Probabilistic Prediction Algorithms

Jon Radoff

Biophysics 101

Fall 2002

Bayesian Decision Theory

• Originally developed by Thomas Bayes in 1763.*

• The general idea is that the likelihood of a future event occurring is based on the past probability that it occurred.

*Thomas Bayes. An essay towards solving a problem in the doctrine of chances. Philosophical Transaction of the Royal Society (London), 53:370-418, 1763.

Bayes Theorem: Basic Example

A simplified Bayes Theorem simply tells us that in the absence of other evidence, the likelihood of an event is equal to its past likelihood. It assumes that the consequences of an incorrect classification are always the same (unlike, for example, a state such as “infected with HIV vs. uninfected with HIV).

Let’s say we have a basket full of apples and oranges. We remove 10 fruit from the basket, and observe that 8 are apples and 2 are oranges. A friend comes along and picks a fruit, but hides it behind their back, challenging us to guess what the fruit is. What’s our guess?

Bayes Theorem: Basic Example

In the format of Bayes Theorem, we’d give each possibility a “class,” typically. We’ll designate this with the set w:

w={w1,w2}w1=applew2=orange

Based on prior information, P(w1)=0.8 and P(w2)=0.2. Since P(w1) > P(w2), we guess that it is an apple.

Bayes: Multiple Evidence

You might have multiple pieces of evidence. For example, in addition to knowing the likelihood of a random fruit from our basket, perhaps we had previously used a colorimeter to that let us associate the wavelength of light emitted from a fruit with the type of fruit it was. In an informal form, you could think of this as:

Posterior = likelihood X prior / evidence

Example: ContinuedLet’s say we had taken those light readings earlier, and generated a graph of the probability density of detecting a particular wavelength of light given a particular fruit. We’ll call the probability density p(x|wj). The wavelength of light will be represented by the random variable x.

Note that since this is a probability density, the area under either curve is always 1. 0

0.05

0.1

0.15

0.2

0.25

0.3

w1 (apple)

w2 (orange)

Bayes: Formal Definition

Let P(x) be the probability mass of an event. Let p(x) be the probability density of an event (lowercase p).

P(wj|x) = p(x|wj) X P(wj) / p(x)

Bayes: Example, Continued

Let’s redo our original experiment. Our friend is going to take a fruit from the basket again, but he’s also going to tell us that the wavelength of light detected by his colorimeter was 575nm. What is our prediction now?

Bayes: Example, Continued

Previous probabilities from our light readings:

p(575|w1) = .05, p(575|w2) = .25

P(w1|575) = 0.05 X 0.8/ ((.05 X 0.8)+(0.25 X 0.2)) = .44

P(w2|575) = 0.25 X 0.2/ ((.05 X 0.8)+(0.25 X 0.2)) = .56

In this case, the additional evidence from the colorimeter leads us to guess that it is an orange (.56 > .44).

Bayesian Belief NetworksA belief network consists of nodes labeled by their discrete states. The links between nodes represent conditional probabilities. Links are directional: when A points at B, A is said to influence B.

A B

C

DE

P(a) P(b)

P(c|a) P(c|b)

P(e|c) P(d|c)

Bayesian Belief Network: ExampleThis over-simplified network illustrates how lung cancer is influenced by other states, and how the presence of particular symptoms might be influenced by lung cancer.

ASmoker

BAsbestosexposure

CLung

Cancer

DChest pain

ECoughing

P(a) P(b)

P(c|a) P(c|b)

P(e|c) P(d|c)

Bayesian Belief Networks: Example

A human expert might provide us with the matrices of all the probabilities in the network, for example:

P(cancer|smoking):

cancer healthyNever 0.001 0.999former 0.005 0.995heavy 0.06 0.94light 0.04 0.96

…and so forth for the 4 other nodes.

Bayesian Belief Networks

If you have complete matrices for your belief network, you can make predictions for any state in the network given a set of input variables. According to Bayes Theorem, if we don’t know a particular state, we can simply default to the overall prior probability.

Bayesian Belief Networks: Example Continued

In our example, we could now answer questions such as:

•What is the likelihood a person will have lung cancer given that they are a heavy smoker and have been exposed to asbestos?•A person has severe coughing, chest pain and is a smoker. What is the likelihood of a cancer diagnosis?•What is the likelihood of past absestos exposure given that a person has been diagnosed with cancer?

Bayesian Belief Networks

The probability of a particular state is the product of the probabilities of all the states, given their prior states.

Remember that p(x) is for probability density, and P(x) is for probability mass.

d

ikik xpp

1

)|()x|(

Markov Chain

A markov chain is a type of belief network where you have a sequence of states (x1, x2 … xi) where the probability of each state is dependent only on the previous state, i.e:

P(x1,x2,…xi,xi+1)=P(xi+1|x1,x2,…xi)P(x1,x2,…xi)

Some content in this section from Matthew Wright, Hidden Markov Models

Markov Chain: Example

Let’s say we know the probability of any particular nucleotide following another nucleotide in a DNA sequence. For example, the probability that a C follows an A might be written as P(AC). If we wanted to find the probability of finding the sequence ACGTC, it would be expressed as follows:

P(ACGTC)=P(A)P(AC)P(CG)P(GT)P(TC)

Hidden Markov Models

In a Hidden Markov Model (HMM) the a Markov Chain is expanded to include the idea of hidden states. Given a set of observations x1, x2…xn and a set of hidden underlying states s1, s2…sn, there is now a transition probability for moving between the hidden states:

…where l and k are the states at positions I and i-1.

)|( 1 kslsPa iikl

Emission Probabilities

At each state, there is a probability of “emitting” a particular observation. We define this as:

)|( ksbxPe iikb

…where e is the probability that the state k at position i emits observation b, and si is the state at position i and xi is the observation at that point.

Probability of a Path

The probability of an individual path through a sequence of hidden states is a restatement of Bayes theorem:

issxss iiii

aeasxP110

),(

In other words, the probability that we observe the sequence of visible states is equal to the product of the conditional probability that the system has made a particular transition multiplied by the probability that it emitted the observation in our target sequence.

HMM Example: CpG IslandLet’s go back to our earlier example of the sequence ACGTC, but now we’ll introduce two hidden states: we may either be “in” or “out” of a CpG island. Let’s designate in by + and out by -. Here is how you would depict all the transitions visually:

A C G T CA+ C+ G+ T+ C+

A- C- G- T- C-

HMM Example: CpG IslandWhat is the probability that we’re in a CpG island at position 3? (G+) We could depict all of the potential paths to this as follows:

A C G T CA+ C+ G+ T+ C+

A- C- G- T- C-

Simplified Total Probability of a Path

In this example, we exclude emission probabilities because there won’t be a case of something such as a C- state emitting an A observation. Algebraically, the recursion of our formula for determining the probability of a path simplifies to:

k

ikkliL 1,, aa

HMM Example: CpG IslandSince this is a recursive algorithm, we need to find position 2 before we can find position 3. The possibility of C- at position 2 may be expressed as:

A C G T CA+ C+ G+ T+ C+

A- C- G- T- C-

)a)(a()a)(a( ,0,,0,2, ACAACAC

HMM Example: CpG IslandLikewise, the likelihood of C+ in position 2 is:

A C G T CA+ C+ G+ T+ C+

A- C- G- T- C-

)a)(a()a)(a( ,0,,0,2, ACAACAC

HMM Example: CpG IslandThe probability of G+ in position 3 is thus:

A C G T CA+ C+ G+ T+ C+

A- C- G- T- C-

)]a)(a()a)(a)[(a()]a)(a()a)(a)[(a(

,0,,0,,

,0,,0,,3,

ACAACAGC

ACAACAGCG