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