its row state to its column state. So, in the matrix, the
cells do the same job that the arrows do in the diagram.
If the state space adds one state, we add one row and one
column, adding one cell to every existing column and row. This means that the
number of cells grow quadratic-ally as we add states as we add states to our
Markov chain.
In the hands of meteorologists, ecologists, computer
scientists, financial engineers and other people who need to model big
phenomenon, Markov chains can get to be quite large and powerful.
So, this was a short introduction to Markov chain. In the
next blog we will try to do explain Markov chain Monte Carlo with the help of
an example.
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