Programme Introduction A Bayesian approach Main results Simulation study

Adaptive Bayes Test for monotonicityENSAI-ENSAE

J-B. Salomond

CREST - Dauphine

22-23 Mars 2012

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Programme Introduction A Bayesian approach Main results Simulation study

Table des matières

1 Introduction

Testing for Monotonicity

Motivations

2 A Bayesian approach

Our Model

The testing procedure

3 Main results

4 Simulation study

Useful remark

Experimental design

Results

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Programme Introduction A Bayesian approach Main results Simulation study

Testing for Monotonicity

Introduction

We consider the regression model with Gaussian residuals

Yi = f (i/n) + ǫi

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Testing for Monotonicity

Introduction

We consider the regression model with Gaussian residuals

Yi = f (i/n) + ǫi

Aim

We want to test f is monotone non increasing versus f is not

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Programme Introduction A Bayesian approach Main results Simulation study

Testing for Monotonicity

Introduction

We consider the regression model with Gaussian residuals

Yi = f (i/n) + ǫi

Aim

We thus construct a Bayesian testing procedure which

has good asymptotic properties

is easy to implement and does not require heavy

computations, even for large datasets

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Motivations

Why ?

Monotonicity appears in many applications (drug response models

for instance)

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Motivations

Why ?

Monotonicity appears in many applications (drug response models

for instance)

Many results in the frequentist literature

but no Bayesian results are known

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Programme Introduction A Bayesian approach Main results Simulation study

Our Model

Our Model

Given a partition (Ii )i of (0, 1) in k steps, we consider a piecewise

constant approximation of the regression function

fω,k(.) =

k∑

i=1

ωi1Ii (.)

and put a prior on f by choosing a prior on ω and k

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Our Model

Our Model cnt’d

When f is monotone, fω,k will be monotone

Idea for a test

We will thus test the monotonicity of the sequence (ωi )i and thus

need

A criteria for monotonicity

Conditions on the prior such that fω,k concentrates around f

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Programme Introduction A Bayesian approach Main results Simulation study

The testing procedure

The testing procedure

We consider H(ω, k) = maxj>i (ωj − ωi). We thus have a test

δπn = 1

{

π(

H(ω, k) > Mkn |Yn

)

> 1/2}

where Mkn is a threshold such that our test is consistent.

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Programme Introduction A Bayesian approach Main results Simulation study

The testing procedure

Required asymp. properties

Let F be the set of monotone non increasing functions. We

would like our test to be consistent

supf ∈F

En0(δπn ) = o(1) (1a)

supf ,d(f ,F)>ρ,f ∈Hα(L)

En0(1− δ

πn ) = o(1) (1b)

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The testing procedure

Required asymp. properties cnt’d

We would like our test to achieve an optimal separation rate

supf ∈F

En0(δπn ) = o(1) (2a)

supf ,d(f ,F)>ρn,f ∈Hα(L)

En0(1 − δ

πn ) = o(1) (2b)

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Main Theorem

Theorem

Let Mkn = M0

√

k log(n)/n and let π a prior on fω,k such that

ωi |kiid∼ g and k ∼ πk . Assume that g puts mass on R and that πk

is such that their exist positive constants Cd and Cu such that

eCdkL(k) ≤ πk (k) ≤ eCukL(k)

Where L(k) is either log(k) or 1. Consider the test

H0 : f ∈ F versus H1 : f 6∈ F , f ∈ Hα(L)

let ρn = M(

n/ log(n))−α/(2α+1)

, then δπn is consistent and

achieve the separation rate ρn

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Useful remark

Some useful results

Some specific choices for the prior can be handy.

Prior

We take k ∼ P(λ) and ω|k ∼ N (m, v2). Then, if ǫi ∼ N (0, σ2)

we get

πk(k |Yn) = C (Y n)e

1/2∑

i

(

Yi/σ2+m/v2

ni/σ2+1/v2

)2k∏

i=1

(

ni/σ2 + 1/v2

)1/2πk(k)

ωi |Yn, k ∼ N

(

m/v2 + Yi/σ2

ni/σ2 + 1/v2;

1

ni/σ2 + 1/v2

)

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Programme Introduction A Bayesian approach Main results Simulation study

Experimental design

Experimental design

We choose nine regression functions and generate N independent

samples yi , i = 1 . . . n. For each sample we perform our test and

approximate the proportion of rejection.

x

Fun

ctio

ns

0.0

0.5

1.0

1.5

2.0

−0.4

−0.2

0.0

0.2

0.4

−1.8

−1.6

−1.4

−1.2

−1.0

F1

F4

F7

0.0 0.2 0.4 0.6 0.8 1.0

0.000.020.040.060.080.100.120.14

−0.15

−0.10

−0.05

0.00

0.05

0.10

−0.5

−0.4

−0.3

−0.2

−0.1

0.0

F2

F5

F8

0.0 0.2 0.4 0.6 0.8 1.0

0.05

0.10

0.15

−0.6

−0.4

−0.2

0.0

0.2

0.4

0

F3

F6

F9

0.0 0.2 0.4 0.6 0.8 1.0

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Experimental design

Experimental design cnt’d

For some empirically realistic value for the parameters, simulate

N = 25 samples for n = 100, 250, 500, 1000.� �Still performing simulation for larger values of N and n

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Results

Results

Good results even for small sample size

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Results

Results cnt’d

We now compare our result with the existing procedures

Function σ2 Bayes Sregn TB

f1 0.01 1.00 0.99 0.99

f2 0.01 1.00 1.00 0.99

f3 0.01 1.00 0.98 0.99

f4 0.01 1.00 0.99 1.00

f5 0.004 1.00 0.99 0.99

f6 0.006 1.00 0.99 0.98

f7 0.01 0.68 0.68 0.76

Table: Comparison with the existing procedures for n = 100

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Programme Introduction A Bayesian approach Main results Simulation study

Results

Results cnt’d

Comparison with the

Bayes Factor approach

We compute the following

Bayes Factor

BF01 =

π(H(ω, k) ≤ 0|Y n)

π(H(ω, k) > 0|Y n)×

π(H(ω, k) > 0)

π(H(ω, k) ≤ 0)

n

100 250 500 1000

f1 0.00 0.00 0.00 0.00

f2 0.00 0.00 0.00 0.00

f3 0.02 0.00 0.00 0.00

f4 0.00 0.00 0.00 0.00

f5 0.00 0.00 0.00 0.00

f6 0.00 0.00 0.00 0.00

f7 0.00 0.00 0.00 0.00

f8 0.94 0.37 0.23 0.10

f9 0.11 0.02 0.00 0.00

Table: B01

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Programme Introduction A Bayesian approach Main results Simulation study

Results

Results cnt’d

Comparison with the

Bayes Factor approach

Satisfying results under

H1, but our procedure

perform better under H0

n

100 250 500 1000

f1 0.00 0.00 0.00 0.00

f2 0.00 0.00 0.00 0.00

f3 0.02 0.00 0.00 0.00

f4 0.00 0.00 0.00 0.00

f5 0.00 0.00 0.00 0.00

f6 0.00 0.00 0.00 0.00

f7 0.00 0.00 0.00 0.00

f8 0.94 0.37 0.23 0.10

f9 0.11 0.02 0.00 0.00

Table: B01

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