PROFESSOR: Hi, everyone.
Welcome back.
So today, I'd like to tackle
a problem in Markov matrices.
Specifically, we're going to
start with this problem which
almost has a physics origin.
If we have a particle that
jumps between positions A and B
with the following
probabilities--
I'll just state it-- if
it starts at A and jumps
to B with probability
0.4 or starts at A
and stays at A with probability
0.6, or if it starts at B
then it goes to A
with probability 0.2
or stays at B with
probability 0.8,
we'd like to know the
evolution of the probability
of this particle over
a long period of time.
So specifically
the problem we're
interested today is:
if we have a particle
and we know that it
starts at position A,
what is the
probability that it is
at position A and
the probability
that it's at position B after
one step, after n steps,
and then finally after an
infinite number of steps?
So I'll let you think about
this problem for a moment
and I'll be back.
Hi everyone.
Welcome back.
So the main difficulty
with this problem
is that it's phrased
as a physics problem.
And we need to convert it into
some mathematical language
to get a handle on it.
So specifically,
what we'd like to do
is to convert this into
a matrix formalism.
So what we can do is we can
write this little graph down
and describe everything in
this graph using a matrix.
So I'm going to
call this matrix A,
and I'm going to associate
the first row of A
with particle position A
and particle position B.
And I'll associate the
first and second columns
with particles
positions A and B.
And then what I'm
going to do is I'm
going to fill in this
matrix with the probability
distributions.
So, specifically what's going
to go in this top left corner?
Well, the number 0.6, which
represents the probability
that I stay at position
A, given that I
start at position A.
What's going to go here
in the bottom left-hand corner?
Well, we're going to put 0.4,
because this is the probability
that I wind up at B, given that
I start at A. And then lastly,
we'll fill in these other two
columns or the second column
with 0.8 and 0.2.
So I'll just state
briefly this is
what's called a Markov matrix.
And it's called Markov, because
first off, every element
is positive or 0.
And secondly, the sum of the
elements in each column is 1.
So if we note 0.4 plus 0.6
is 1, 0.8 plus 0.2 is 1.
And these matrices
come up all the time
when we're talking about
probabilities and the evolution
of probability distributions.
OK.
So now, once we've encoded
this graph using this matrix A,
we now want to
tackle this problem.
So I'm going to
introduce the vector p,
and I'm going to
use a subscript 0
is to denote the probability
that the particle is at time 0.
So we're told that the
particle starts at position A.
So at time 0, I'm going
to use the vector [1, 0].
Again, I'm going to match the
top component of this vector
with the top component
of this matrix
and the first column
of this matrix.
And then likewise, the second
component of this vector
with the second row and
second column of this matrix.
And we're interested
in: how does
this probability evolve as
the particle takes many steps?
So for one step, what's the
probability of the particle
going to be?
Well, this is the beauty of
introducing matrix notation.
I'm going to denote p_1
to be the probability
of the particle after one step.
And we see that we can write
this as the matrix A multiplied
by p_0.
So the answer is 0.6 and 0.4.
And I achieve this just
by multiplying this matrix
by this vector.
OK?
So this concludes part one.
Now part two is a
little trickier.
So part two is n steps.
And to tackle this
problem, we need
to use a little more machinery.
So first off, I'm going to
note that p_1 is A times p_0.
Likewise, p_2-- so
this is the position
of the-- the probability
distribution of the particle
after two steps.
This is A times p_0, which
is A squared times p_0.
And we note that
there's a general trend.
After n steps-- so
P_n-- the general trend
is, it's going to be this matrix
A raised to the n-th power,
multiply the vector P0.
So how do we take the
n-th power of a matrix?
Well, this is where we use
eigenvectors and eigenvalues.
So recall, that we can take any
matrix A that's diagonalizable
and write it as U D U inverse,
where D is a diagonal matrix
and this matrix U is a matrix
whose columns correspond
to the eigenvectors of A.
So for this problem,
I'm just going
to state what the eigenvalues
and eigenvectors are.
And I'll let you work them out.
So because it's a
Markov matrix, we always
have an eigenvalue which is 1.
And in this case, we have an
eigenvector u which is 1 and 2.
In addition, the second
eigenvalue is 0.4.
And the eigenvector
corresponding to this one
is [1, -1].
And I'll just call these
u_1 and u_2, like that.
OK, we can now write this
big matrix U as 1, 2; 1, -1.
D is going to be-- now I
have to match things up.
If I'm going to put
the first eigenvector
in the first column, we have
to stick 1 in the first column
as well and then 0.4 like this.
And then lastly, we also have
U inverse which I can just
work out to be minus 1/3,
one over the determinant,
times -1, -1; -2, and 1,
which simplifies to this.
OK, so now if we take A and
raise it to the power of n,
we have this nice identity
that all the U and U inverses
collapse in the middle.
And we're left with U, D
to the n, U inverse, p_0.
Now raising the a diagonal
matrix to the power of n
is a relatively
simple thing to do.
We just take the eigenvalues and
raise them to the power of n.
So when we compute this
product, there's a question
of what order do we do things?
Now these are 2 by 2
matrices, so in theory we
could just multiply
out, 2 by 2 matrix, 2
by 2 matrix, 2 by 2 matrix, and
then on a vector which is a 2
by 1 matrix.
But if you're in a test and
you're cramped for time,
you want to do as little
computations as possible.
So what you want
to do is you want
to start on the right-hand
side and then work backwards.
So if we do this, we
end up obtaining 1, 2,
this is going to be to the
power of n, 1/3, [1, 2].
OK, so for this
last part, I'm just
going to write down
the final answer.
And I'll let you work out the
multiplication of matrices.
So we have for p_n: 1/3, 2 times
0.4 to the n plus 1, -2 0.4
to the n plus 2.
And this is the final
vector for p of n.
So this finishes up Part 2.
And then lastly,
for Part 3, what
happens when n goes to infinity?
Well, we have the
answer for any n.
So we can just take the
limit as n goes to infinity.
Now, specifically as
n goes to infinity,
0.4 raised to some very
large power vanishes.
So these two terms drop off.
And at the end of the day,
we're left with p_infinity
is 1/3 [1, 2].
OK?
So just to recap, we started off
with a particle starting at A,
and then after a very
long time, the particle
winds up with a
probability distribution
which is 1/3, 1 and 2.
And this is quite characteristic
of Markov matrix chains.
Specifically, we note
that 1/3 * [1, 2]
is a multiple of the eigenvector
corresponding to eigenvalue 1.
So even though the particle
started at position A,
after a long period
of time, it tended
to forget where it started
and approached, diffused
into this uniform distribution.
OK.
I'd like to finish up here.
And I'll see you next time.
