Conjugate Priors, Uninformative Priors Priors, Uninformative Priors Nasim Zolaktaf UBC Machine...

Conjugate Priors, Uninformative Priors

Nasim Zolaktaf

UBC Machine Learning Reading Group

January 2016

Outline

Exponential Families

Conjugacy

Conjugate priorsMixture of conjugate prior

Uninformative priors

Jeffreys prior

The Exponential Family

A probability mass function (pmf) or probability distributionfunction (pdf) p(X|θ), for X = (X1, ..., Xm) ∈ Xm and θ ⊆ Rd, issaid to be in the exponential family if it is the form:

p(X|θ) =1

Z(θ)h(X) exp[θTφ(X)] (1)

= h(X) exp[θTφ(X)−A(θ)] (2)

whereZ(θ) =

h(X) exp[θTφ(X)]dx (3)

A(θ) = logZ(θ) (4)

Z(θ) is called the partition function, A(θ) is called the log partitionfunction or cumulant function, and h(X) is a scaling constant.Equation 2 can be generalized by writing

p(X|θ) = h(X) exp[η(θ)Tφ(X)−A(η(θ))] (5)

p(X|θ) =1

whereZ(θ) =

A(θ) = logZ(θ) (4)

p(X|θ) =1

whereZ(θ) =

A(θ) = logZ(θ) (4)

p(X|θ) =1

whereZ(θ) =

A(θ) = logZ(θ) (4)

Z(θ) is called the partition function, A(θ) is called the log partitionfunction or cumulant function, and h(X) is a scaling constant.

Equation 2 can be generalized by writing

p(X|θ) =1

whereZ(θ) =

A(θ) = logZ(θ) (4)

Binomial Distribution

As an example of a discrete exponential family, consider theBinomial distribution with known number of trials n. The pmf forthis distribution is

p(x|θ) = Binomial(n, θ) =

)θx(1− θ)n−x, x ∈ {0, 1, .., n} (6)

This can equivalently be written as

p(x|θ) =

)exp(x log(

1− θ) + n log(1− θ)) (7)

which shows that the Binomial distribution is an exponentialfamily, whose natural parameter is

η = logθ

1− θ(8)

Binomial Distribution

As an example of a discrete exponential family, consider theBinomial distribution with known number of trials n. The pmf forthis distribution is

p(x|θ) = Binomial(n, θ) =

)θx(1− θ)n−x, x ∈ {0, 1, .., n} (6)

This can equivalently be written as

p(x|θ) =

)exp(x log(

1− θ) + n log(1− θ)) (7)

which shows that the Binomial distribution is an exponentialfamily, whose natural parameter is

η = logθ

1− θ(8)

Conjugacy

Consider the posterior distribution p(θ|X) with prior p(θ) andlikelihood function p(x|θ), where p(θ|X) ∝ p(X|θ)p(θ).

If the posterior distribution p(θ|X) are in the same family as theprior probability distribution p(θ), the prior and posterior are thencalled conjugate distributions, and the prior is called a conjugateprior for the likelihood function p(X|θ).

All members of the exponential family have conjugate priors.

Conjugacy

Brief List of Conjugate Models

The Conjugate Beta Prior

The Beta distribution is conjugate to the Binomial distribution.

p(θ|x) = p(x|θ)p(θ) = Binomial(n, θ) ∗ Beta(a, b) =(n

)θx(1− θ)n−x Γ(a+ b)

Γ(a)Γ(b)θ(a−1)(1− θ)b−1

∝ θx(1− θ)n−xθ(a−1)(1− θ)b−1

p(θ|x) ∝ θ(x+a−1)(1− θ)n−x+b−1 (10)

The posterior distribution is simply a Beta(x+ a, n− x+ b)

distribution.

Effectively, our prior is just adding a− 1 successes and b− 1

failures to the dataset.

Γ(a)Γ(b)θ(a−1)(1− θ)b−1

∝ θx(1− θ)n−xθ(a−1)(1− θ)b−1

p(θ|x) ∝ θ(x+a−1)(1− θ)n−x+b−1 (10)

distribution.

Γ(a)Γ(b)θ(a−1)(1− θ)b−1

∝ θx(1− θ)n−xθ(a−1)(1− θ)b−1

p(θ|x) ∝ θ(x+a−1)(1− θ)n−x+b−1 (10)

distribution.

Coin Flipping Example: Model

Use a Bernoulli likelihood for coin X landing ‘heads’,

p(X = ‘H ′|θ) = θ, p(X = ‘T ′|θ) = 1− θ,

p(X|θ) = θI(X=‘H′)(1− θ)I(X=‘T ′).

Use a Beta prior for probability θ of ‘heads’, θ ∼ Beta (a, b),

p(θ|a, b) =θa−1(1− θ)b−1

B(a, b)∝ θa−1(1− θ)b−1

with B(a, b) = Γ(a)Γ(b)Γ(a+b) .

Remember that probabilities sum to one so we have

p(θ|a, b)dθ =

θa−1(1− θ)b−1

B(a, b)dθ =

B(a, b)

θa−1(1− θ)b−1dθ

which helps us compute integrals since we have∫ 1

θa−1(1− θ)b−1dθ = B(a, b).

p(X = ‘H ′|θ) = θ, p(X = ‘T ′|θ) = 1− θ,

p(X|θ) = θI(X=‘H′)(1− θ)I(X=‘T ′).

p(θ|a, b) =θa−1(1− θ)b−1

B(a, b)∝ θa−1(1− θ)b−1

p(θ|a, b)dθ =

θa−1(1− θ)b−1

B(a, b)dθ =

B(a, b)

θa−1(1− θ)b−1dθ = B(a, b).

p(X = ‘H ′|θ) = θ, p(X = ‘T ′|θ) = 1− θ,

p(X|θ) = θI(X=‘H′)(1− θ)I(X=‘T ′).

p(θ|a, b) =θa−1(1− θ)b−1

B(a, b)∝ θa−1(1− θ)b−1

p(θ|a, b)dθ =

θa−1(1− θ)b−1

B(a, b)dθ =

B(a, b)

θa−1(1− θ)b−1dθ = B(a, b).

p(X = ‘H ′|θ) = θ, p(X = ‘T ′|θ) = 1− θ,

p(X|θ) = θI(X=‘H′)(1− θ)I(X=‘T ′).

p(θ|a, b) =θa−1(1− θ)b−1

B(a, b)∝ θa−1(1− θ)b−1

p(θ|a, b)dθ =

θa−1(1− θ)b−1

B(a, b)dθ =

B(a, b)

θa−1(1− θ)b−1dθ = B(a, b).

Coin Flipping Example: Posterior

Our model is:

X ∼ Ber(θ), θ ∼ Beta (a, b).

If we observe ‘HHH’ then our posterior distribution is

p(θ|HHH) =p(HHH|θ)p(θ)p(HHH)

(Bayes’ rule)

∝ p(HHH|θ)p(θ) (p(HHH) is constant)

= θ3(1− θ)0p(θ) (likelihood def’n)

= θ3(1− θ)0θa−1(1− θ)b−1 (prior def’n)

= θ(3+a)−1(1− θ)b−1.

Which we’ve written in the form of a Beta distribution,

θ | HHH ∼ Beta(3 + a, b),

which let’s us skip computing the integral p(HHH).

Our model is:

(Bayes’ rule)

= θ(3+a)−1(1− θ)b−1.

Our model is:

(Bayes’ rule)

= θ(3+a)−1(1− θ)b−1.

Coin Flipping Example: Estimates

If we observe ‘HHH’ with Beta(1, 1) prior, thenθ | HHH ∼ Beta (3 + a, b) and our different estimates are:

Maximum likelihood:

θ̂ =nHn

MAP with uniform,

θ̂ =(3 + a)− 1

(3 + a) + b− 2=

Posterior predictive,

p(H|HHH) =

p(H|θ)p(θ|HHH)dθ

Ber(H|θ)Beta (θ|3 + a, b)dθ

θBeta(θ|3 + a, b)dθ = E[θ]

=(3 + a)

(3 + a) + b=

5= 0.8.

Maximum likelihood:

θ̂ =nHn

MAP with uniform,

θ̂ =(3 + a)− 1

(3 + a) + b− 2=

p(H|HHH) =

p(H|θ)p(θ|HHH)dθ

=(3 + a)

(3 + a) + b=

5= 0.8.

Maximum likelihood:

θ̂ =nHn

MAP with uniform,

θ̂ =(3 + a)− 1

(3 + a) + b− 2=

p(H|HHH) =

p(H|θ)p(θ|HHH)dθ

=(3 + a)

(3 + a) + b=

5= 0.8.

Maximum likelihood:

θ̂ =nHn

MAP with uniform,

θ̂ =(3 + a)− 1

(3 + a) + b− 2=

p(H|HHH) =

p(H|θ)p(θ|HHH)dθ

=(3 + a)

(3 + a) + b=

5= 0.8.

Coin Flipping Example: Effect of Prior

We assume all θ equally likely and saw HHH,ML/MAP predict it will always land heads.Bayes predict probability of landing heads is only 80%.

Takes into account other ways that HHH could happen.

Beta(1, 1) prior is like seeing HT or TH in our mind before we flip,Posterior predictive would be 3+1

3+1+1= 0.80.

Beta(3, 3) prior is like seeing 3 heads and 3 tails (strongeruniform prior),

Posterior predictive would be 3+33+3+3

= 0.667.

Beta(100, 1) prior is like seeing 100 heads and 1 tail (biased),Posterior predictive would be 3+100

3+100+1= 0.990.

Beta(0.01, 0.01) biases towards having unfair coin (head or tail),Posterior predictive would be 3+0.01

3+0.01+0.01= 0.997.

Dependence on (a, b) is where people get uncomfortable:But basically the same as choosing regularization parameter λ.If your prior knowledge isn’t misleading, you will not overfit.

3+1+1= 0.80.

= 0.667.

3+100+1= 0.990.

3+0.01+0.01= 0.997.

3+1+1= 0.80.

= 0.667.

3+100+1= 0.990.

3+0.01+0.01= 0.997.

3+1+1= 0.80.

= 0.667.

3+100+1= 0.990.

3+0.01+0.01= 0.997.

Mixtures of Conjugate Priors

A mixture of conjugate priors is also conjugate.

We can use a mixture of conjugate priors as a prior.

For example, suppose we are modelling coin tosses, and wethink the coin is either fair, or is biased towards heads. Thiscannot be represented by a Beta distribution. However, we canmodel it using a mixture of two Beta distributions. For example,we might use:

p(θ) = 0.5 Beta(θ|20, 20) + 0.5 Beta(θ|30, 10) (11)

If θ comes from the first distribution, the coin is fair, but if it comesfrom the second it is biased towards heads.

p(θ) = 0.5 Beta(θ|20, 20) + 0.5 Beta(θ|30, 10) (11)

Mixtures of Conjugate Priors (Cont.)

The prior has the form

p(θ) =∑k

p(z = k)p(θ|z = k) (12)

where z = k means that θ comes from mixture component k,p(z = k) are called the prior mixing weights, and each p(θ|z = k)

is conjugate.

Posterior can also be written as a mixture of conjugatedistributions as follows:

p(θ|X) =∑k

p(z = k|X)p(θ|X, z = k) (13)

where p(z = k|X) are the posterior mixing weights given by

p(z = k|X) =p(z = k)p(X|z = k)∑

k′ p(z = k′|X)p(θ|X, z = k′)(14)

Mixtures of Conjugate Priors (Cont.)

The prior has the form

p(θ) =∑k

p(z = k)p(θ|z = k) (12)

where z = k means that θ comes from mixture component k,p(z = k) are called the prior mixing weights, and each p(θ|z = k)

is conjugate.

Posterior can also be written as a mixture of conjugatedistributions as follows:

p(θ|X) =∑k

p(z = k|X)p(θ|X, z = k) (13)

where p(z = k|X) are the posterior mixing weights given by

p(z = k|X) =p(z = k)p(X|z = k)∑

k′ p(z = k′|X)p(θ|X, z = k′)(14)

Uninformative Priors

If we don’t have strong beliefs about what θ should be, it iscommon to use an uninformative or non-informative prior, and tolet the data speak for itself.

Designing uninformative priors is tricky.

Uninformative Prior for the Bernoulli

Consider the Bernoulli parameter

p(x|θ) = θx(1− θ)n−x (15)

What is the most uninformative prior for this distribution?The uniform distribution Uniform(0, 1)?.

This is equivalent to Beta(1, 1) on θ.We can predict the MLE is N1

N1+N0, whereas the posterior mean is

E[θ|X] = N1+1N1+N0+2

. Prior isn’t completely uninformative!

By decreasing the magnitude of the pseudo counts, we canlessen the impact of the prior. By this argument, the mostuninformative prior is Beta(0, 0), which is called Haldane prior.

Haldane prior is an improper prior; it does not integrate to 1.Haldane prior results in the posterior Beta(x, n− x) which will beproper as long as n− x 6= 0 and x 6= 0.

We will see that the ”right” uninformative prior is Beta( 12 ,

p(x|θ) = θx(1− θ)n−x (15)

What is the most uninformative prior for this distribution?

The uniform distribution Uniform(0, 1)?.This is equivalent to Beta(1, 1) on θ.We can predict the MLE is N1

E[θ|X] = N1+1N1+N0+2

p(x|θ) = θx(1− θ)n−x (15)

E[θ|X] = N1+1N1+N0+2

p(x|θ) = θx(1− θ)n−x (15)

This is equivalent to Beta(1, 1) on θ.

We can predict the MLE is N1N1+N0

, whereas the posterior mean isE[θ|X] = N1+1

N1+N0+2. Prior isn’t completely uninformative!

p(x|θ) = θx(1− θ)n−x (15)

E[θ|X] = N1+1N1+N0+2

p(x|θ) = θx(1− θ)n−x (15)

E[θ|X] = N1+1N1+N0+2

p(x|θ) = θx(1− θ)n−x (15)

E[θ|X] = N1+1N1+N0+2

p(x|θ) = θx(1− θ)n−x (15)

E[θ|X] = N1+1N1+N0+2

Haldane prior is an improper prior; it does not integrate to 1.

Haldane prior results in the posterior Beta(x, n− x) which will beproper as long as n− x 6= 0 and x 6= 0.

p(x|θ) = θx(1− θ)n−x (15)

E[θ|X] = N1+1N1+N0+2

p(x|θ) = θx(1− θ)n−x (15)

E[θ|X] = N1+1N1+N0+2

Jeffreys Prior

Jeffrey argued that a uninformative prior should be invariant tothe parametrization used. The key observation is that if p(θ) isuninformative, then any reparametrization of the prior, such asθ = h(φ) for some function h, should also be uninformative.

Jeffreys prior is the prior that satisfies p(θ) ∝√detI(θ), where

I(θ) is the Fisher information for θ, and is invariant underreparametrization of the parameter vector θ.The Fisher information is a way of measuring the amount ofinformation that an observable random variable X carries aboutan unknown parameter θ which the probability of X depends.

I(θ) = E[(∂

∂θlog f(X; θ))

|θ] (16)

If log f(X; θ) is twice differentiable with respect to θ, then

I(θ) = −E[∂2

∂θ2log f(X; θ)|θ] (17)

Jeffreys Prior

Jeffrey argued that a uninformative prior should be invariant tothe parametrization used. The key observation is that if p(θ) isuninformative, then any reparametrization of the prior, such asθ = h(φ) for some function h, should also be uninformative.Jeffreys prior is the prior that satisfies p(θ) ∝

√detI(θ), where

I(θ) is the Fisher information for θ, and is invariant underreparametrization of the parameter vector θ.

The Fisher information is a way of measuring the amount ofinformation that an observable random variable X carries aboutan unknown parameter θ which the probability of X depends.

I(θ) = E[(∂

∂θlog f(X; θ))

|θ] (16)

I(θ) = −E[∂2

∂θ2log f(X; θ)|θ] (17)

Jeffreys Prior

√detI(θ), where

I(θ) = E[(∂

∂θlog f(X; θ))

|θ] (16)

I(θ) = −E[∂2

∂θ2log f(X; θ)|θ] (17)

Jeffreys Prior

√detI(θ), where

I(θ) = E[(∂

∂θlog f(X; θ))

|θ] (16)

I(θ) = −E[∂2

∂θ2log f(X; θ)|θ] (17)

Reparametrization for Jeffreys Prior: One parameter

For an alternative parametrization φ we can derive p(φ) =√I(φ)

from p(θ) =√I(θ), using the change of variables theorem and

the definition of Fisher information:

p(φ) = p(θ)| dθdφ| ∝

√I(θ)(

dφ)2 =

√E[(

dθ)2](

√E[(

dφ)2] =

√E[(

dφ)2] =

√I(φ)

Jeffreys Prior for the Bernoulli

Suppose X ∼ Ber(θ). The log-likelihood for a single sample is

log p(X|θ) = X logθ + (1−X) log(1− θ) (19)

The score function is gradient of log-likelihood

s(θ) =d

dθlog(pX|θ) =

θ− 1−X

1− θ(20)

The observed information is

J(θ) = − d2

dθ2log p(X|θ) = −s′(θ|X) =

1−X(1− θ)2 (21)

The Fisher information is the expected information

I(θ) = E[J(θ|X)|X ∼ θ] =θ

1− θ(1− θ)2 =

θ(1− θ)(22)

Hence Jeffreys prior is

p(θ) ∝ θ− 12 (1− θ)−

12 ∝ Beta(

2). (23)

s(θ) =d

dθlog(pX|θ) =

θ− 1−X

1− θ(20)

J(θ) = − d2

1−X(1− θ)2 (21)

I(θ) = E[J(θ|X)|X ∼ θ] =θ

1− θ(1− θ)2 =

θ(1− θ)(22)

p(θ) ∝ θ− 12 (1− θ)−

12 ∝ Beta(

2). (23)

s(θ) =d

dθlog(pX|θ) =

θ− 1−X

1− θ(20)

J(θ) = − d2

1−X(1− θ)2 (21)

I(θ) = E[J(θ|X)|X ∼ θ] =θ

1− θ(1− θ)2 =

θ(1− θ)(22)

p(θ) ∝ θ− 12 (1− θ)−

12 ∝ Beta(

2). (23)

s(θ) =d

dθlog(pX|θ) =

θ− 1−X

1− θ(20)

J(θ) = − d2

1−X(1− θ)2 (21)

I(θ) = E[J(θ|X)|X ∼ θ] =θ

1− θ(1− θ)2 =

θ(1− θ)(22)

p(θ) ∝ θ− 12 (1− θ)−

12 ∝ Beta(

2). (23)

Selected Related Work

[1] Kevin P Murphy(2012)

Machine learning: a probabilistic perspective

MIT press

[2] Jarad Niemi

Conjugacy of prior distributions:https://www.youtube.com/watch?v=yhewYFqGjFA

[3] Jarad Niemi

Noninformative prior distributions:https://www.youtube.com/watch?v=25-PpMSrAGM

Fisher information:https://en.wikipedia.org/wiki/Fisher_information

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