probability and markov models

7/31/2019 Probability and Markov Models

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Timothy L. BaileyBIOL3014

1

Probability and Markov

Models


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2

Reading

Chapter 1 in the book.

Chapter 3 pages 46-51.


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3

Definition: Random Process

A RANDOM PROCESS is something thathas a random outcome:

Roll a die, flip a coin, roll 2 dice

Observe orthologous base pair in 2 seqs

Measure an mRNA level

Weigh a person


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Definition: Experiment

In probability theory, an EXPERIMENT is asingle observation of a random process.


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Definition: Event

An EVENT is a set of possible outcomes of anexperiment.

An ELEMENTARY EVENT is whatever you

decide it is. For example: The outcome of 1 roll of a die The outcomes of nrolls of a die

The residue at position 237 in a protein

The residues at position 237 in a family of proteins The weight of a person

Elementary events must be non-overlapping!


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Compound Events

A COMPOUND EVENT is a set of one ormore elementary events.

For example, you might define twocompound events in a die-rollingexperiment: E=roll less than 3, F=roll

greater than or equal to 3.

Then,

E = {1, 2} and F = {3, 4, 5,6}.


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Defn: Sample Space

The SAMPLE SPACE is the set of allELEMENTARY EVENTS.

So the sample space is the universe of

all possible outcomes of the experiment. This is written:

= { Ei}

For example, for rolls of a die, you mighthave: = {1, 2, 3, 4, 5, 6}


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Discrete vs. Continuous Events

The sample space might be INFINITE.For example, the weight of person can beany real number greater than 0.

Some events are DISCRETE: countable

Base pairs, residues, die rolls

Other events are CONTINUOUS: eg, realnumbers

Weights, alignment scores, mRNA levels


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The Axioms of Probability

Let E and F be events. Then the axioms ofprobability are:

1. Pr(E) 0

2. Pr() = 1

3. Pr(E U F) = Pr(E) + Pr(F) if (E F) = empty set

4. Pr(E | F) Pr(F) = Pr (E F)

E

F

E U F

E

F

E F

ProbabilityIs likearea

in Venndiagrams


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Notation

Joint Probability: Pr(E,F)

The probability of EandF

Conditional probability: Pr(E | F)

The probability of EgivenF


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Conditional Probability and Bayes

Rule

Conditional probability can be defined as:

Pr(E | F) = Pr (E,F) / Pr(F)

Bayes Rule can be used to reverse theroles of E and F:

Pr(F | E) = Pr (E|F) Pr(F) / Pr(E)


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Sequence Models

Observed biological sequences (DNA,RNA, protein) can be thought of as theoutcomes of random processes.

So, it makes sense to model sequencesusing probabilistic models.

You can think of a sequence model as alittle machine that randomly generatessequences.


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A Simple Sequence Model

Imagine a tetrahedral (four-sided) die withthe letters A, C, G and T on its sides.

You roll the die 100 times and write downthe letters that come up (down, actually).

This is a simple random sequence model.


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Zero-order Markov Model

The four-sided die model is called a

0-order Markov model.

It can be drawn thus:

M

qAqCqG

qT

0-order MarkovSequence model

Emission Probabilites

p=1 Transition probability


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Complete 0-order Markov Model

To model the length of the sequences thatthe model can generate, we need to addstart and end states.

Complete 0-order MarkovSequence model

M

qAqCqGqT

S E1 1-p

p


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Generating a Sequence

This Markov model can generateany DNA sequence.Associated with eachsequence is a path and aprobability.

1. Start in state S: P = 1

2. Move to state M: P=1P

3. Print x: P = qXP

4. Move to state M: P=pP

or to state E: P=(1-p) P5. If in state M, go to 3. If in

state E, stop.Sequence: GCAGCT

Path: S, M, M, M, M, M, M, E

P=1qGpqCpqApqGpqCpqT(1-p)

M

qAqCqGqT

S E1 1-p

p


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Using a 0-order Markov Model

This model can generate any DNA sequence, so it canbe used to model DNA.

We used it when we created scoring matrices forsequence alignment as the background model.

Its a pretty dumb model, though. DNA is not very well modeled by a 0-order Markov

model because the probability of seeing, say, a Gfollowing a C is usually different than a Gfollowing an A, (e.g, in CpG islands.)

So we need a better models: higher order Markovmodels.


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Markov Model Order

This simple sequence model iscalled a 0-order Markovmodelbecause the probabilitydistribution of the next letter tobe generated doesnt depend on

any (zero) of the letterspreceding it.

The Markov Property:

Let X = X1X2XL be a sequence.

In an n-order Markov sequence model, the probability distributionof the next letter depends on the previous n letters generated.

0-order: Pr(Xi|X1X2Xi-1)=Pr(Xi)

1-order: Pr(Xi|X1X2Xi-1)=Pr(Xi|Xi-1)

n-order: Pr(Xi|X1X2Xi-1)=Pr(Xi|Xi-1Xi-2Xi-n)

M

qAqCqGqT

S E1 1-p

p


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A 1-order Markov Sequence Model

In a first-order Markov sequence model, the probability of the next letterdepends on what the previous letter generated was.

We can model this by making a state for each letter. Each state alwaysemits the letter it is labeled with. (Not all transitions are shown.)

S E

A

Pr(A|A)

T

Pr(T|T)

G

Pr(G|G)

C

Pr(C|C)

Pr(T|A)

Pr(C|G)


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A 2-order Markov Model

To make a second order Markov sequencemodel, each state is labelled with two letters. Itemits the second letter in its label.

There would have to be sixteen states: AA, AC,AG, AT, CA, CG, CT etc., plus four states for thefirst letter in the sequence: A, C, G, T

Each state would have transitions only to stateswhose first letter matched their second letter.


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Timothy L. BaileyBIOL3014 21

Part of a 2-order Model

Each state remembers what the previous

letter emitted was in its label.

E

AA

Pr(A|AA)

AT

AG AC

Pr(T|AA)

Pr(G|AA)

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