Review of statistical modeling and probability theory

Alan Moses

ML4bio

What is modeling?

• Describe some observations in a simple, more compact way

X = (X1,X2)

What is modeling?

• Describe some observations in a simple, more compact way

Model: a = - G m

r2

Instead of all the observations, we only need to remember a constant ‘G’ and measure some parameters ‘m’ and ‘r’.

What is statistical modeling?

• Deals also with the ‘uncertainty’ in observations

Expectation

Deviation or Variance

• Also use the term ‘probabilistic’ modeling• Mathematics is more complicated

What kind of questions will we answer in this course?

What’s the best linear model to explain some data?

What kind of questions will we answer in this course?

Are there multiple groups? What are they?

What kind of questions will we answer in this course?

Given new data, which group do we assign it to?

3 major areas of machine learning

What’s the best linear model to explain some data?

Are there multiple groups? What are they?

Given new data, which group do we assign it to?

• Regression

• Clustering

• Classification

(that have proven useful in biology)

Molecular Biology exampleE

xpre

ssio

n Le

vel

Expectation

Variance

disease

X = (L,D)

Molecular Biology exampleE

xpre

ssio

n Le

vel

Expectation

Variance

E1 V1

E2 V2

“clustering”

Exp

ress

ion

Leve

l

disease

Class 2 is “enriched” for disease

Molecular Biology exampleE

xpre

ssio

n Le

vel

Expectation

Variance

E1 V1

E2 V2

“clustering”

“regression”

Genotype

AA Aa aa

Exp

ress

ion

Leve

l

Exp

ress

ion

Leve

l

disease

Class 2 is “enriched” for disease

Molecular Biology exampleE

xpre

ssio

n Le

vel

Expectation

Variance

E1 V1

E2 V2

“clustering”

“regression”

Genotype

AA Aa aa

Exp

ress

ion

Leve

l

Exp

ress

ion

Leve

l “classification”

Genotype

AA Aa aa

Exp

ress

ion

Leve

l

disease

disease?

Aa

Class 2 is “enriched” for disease

Probability theory

• Probability theory quantifies uncertainty using ‘distributions’

• Distributions are the ‘models’ and they depend on constants and parameters

E.g., in one dimension, the Gaussian or Normal distribution depends on two constants e and π and two parameters that have to be measured, μ and σ

P(X|μ,σ) = e1√2πσ2

(X–μ)2

2σ2–

‘X’ are the possible datapoints that could come from the distribution. In statistics jargon ‘X’ is called a random variable

Probability theory

• Probability theory quantifies uncertainty using ‘distributions’

• Choosing the distribution or ‘model’s the first step in a statistical model

• E.g., data: mRNA expression levels, counts of sequencing reads, presence or absence of protein domains or ‘A’ ‘C’ ‘G’ and ‘T’ s

• We will use different distributions to describe these different types of data.

Typical data and distributions

• Data is categorical (yes or no, A,C,G,T)

• Data is a fraction (e.g., 13 out of 5212)

• Data is a continuous number (e.g., -6.73)

• Data is a ‘natural’ number (0,1,2,3,4…)

• It’s also possible to do regression, clustering and classification without specifying a distribution

Molecular Biology example

“classification”

Genotype

AA Aa aa

Exp

ress

ion

Leve

l

disease?

Aa

• In this example, we might try to combine a Bernoulli for the disease data, Poisson for the genotype and Gaussian for the expression level

• We also might try to classify without specifying distributions

Molecular Biology exampleG

ene

2 E

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• genomics era means we will almost never have the expression level for just one gene or the genotype at just one locus

• Each gene’s expression level can be considered another ‘dimension’

Gene 1 Expression Level

• for two genes, if each point is data for one person, we can make a graph of this type of data

• for 1000s of genes….

Gen

e 2

Exp

ress

ion

Leve

l

Gene 1 Expression Level

Gene 3

Gene 4Gene 5…

Molecular Biology exampleG

ene

2 E

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n Le

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• genomics era means we will almost never have the expression level for just one gene or the genotype at just one locus

• We’ll usually make 2-D plots, but anything we say about 2-D can usually be generalized to n-dimensions

Gene 1 Expression Level

Each “observation” , X, contains expression level for Gene 1 and Gene 2

X = (1.3, 4.6)

Represent this as a vector:

X = (X1, X2)

e.g.,

Or generally

Molecular Biology exampleG

ene

2 E

xpre

ssio

n Le

vel

• genomics era means we will almost never have the expression level for just one gene or the genotype at just one locus

• We’ll usually make 2-D plots, but anything we say about 2-D can usually be generalized to n-dimensions

Gene 1 Expression Level

Each “observation” , X, contains expression level for Gene 1 and Gene 2

X = (1.3, 4.6)

Represent this as a vector:

X = (X1, X2)

e.g.,

Or generally

This gives a geometric interpretation to multivariate statistics

Probability theory

• Probability theory quantifies uncertainty using ‘distributions’

• Distributions are the ‘models’ and they depend on constants and parameters

E.g., in two dimensions, the Gaussian or Normal distribution depends on two constants e and π and 5 parameters that have to be measured, μ and Σ

P(X|μ,σ) = e12π

(X–μ)T Σ-1 (X–μ)2

–

‘X’ are the possible datapoints that could come from the distribution. In statistics jargon ‘X’ is called a random variable

√|Σ|

1

What does the mean mean in 2 dimensions?

What does the standard deviation mean?

Bivariate Gaussian

Molecular Biology exampleG

ene

2 E

xpre

ssio

n Le

vel

• genomics era means we will almost never have the expression level for just one gene or the genotype at just one locus

• We’ll usually make 2-D plots, but anything we say about 2-D can usually be generalized to n-dimensions

Gene 1 Expression Level

Each “observation” , X, contains expression level for Gene 1 and Gene 2

Represent this as a vector: X = (X1, X2)

The mean is also a vector: µ = (µ1, µ2)µ

The variance is a matrix: Σ σ11 σ12σ21 σ22

=

-4 -2 0 2 4

-4-2

02

4

rmvnorm(n = 300, mean = c(1, 1), sigma = matrix(c(1, -1.9, -1.9, [,1] 4), ncol = 2))[,1]rm

vno

rm(n

= 3

00

, me

an

= c

(1, 1

), s

igm

a =

ma

trix

(c(1

, -1

.9, -

1.9

, [,2

]

4),

nco

l = 2

))[,2

]

1.0 -1.9-1.9 4.0Σ =

-4 -2 0 2 4

-4-2

02

4

rmvnorm(n = 300, mean = c(1, 1), sigma = matrix(c(1, 0, 0, 4), [,1] ncol = 2))[,1]

rmvn

orm

(n =

300

, mea

n =

c(1,

1),

sigm

a =

mat

rix(c

(1, 0

, 0, 4

), [,2

]

nco

l = 2

))[,2

]

1 0 0 4Σ =

-4 -2 0 2 4

-4-2

02

4

rmvnorm(n = 300, mean = c(1, 1), sigma = matrix(c(1, 0, 0, 1), [,1] ncol = 2))[,1]rm

vnor

m(n

= 3

00, m

ean

= c(

1, 1

), si

gma

= m

atrix

(c(1

, 0, 0

, 1),

[,2]

n

col =

2))

[,2]

1 0 0 1Σ =

-4 -2 0 2 4

-4-2

02

4

rmvnorm(n = 300, mean = c(1, 1), sigma = matrix(c(1, 0.5, 0.5, [,1] 1), ncol = 2))[,1]

rmvn

orm

(n =

300

, mea

n =

c(1,

1),

sigm

a =

mat

rix(c

(1, 0

.5, 0

.5, [

,2]

1

), nc

ol =

2))

[,2]

1.0 0.5 0.5 1.0Σ =

µ

“correlated data”

“axis-aligned, diagonal covariance” “full covariance”

“spherical covariance”Σ = σ2I

Probability theory

• Probability theory quantifies uncertainty using ‘distributions’

• Distributions are the ‘models’ and they depend on constants and parameters

• Once we chose a distribution, the next step is to chose the parameters

• This is called “estimation” or “inference”

P(X|μ,σ) = e1√2πσ2

(X–μ)2

2σ2–

• Choose the parameters so the model ‘fits the data’

• There are many ways to measure how well a model fits that data • Different “Objective functions” will produce different “estimators” (E.g., MSE, ML, MAP)

EstimationE

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Expectation

Variance

We want to make a statistical model.

1.Choose a model (or probability distribution)

2.Estimate its parameters

P(X|μ,σ) = e1√2πσ2

(X–μ)2

2σ2–

How do we know which parameters fit the data?

Laws of probability

• If X1 … XN are a series of random variables (think datapoints)

P(X1 , X2) is the “joint probability” and is equal to

P(X1) P(X2) if X1 and X2 are independent.

P(X1 | X2), is the “conditional probability” of event X1 given X2

Conditional probabilities are related by Bayes’ theorem:

P(X1 … XN ) = P(Xi)Πi=1

i=N

P(X1| X2) = P(X2 |X1)P(X1)

P(X2)

(True for all distributions)

Likelihood and MLEs

• Likelihood is the probability of the data (say X) given certain parameters (say θ)

• Maximum likelihood estimation says: choose θ, so that the data is most probable.

• In practice there are many ways to maximize the likelihood.

L = P(X|θ)

L = 0θ

Example of ML estimation

Xi P(Xi|μ=6.5, σ=1.5)

Data:

= P(Xi|μ=6.5, σ=1.5) = 6.39 x 10-5 Πi=1

i=5

L = P(X|θ) = P(X1 … XN | μ, σ)

Mean, μ

L

0.1827373040.0592273220.139963680.2307610960.182737304

5.29.18.27.37.8

Example of ML estimation

Mean, μ

Log(L)

In practice, we almost always use the log likelihood, which becomes a very large negative number when there is a lot of data

Mean, μStan

dard dev

iation, σ

Log

(L)

Example of ML estimation

ML Estimation• In general, the likelihood is a function of multiple variables, so the

derivatives with respect to all of these should be zero at a maximum

• In the example of the Gaussian, we have two parameters, so that

• In general, finding MLEs means solving a set of coupled equations, which usually have to be solved numerically for complex models.

L = 0μL = 0σ

and

MLEs for the Gaussian

• The Gaussian is the symmetric continuous distribution that has as its “centre” a parameter given by what we consider the “average” (the expectation).

• The MLE for the for variance of the Gaussian is like the squared error from the mean, but is actually a biased (but still consistent!?) estimator

μML = X VML = (X - μML)2 ΣXN

1 ΣXN

1

Other estimators• Instead of likelihood, L = P(X|θ) we can choose

parameters to maximize posterior probability:

• Or sum of squared errors:

• Or a penalized likelihood: L* = P(X|θ)

• In each case, estimation involves a mathematical optimization problem that usually has to be solved on computer

• How do we choose?

P(θ|X)

(X – μMSE)2 ΣX

e– θ2

x