15 - Probability Matrices

15 - Probability Matrices - 1 Math 1b Practical February...

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1 Math 1b Practical — February 25, 2011 — revised February 28, 2011 Probability matrices A probability vector is a nonnegative vector whose coordinates sum to 1. A square matrix P is called a probability matrix (or a left-stochastic matrix or a column-stochastic matrix) when all of its columns are probability vectors. [Caution: Some references use probability matrix to mean a row-stochastic matrix.] Probability matrices arise as “tran- sition matrices” in Markov chains. Let A =( a ij ) be a probability matrix and u u 1 ,...,u n ) a probability vector. If we think of u j as the proportion of some commodity or population in ‘state’ j at a given moment (or as the probability that a member of the population is in state j ), and of a ij as the proportion of (or probability that) the commodity or population in state j that will change to state i after a unit of time, we would ±nd the proportion of the commodity or population in state i after one unit of time to be a i 1 u 1 + ... + a in u n . That is, the vector that describes the new proportions of the commodity or population in various states after one unit of time is A u .A f te r k units of time, the vector that describes the new proportions of the commodity or population in various states is A k u . If a ij 6 = 0, the edge directed from i to j in the digraph of a probability matrix A may be labeled with the number a ij ; this may help to ‘visualize’ the matrix and its meaning. Here is an example (taken from Wikipedia) similar to Story 1 about smoking. Assume that weather observations at some location show that a sunny day is 90% likely to be followed by another sunny day, and a rainy day is 50% likely to be followed by another rainy day. We are asked to predict the proportion of sunny days in a year. Let x n a n ,b n ) > where a n is the probability that it is rainy on day n ,and b n =1 a n .Then ± a n +1 b n +1 ² = A ± a n b n ² where A = ± . 9 . 5 . 1 . 5 ² . The digraph of A is Theorem 1. Any probability matrix A has 1 as an eigenvalue. Proof: Let 1 1 be the row vector of all ones. We have 1 1 P = 1 1 .(W ec o u l dc a l l 1 1 a left eigenvector and 1 a left eigenvalue.) This means 1 1 ( P I )= 0 ,sotherowso f P I are linearly dependent, so the columns of P I are linearly dependent, so ( P I ) u = 0 for some u ,or P u = u . ut (In general, the left eigenvalues of a matrix are the same as the right eigenvalues, and they have both the same left and right geometric and algebraic multiplicities.)
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2 A probability vector s
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15 - Probability Matrices - 1 Math 1b Practical February...

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