APM 541: Probability Notes

77 Pages·2013·1.96 MB·English

by  Jay Taylor

APM 541: Stochastic Modelling in Biology
Probability Notes
Jay Taylor
Fall 2013
JayTaylor (ASU) APM541 Fall2013 1/77
;
Outline
Outline
1 Motivation
2 Probability and Uncertainty
3 Conditional Probability
4 Discrete Distributions
5 Continuous Distributions
6 Multivariate Distributions
7 Limit Theorems for Sums of Random Variables
JayTaylor (ASU) APM541 Fall2013 2/77
Motivation
Chance and Contingency Influence Biological Processes at all Scales
quantum biology: stochastic tunneling in photosynthetic pathways
stochastic molecular dynamics: molecular motors as Brownian ratchets
noise in biochemical and gene regulatory networks
developmental biology: stochastic X-inactivation leads to mosaic phenotypes in
female mammals (tortoiseshell cats)
behavioral ecology: optimal foraging in uncertain environments
population ecology: extinction risk of small populations depends on demographic
and environmental stochasticity
population genetics: allele frequencies fluctuate because of random genetic drift
and random mutation
phylogenetics: random DNA substitutions are used for phylogenetic inference
JayTaylor (ASU) APM541 Fall2013 3/77
Motivation
Stochastic X-inactivation produces the Mass extinctions have had profound and
mottled coat pattern on tortoiseshell often unpredictable consequences for life
cats. on earth.
JayTaylor (ASU) APM541 Fall2013 4/77
ProbabilityandUncertainty
Multiple Interpretations of Probability
Frequentist interpretation (J. Venn): The probability of an event is equal to its
limiting frequency in an infinite series of independent, identical trials.
Propensity interpretation (C. Pierce, K. Popper): The probability of an event is
equal to the physical propensity of the event. (Confession: I have no idea what
this means.)
Logical interpretation (J. M. Keynes, R. Cox, E. Jaynes): Probabilities quantify
the degree of support that some evidence E gives to a hypothesis H.
Subjective interpretation (F. Ramsey, J. von Neumann, B. de Finetti): The
probability of a proposition is equal to the degree of an individual’s belief that the
proposition is true.
JayTaylor (ASU) APM541 Fall2013 5/77
ProbabilityandUncertainty
A Behavioristic Derivation of the Laws of Probability
We consider a two-player game, with players P and P , that make wagers on an event
1 2
with a then-unknown outcome. Here we will assume that the event can result in one of
n mutually-exclusive and exhaustive outcomes, e ,··· ,e . The game takes place in
1 n
three steps.
1 First, player P1 assigns a unit price Pr(A) to each possible event A⊂S =
{e ,··· ,e }. This is the amount that P is willing to pay for a $1 wager on the
1 n 1
event that A happens, as well as the amount that P is willing to accept from P
1 2
in exchange for such a wager. The player selling such a wager agrees to pay the
purchaser $1 if A occurs and $0 if it does not occur.
2 PlayerP2 thendecidestheamountS(A)thattheywishtowageroneacheventA.
If S(A) is positive, then P purchases a wager worth $S for S(A)·Pr(A) from P ,
2 1
while if S(A) is negative, then P sells such a wager to P for the same amount.
2 1
3 The outcome of the event is then determined and the two players settle their
wagers.
JayTaylor (ASU) APM541 Fall2013 6/77
ProbabilityandUncertainty
Although player P may assign any set of prices to the various possible wagers, many
1
assignments can be considered irrational insofar as they lead to certain loss for P .
1
Indeed, a set of prices {Pr(A):A⊂S} and a set of wagers {S(A):A⊂S} is said to
be a Dutch book if these will guarantee that P will earn a profit from P no matter
2 1
which outcome occurs. The following are examples of Dutch books.
Case 1: Suppose that Pr(A)<0 for some event A. In this case, P can agree to
2
purchase a $1 wager from P at a cost of Pr(A), i.e., P will pay P an amount |Pr(A)|
1 1 2
to take such a wager. Then P will have earned 1+|Pr(A)| if the event A occurs and
2
will have earned |Pr(A)| if the event A does not occur. Thus P makes a profit no
2
matter what occurs and thus this is a Dutch book.
Case 2: SupposethatPr(S)<1. Inthiscase, P canearnaprofitof1−Pr(S)>0by
2
purchasing a $1 wager on S. Since one of the events e ,··· ,e is certain to happen, it
1 n
follows that S will occur no matter which event occurs so that P is certain to make a
2
profit of 1−Pr(S)>0. Similarly, if Pr(S)>1, then P should sell a $1 stake in S to
2
P and this will guarantee that P will earn a profit of Pr(S)−1>0.
1 2
JayTaylor (ASU) APM541 Fall2013 7/77
ProbabilityandUncertainty
Case 3: Now suppose that A and B are disjoint subsets of S, i.e., AB ≡A∩B =∅,
and that Pr(A∪B)<Pr(A)+Pr(B). Assume that P purchases from P a $1 wager
2 1
on the event A∪B and sells to P both a $1 wager on A and a $1 wager on B. Then
1
the profit/losses incurred by P are summarized in the following table:
2
A B A∪B
-1 -1 1 total
outcome
AB¯ Pr(A)−1 Pr(B) 1−Pr(A∪B) Pr(A)+Pr(B)−Pr(A∪B)
A¯B Pr(A) Pr(B)−1 1−Pr(A∪B) Pr(A)+Pr(B)−Pr(A∪B)
A¯B¯ Pr(A) Pr(B) −Pr(A∪B) Pr(A)+Pr(B)−Pr(A∪B)
Notice that we do not consider the outcome AB since we are assuming that A and B
represent mutually exclusive events. It follows that P earns a profit no matter which
2
outcome occurs and so P has made Dutch book against P . Similarly, if Pr(A∪B)
2 1
>Pr(A)+Pr(B), then P can make a Dutch book by selling a $1 wager on A∪B and
2
purchasing a $1 wager on each event A and B.
JayTaylor (ASU) APM541 Fall2013 8/77
ProbabilityandUncertainty
The above examples show that a Dutch book can be made against P whenever the
1
prices assigned by P to events in S satisfy certain inequalities. This motivates the
1
following definition. The set of prices (Pr(A):A⊂S) is said to be coherent if it
satisfies the following conditions:
1 Pr(A)≥0 for each A⊂S;
2 Pr(S)=1;
3 Pr(A∪B)=Pr(A)+Pr(B) whenever A and B are disjoint subsets of S.
Thusourpreviousexamplesshowthatanysetofpricesthatisnotcoherentisvulnerable
to a Dutch book. In fact, the converse is also true: under the rules of the above game,
P cannot make a Dutch book against P if the prices assigned by P are coherent.
2 1 1
JayTaylor (ASU) APM541 Fall2013 9/77
ProbabilityandUncertainty
Condition (3) implies that a coherent price assignment is finitely-additive. If
A ,··· ,A are disjoint subsets of S, then
1 m
m ! m
[ X
Pr A = Pr(A).
i i
i=1 i=1
Indeed, this follows by recursion on m since we can use (3) to peel off sets
A ,A ,··· one by one from the union inside the price function:
m m−1
m ! m−1 ! m−1 !
[ [ [
Pr A = Pr A ∪A =Pr A +Pr(A )
i i m i m
i=1 i=1 i=1
m−2 !
[
= Pr A +Pr(A )+Pr(A )
i m−1 m
i=1
···
m
X
= Pr(A).
i
i=1
JayTaylor (ASU) APM541 Fall2013 10/77