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Model Selection and Related Topics

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Title: Model Selection and Related Topics


1
Overview Model Selection Theory Computing Continuo
us Model Selection
Model Selection and Related Topics
A mostly Bayesian perspective
David Madigan Rutgers
2
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian basics Bayes factors
Bayesian Basics
  • The Bayesian approach to statistical inference
    computes probability distributions for all
    unknowns (model parameters, future observables,
    etc.) conditional on the observed data
  • Thus, denoting by q the unknowns, we compute
  • p(q data) ? p(data q) x p(q)
  • To play this game you need a prior, p(q), and a
    likelihood, p(data q).

3
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian basics Bayes factors
Bayesian Priors (per D.M. Titterington)
4
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian basics Bayes factors
Bayesian Basics
  • This presentation will focus mostly on the
    model p(data q)
  • An idealization of the probabilistic process by
    which mother nature generates the data
  • Frequently, a data analyst will entertain several
    models for the data p(data q1,M1), p(data
    q2,M2), etc.
  • This gives rise the model selection problem
    (including model composition)

5
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian basics Bayes factors
Bayes Factors comparing two models/hypotheses
  • Bayes Factors compare the posterior to prior odds
    of one hypothesis to the posterior to prior odds
    of another hypothesis

6
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian basics Bayes factors
Interpretation of Bayes Factors
  • Jeffreys suggested the following scale for
    interpreting the numerical value of a Bayes
    Factor

(0.03)
7
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian basics Bayes factors
Interpretation of Bayes Factors
  • Note that the Bayes Factor involves model
    probabilities both prior, p(M), and posterior,
    p(Mdata)
  • p(M) is the probability that model M generated
    the data
  • What if we dont believe that any of the model
    generated the data?

8
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian basics Bayes factors
Bayes Factors Example (Draper)
  • Here is density estimate for 100 univariate
    observations, y1,…,y100
  • M0 yi N(m,t)
  • M1 yi t(m,t,n)

9
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian basics Bayes factors
Bayes Factors Example (cont.)
  • Need to specify priors for everything

10
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian basics Bayes factors
Bayes Factors Example (Draper)
  • M0 yi N(m,t)
  • M1 yi t(m,t,n)
  • K is about 0.04
  • Interesting tidbit
  • posterior standard deviation of m given M0 0.165
  • posterior standard deviation of m given M1 0.153
  • (so model averaging can reduce the posterior
    standard deviation)

11
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian basics Bayes factors
Bayes Factors and Improper Priors
  • Using improper parameter priors means the Bayes
    factor contains a ratio of unknown constants
  • Lots of work trying to get around this
    fractional Bayes factors, partial Bayes factors,
    intrinsic Bayes factors, etc.
  • Simpler solution use proper priors

12
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian basics Bayes factors
Bayes Factors and Model Probabilities
  • Note that posterior models probabilities can
    derive from all pairwise Bayes factors and
    pairwise prior odds

13
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Bayesian Model Selection
  • If we believe that one of the candidate models
    generated the data, then the predictively optimal
    model has highest posterior probability
  • This is also true for variable selection with
    standard linear models when XTX is diagonal, s2
    is known, and suitable priors are used (Clyde and
    Parmigiani, 1996)

14
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
The Median Probability Model (Barbieri and
Berger, 2004)
15
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
The Median Probability Model (Barbieri and
Berger, 2004)
16
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Deviance Information Criterion (DIC)
  • Deviance is a standard measure of model fit
  • Can summarize in two ways…at posterior mean or
    mode
  • (1)
  • or by averaging over the posterior
  • (2)
  • (2) will be bigger (i.e., worse) than (1)

17
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Deviance Information Criterion (DIC)
  • is a measure of model complexity.
  • In the normal linear model pD(1) equals the
    number of parameters
  • More generally pD(1) equals the number of
    unconstrained parameters
  • DIC
  • Approximately optimizes predictive log loss

18
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Other Selection Criteria
  • The training error rate
  • will usually be less than the true error
  • Typically work with error estimates of the form
  • where is an estimate of the optimism

19
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Specific Selection Criteria
(squared error loss)
20
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Selection Criteria - Theory
  • BIC is consistent when the true model is fixed
  • AIC is consistent if the dimensionality of the
    true model increases with N at an appropriate
    rate
  • For standard linear models with known variance
    AIC and Cp are essentially equivalent
  • Folklore is that AIC tends to overfit

21
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Cross-validation
  • Since we dont usually believe that one of the
    candidate models generated the data and
    predictive accuracy on future data is key, many
    authors argue in favor of cross-validation
  • For example (Bernardo and Smith, 1984, 6.1.6)
    select the model that maximizes
  • where xn-1(j) represents the data with
    observation xj removed and x1,…xk is a random
    sample from the data

22
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Cross-validation
  • Cross-validation give a slightly biased estimate
    of future accuracy because it does not use all
    the data
  • Burman (1989) provides a bias correction

23
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Variable Selection
  • Important special case of model selection
  • Which subset of X1,…,Xd to use as predictors of
    a response variable Y ?
  • 2d possible models. For d30, there are 109
    models. For d50, there are gt1015

24
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Variable Selection for Linear Models
Here the Xs might be
  • Raw predictor variables (continuous or
    coded-categorical)
  • Transformed predictors (X4log X3)
  • Basis expansions (X4X32, X5X33, etc.)
  • Interactions (X4X2 X3 )

Popular choice for estimation is least squares
25
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Variable Selection for Linear Models
  • Standard all-subsets finds the subset of size
    k, k1,…,p, that minimizes RSS

  • Choice of subset size requires tradeoff AIC,
    BIC, marginal likelihood, cross-validation, etc.
  • Leaps and bounds is an efficient algorithm to
    do all-subsets up to about 40 variables

26
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Bayesian Variable Selection
  • Two key challenges
  • - Exploring a space of 2d models (more about this
    later)
  • - Choosing a p(M) ? p(g) where g indexes models
  • Many applications use p(M) ? 1 but this induces a
    binomial distribution over model size
  • Denison et al (1998) use a truncated Poisson
    prior distribution for model size

27
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Selection Bias
  • Selection bias is a significant unresolved issue
  • Searching model space to find the best model
    tends to overfit the data
  • This holds even when using the close-to-unbiased
    estimate of predictive performance that
    cross-validation provides

28
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Vehtari and Lampinen example
  • Data
  • Model

29
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Vehtari and Lampinen example
30
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Vehtari and Lampinens Solution
  • Select the simplest model that gives a predictive
    distribution that is close to the BMA
    predictive distribution
  • Not obvious how to conduct this search in
    high-dimensional problems

31
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Cross-validation and model complexity

One standard error rule pick the simplest model
within one standard error of the minimum
32
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Post-Model Selection Statistical Inference
  • Conducting a data-driven model search and then
    proceeding as if the search never took place
    leads to biased and overconfident inferences
  • Some non-Bayesian work on adjustment for model
    selection (e.g., current issue of JASA)

33
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
34
Overview Model Selection Theory Computing Continuo
us Model Selection
Bayesian model selection Model scores Variable
selection Model averaging
Bayesian Model Averaging
  • If we believe that one of the candidate models
    generated the data, then the predictively optimal
    strategy is to average over all the models.
  • If Q is the inferential target, Bayesian Model
    Averaging (BMA) computes
  • Substantial empirical evidence that BMA provides
    better prediction than model selection

35
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Laplaces Method for p(DM) ? p(Dq,M)p(qM)dq
(i.e., the log of the integrand divided by n)
then
and
where
is the posterior mode
36
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
  • Tierney Kadane (1986, JASA) show the
    approximation is O(n-1)
  • Using the MLE instead of the posterior mode is
    also O(n-1)
  • Using the expected information matrix in ? is
    O(n-1/2) but convenient since often computed by
    standard software
  • Raftery (1993) suggested approximating by a
    single Newton step starting at the MLE
  • Note the prior is explicit in these approximations

37
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Monte Carlo Estimates of p(DM)
Draw iid ?1,…, ?m from p(?)
In practice has large variance
38
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Better Monte Carlo Estimates of p(DM)
Draw iid ?1,…, ?m from p(?D)
Importance Sampling
39
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
  • Newton and Rafterys Harmonic Mean Estimator
  • Unstable in practice and needs modification

40
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
p(DM) from Gibbs sampler output (Chibs method)
First note the following identity (for any ? )
p(D?) and p(?) are usually easy to evaluate.
What about p(?D)?
Suppose we decompose ? into (?1,?2) such that
p(?1D,?2) and p(?2D,?1) are available in
closed-form…
41
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
The Gibbs sampler gives (dependent) draws from
p(?1, ?2 D) and hence marginally from p(?2 D)…
Rao-Blackwellization
42
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
What about three parameter blocks?
OK
OK
?
To get these draws, continue the Gibbs sampler
sampling in turn from
and
43
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
p(DM) from Metropolis sampler output (Chib
Jeliazkov)
44
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
E1 with respect to ?y
E2 with respect to q(?, ?)
45
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Savage-Dickey Density Ratio
  • Suppose M0 simplifies M1 by setting one parameter
    (say q1) to some constant (typically zero)
  • If p1(q2 q1 0) p0(q2) then

p(data M0)
p(q1 0 M1, data)

p(data M1)
p(q1 0 M1)
46
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Bayesian Information Criterion (BIC)
(SL is the negative log-likelihood)
  • BIC is an O(1) approximation to p(DM)
  • Circumvents explicit prior
  • Approximation is O(n-1/2) for a class of priors
    called unit information priors.
  • No free lunch (Weakliem (1998) example)

47
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
48
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
49
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
50
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Computing Variable Selection via Stepwise Methods
  • Efroymsons 1960 algorithm still the most widely
    used

51
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Efroymson
  • F-to-Enter
  • F-to-Remove
  • Guaranteed to converge
  • Not guaranteed to converge to the right model…

Distribution not even remotely like F
52
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Trouble
  • Y X1 X2
  • Y almost orthogonal to X1 and X2
  • Forward selection and Efroymson pick X3 alone

53
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
More Trouble
  • Berk Example with 4 variables
  • The forward and backward sequence is (X1, X1X2,
    X1X2 X4)
  • The R2 for X1X2 is 0.015

54
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Even More Trouble
  • Detroit example, N13, d11
  • First variable selected in forward selection is
    the first variable eliminated by backward
    elimination
  • Best subset of size 3 gives RSS of 6.8
  • Forwards best set of 3 has RSS 21.2
    Backwards gets 23.5

55
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Alternatives to all-subsets
  • Simulated Annealing, Tabu Search, etc.

56
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
MCMC for Bayesian Variable Selection (Ioannis
Ntzoufras)
http//www.ba.aegean.gr/ntzoufras/courses/bugs2/ha
ndouts/modelsel/4_1_tutorial_handouts.pdf
57
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Why MCMC?
58
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Reversible Jump MCMC
59
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Reversible Jump MCMC
60
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Stochastic Search Variable Selection (George
McCulloch)
61
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
SSVS Procedure
62
Estimating Spina Bifida Numbers with
Capture-Recapture Methods
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
(Regal Hook, 1991, Madigan York, 1996)

Model Pr(Model) N
95 HPD
B D R 0.37 731 (701,767)
B D R 0.30 756 (714,811)
B R D 0.28 712 (681,751)
BMA 728 (682,797)
63
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Spina Bifida 95 HPDs
M3
M2
M1
BMA
64
Overview Model Selection Theory Computing Continuo
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Model probabilities BIC Variable selection Model
averaging
65
Overview Model Selection Theory Computing Continuo
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Model probabilities BIC Variable selection Model
averaging

66
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
67
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
68
Overview Model Selection Theory Computing Continuo
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Model probabilities BIC Variable selection Model
averaging
69
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
70
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
71
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
72
Model Uncertainty
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
  • Posterior Variance Within-Model
    Variance Between-Model Variance
  • Data-driven model selection is risky Part of
    the evidence is spent specify the model (Leamer,
    1978)
  • Model-based inferences can be over-precise
  • Model-based predictions can be badly calibrated
  • Draper (1995), Chatfield (1995)

Bayesian Model Averaging (BMA) can help
73
Bayesian Model Averaging
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
74
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
75
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
76
Out-of-Sample Predictive Performance
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
  • Average Improvement
  • Data
    Model Class in Predictive Probability

  • over MAP Model
  • 1. Coronary Heart Disease Decomposable UDGs 2.2
  • 2. Women and Mathematics Decomposable UDGs 0.6
  • 3. Scrotal Swellings Decomposable UDGs 5.1
  • 4. Crime and Punishment Linear Regression 61.3
  • 5. Lung Cancer Exponential Regression 1.8
  • 6. Cirrhosis Cox Survival Regression 1.8
  • 7. Coronary Heart Disease Essential graphs 1.5
  • 8. Women and Mathematics Essential graphs 3.0
  • 9. Stroke Cox Survival Regression 15.0

77
BMA Computing
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Occams Window Find parsimonious models with
large Pr(MD) and average over those. Importance
Samping (Clyde et al., JASA, 1996) MCMCMC Use
MCMC to draw from Pr(MD).
Madigan and Raftery (1991)
Madigan and York (1992)
78
Gibbs MC3
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
e.g. Undirected graphical models
SSVS (George and McCulloch, 1993)
  • Choose vi, vj at random (or systematically) and
    draw from

79
Metropolis MC3
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
e.g. SVO Regression Outliers (Hoeting,
Raftery, Madigan, 1994,5,6)
Possible Predictors a,b,c,d Possible Outliers
13,20,40 Current model
(b,c)(13,20) Candidate Models (b)(13,20) (
b,c)(13) (c)(13,20) (b,c)(20) (b,c,d)(13,20
) (b,c)(13,20,40) (a,b,c)(13,20)
Accept the Candidate Model with Prob
80
Augmented MC3
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
e.g. Bayesian Networks
  • A total ordering T of V is said to be compatible
    with a directed graph M, if the orientation of
    the arrows in M is consistent with T.
  • Draw from Pr(T, M D)
  • Pr(M T, D) Pr(T M, D)
  • Uniform on compatible Ts
  • Metropolis accept/reject
  • Generate M by adding or deleting an edge from M
    consistent with T
  • Metropolis accept/reject

81
More Augmented MC3
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
  • Draw from Pr(Z, M D)
  • Pr(M Z, D) Pr(Z M, D)

e.g. Double Sampling Missing Data (York et al,
1994)
Pr(Z, qM M, D)
Pr(Z qM, M, D) Pr(qM Z, M, D)
Reversible Jump MCMC (Green, 1995)
82
Linear Regression SVT, SVO, SVOT
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Hoeting, Raftery, Madigan (1994,5,6)
  • Normal-gamma conjugate priors
  • Box-Cox and ACE Transformations
  • Outliers (pre-process with LMS
    regression)

83
Overview Model Selection Theory Computing Continuo
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Model probabilities BIC Variable selection Model
averaging
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Model probabilities BIC Variable selection Model
averaging
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Model probabilities BIC Variable selection Model
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Model probabilities BIC Variable selection Model
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Pr(Bi0D)
94
Generalized Linear Models
Overview Model Selection Theory Computing Continuo
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Model probabilities BIC Variable selection Model
averaging
Raftery (1996)
  • Laplace Approximation

is minus the inverse Hessian of
evaluated at
  • Idea approximate by one Newton step starting
    from
  • approximation using only GLIM output

95
Prior Distribution on Models
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
Madigan, Gavrin, Raftery (1995)
  • Start with uniform pre-prior, PPr(M)
  • Elicit Imaginary Data from the Expert
  • Update pre-prior using imaginary data to get the
    real prior, Pr(M)
  • Provided improved predictive performance in a
    particular medical example
  • Ibrahim and Laud (1994)

96
Predicting Strokes
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
(Volinsky, Madigan, Raftery, Kronmal, 1996)
  • Stroke is the third leading cause of death
    amongst adults in the US
  • Gorelick (1995) estimated that 80 of strokes are
    preventable
  • Cardiovascular Health Study (CHS)
  • On-going Started 1989 5,201 in four counties
  • 65 risk factors different for older people?
  • 172/4501 strokes

97
Measured Covariates
Overview Model Selection Theory Computing Continuo
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Model probabilities BIC Variable selection Model
averaging
Follow-up ranged from 3.5 to 4.5 years (Average
4.1)
98
Cox Model
Overview Model Selection Theory Computing Continuo
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Model probabilities BIC Variable selection Model
averaging
Estimation usually based on the Partial
Likelihood
(Taplin, 1993, Draper, 1995)
99
Finding the Models in Occams Window
Overview Model Selection Theory Computing Continuo
us Model Selection
Model probabilities BIC Variable selection Model
averaging
  • Need models with posterior probability within a
    factor of C of MAP model
  • Approximate Leaps and Bounds
  • Furnival and Wilson (1974) Lawless and Singhal
    (1978)
  • Finds top q models of each size
  • Find models within factor of C2
  • Compute Exact BIC for these models

100
Picture of Probs vs P-values
Overview Model Selection Theory Computing Continuo
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Model probabilities BIC Variable selection Model
averaging
101
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Model probabilities BIC Variable selection Model
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102
Predictive Performance
Overview Model Selection Theory Computing Continuo
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Model probabilities BIC Variable selection Model
averaging
Model Averaging Stepwise Top PMP
Stroke Stroke
Stroke
Low 751 7
750 8 724 10
Medium 770 24
799 27 801 28
High 645
617 51 641 48
Assigned Risk Group
103
Overview Model Selection Theory Computing Continuo
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Model probabilities BIC Variable selection Model
averaging
104
Overview Model Selection Theory Computing Continuo
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Ridge Lasso LARS Case Study
Shrinkage Methods
  • Subset selection is a discrete process
    individual variables are either in or out.
    Combinatorial nightmare.
  • This method can have high variance a different
    dataset from the same source can result in a
    totally different model
  • Shrinkage methods allow a variable to be partly
    included in the model. That is, the variable is
    included but with a shrunken co-efficient


105
Overview Model Selection Theory Computing Continuo
us Model Selection
Ridge Lasso LARS Case Study
Ridge Regression
subject to
Equivalently
This leads to Choose ? by cross-validation.
works even when XTX is singular
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Ridge Regression Bayesian MAP Regression
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Least Absolute Shrinkage Selection Operator
(LASSO)
subject to
Quadratic programming algorithm needed to solve
for the parameter estimates
q0 var. sel. q1 lasso q2 ridge Learn q?
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Ridge LASSO - Theory
  • Lasso estimates are consistent
  • But, Lasso does not have the oracle property.
    That is, it does not deliver the correct model
    with probability 1
  • Fan Lis SCAD penalty function has the Oracle
    property


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LARS
  • New geometrical insights into Lasso and
    Stagewise
  • Leads to a highly efficient Lasso algorithm for
    linear regression


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LARS
  • Start with all coefficients bj 0
  • Find the predictor xj most correlated with y
  • Increase bj in the direction of the sign of its
    correlation with y. Take residuals ry-yhat along
    the way. Stop when some other predictor xk has as
    much correlation with r as xj has
  • Increase (bj,bk) in their joint least squares
    direction until some other predictor xm has as
    much correlation with the residual r.
  • Continue until all predictors are in the model


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Statistical Analysis of Textual Data
  • Statistical text analysis has a long history in
    literary analysis and in solving disputed
    authorship problems
  • First (?) is Thomas C. Mendenhall in 1887

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  • Mendenhall was Professor of Physics at Ohio State
    and at University of Tokyo, Superintendent of the
    USA Coast and Geodetic Survey, and later,
    President of Worcester Polytechnic Institute

Mendenhall Glacier, Juneau, Alaska
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X2 127.2, df12
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  • Used Naïve Bayes with Poisson and Negative
    Binomial model
  • Out-of-sample predictive performance

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today
  • Statistical methods routinely used for textual
    analyses of all kinds
  • Machine translation, part-of-speech tagging,
    information extraction, question-answering, text
    categorization, etc.
  • Not reported in the statistical literature (no
    statisticians?)

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Text categorization
  • Automatic assignment of documents with respect to
    manually defined set of categories
  • Applications automated indexing, spam filtering,
    content filters, medical coding, CRM, essay
    grading
  • Dominant technology is supervised machine
    learning
  • Manually classify some documents, then learn a
    classification rule from them (possibly with
    manual intervention)

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Document Representation
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  • Documents usually represented as bag of words
  • xis might be 0/1, counts, or weights (e.g.
    tf/idf, LSI)
  • Many text processing choices stopwords,
    stemming, phrases, synonyms, NLP, etc.

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Classifier Representation
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  • For instance, linear classifier
  • xis derived from text of document
  • yi indicates whether to put document in category
  • ßj are parameters chosen to give good
    classification effectiveness

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Logistic Regression Model
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  • Linear model for log odds of category membership
  • Equivalent to
  • Conditional probability model

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Logistic Regression as a Linear Classifier
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  • If estimated probability of category membership
    is greater than p, assign document to category
  • Choose p to optimize expected value of your
    effectiveness measure
  • Can change measure w/o changing model

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Maximum Likelihood Training
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  • Choose parameters (ßj's) that maximize
    probability (likelihood) of class labels (yi's)
    given documents (xis)
  • Maximizing (log-)likelihood can be viewed as
    minimizing a loss function

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Hastie, Friedman Tibshirani
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Avoiding Overfitting
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  • Text is high dimensional
  • Maximum likelihood gives infinite parameter
    values, poor effectiveness
  • Solution penalize large ßj's, e.g. maximize
  • Called ridge logistic regression

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A Bayesian Interpretation of Ridge Logistic
Regression
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  • Suppose
  • we believe each ßj is a small value near 0
  • and encode this belief as separate Gaussian
    probability distributions over values of ßj
  • Bayes rule specifies our new (posterior) belief
    about ß after seeing training data
  • Choosing maximum a posteriori value of the ß
    gives same result as ridge logistic regression

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Zhang Oles Results
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  • Reuters-21578 collection
  • Ridge logistic regression plus feature selection

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Bayes!
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  • MAP logistic regression with Gaussian prior gives
    state of the art text classification
    effectiveness
  • But Bayesian framework more flexible than SVM for
    combining knowledge with data
  • Feature selection
  • Stopwords, IDF
  • Domain knowledge
  • Number of classes
  • (and kernels.)

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Bayesian Supervised Feature Selection
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  • Results on ridge logistic regression for text
    classification use ad hoc feature selection
  • Use of feature selection ? belief (before seeing
    data) that many coefficients are 0
  • Put that belief into our prior on coefficients...
  • Laplace prior, i.e. lasso logistic regression

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Data Sets
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  • ModApte subset of Reuters-21578
  • 90 categories 9603 training docs 18978 features
  • Reuters RCV1-v2
  • 103 cats 23149 training docs 47152 features
  • OHSUMED heart disease categories
  • 77 cats 83944 training docs 122076 features
  • Cosine normalized TFxIDF weights

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Dense vs. Sparse Models (Macroaveraged F1,
Preliminary)
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Bayesian Unsupervised Feature Selection and
Weighting
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  • Stopwords low content words that typically are
    discarded
  • Give them a prior with mean 0 and low variance
  • Inverse document frequency (IDF) weighting
  • Rare words more likely to be content indicators
  • Make variance of prior inversely proportional to
    frequency in collection
  • Experiments in progress

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Bayesian Use of Domain Knowledge
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  • Often believe that certain words are positively
    or negatively associated with category
  • Prior mean can encode strength of positive or
    negative association
  • Prior variance encodes confidence

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First Experiments
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  • 27 RCV1-v2 Region categories
  • CIA World Factbook entry for country
  • Give content words higher mean and/or variance
  • Only 10 training examples per category
  • Shows off prior knowledge
  • Limited data often the case in applications

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Results (Preliminary)
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Polytomous Logistic Regression
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  • Logistic regression trivially generalizes to
    1-of-k problems
  • Cleaner than SVMs, error correcting codes, etc.
  • Laplace prior particularly cool here
  • Suppose 99 classes and a word that predicts class
    17
  • Word gets used 100 times if build 100 models, or
    if use polytomous with Gaussian prior
  • With Laplace prior and polytomous it's used only
    once
  • Experiments in progress, particularly on author
    id

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1-of-K Sample Results brittany-l
89 authors with at least 50 postings. 10,076
training documents, 3,322 test documents.
BMR-Laplace classification, default
hyperparameter
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1-of-K Sample Results brittany-l
4.6 million parameters
89 authors with at least 50 postings. 10,076
training documents, 3,322 test documents.
BMR-Laplace classification, default
hyperparameter
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Future
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  • Choose exact number of features desired
  • Faster training algorithm for polytomous
  • Currently using cyclic coordinate descent
  • Hierarchical models
  • Sharing strength among categories
  • Hierarchical relationships among features
  • Stemming, thesaurus classes, phrases, etc.

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Text Categorization Summary
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  • Conditional probability models (logistic,
    probit, etc.)
  • As powerful as other discriminative models (SVM,
    boosting, etc.)
  • Bayesian framework provides much richer ability
    to insert task knowledge
  • Code http//stat.rutgers.edu/madigan/BBR
  • Polytomous, domain-specific priors soon

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For high-dimensional predictive modeling…
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  • Regularized regression methods are better than
    ad-hoc variable selection algorithms
  • Regularized regression methods are more practical
    than discrete model averaging (and probably make
    more sense)
  • L1-regularization is the best way to variable
    selection

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For high-dimensional predictive modeling…
Overview Model Selection Theory Computing Continuo
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  • Regularized regression methods are better than
    ad-hoc variable selection algorithms
  • Regularized regression methods are more practical
    than discrete model averaging (and probably make
    more sense)
  • L1-regularization is the best way to variable
    selection

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For high-dimensional predictive modeling…
Overview Model Selection Theory Computing Continuo
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  • Regularized regression methods are better than
    ad-hoc variable selection algorithms
  • Regularized regression methods are more practical
    than discrete model averaging (and probably make
    more sense)
  • L1-regularization is the best way to variable
    selection
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