A Frequentist Assessment of Bayesian Inclusion Probabilities
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1 A Frequentist Assessment of Bayesian Inclusion Probabilities Department of Statistical Sciences and Operations Research October 13, 2008
2 Outline 1 Quantitative Traits Genetic Map The model 2 Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space 3 Theorem Performance 4
3 Quantitiative Traits Quantitative Traits Genetic Map The model Biologists are often interested in which genes control a quantitative trait. Examples: Height Weight Yeild Cotyledon Opening Angle
4 Quantitiative Traits Quantitative Traits Genetic Map The model Biologists are often interested in which genes control a quantitative trait. Examples: Height Weight Yeild Cotyledon Opening Angle
5 Quantitiative Traits Quantitative Traits Genetic Map The model Biologists are often interested in which genes control a quantitative trait. Examples: Height Weight Yeild Cotyledon Opening Angle
6 Quantitiative Traits Quantitative Traits Genetic Map The model Biologists are often interested in which genes control a quantitative trait. Examples: Height Weight Yeild Cotyledon Opening Angle
7 Quantitive Trait Loci Quantitative Traits Genetic Map The model Genetic Map of the Arabidopsis thaliana. 0 I II III IV V T1G11 MSAT2-5 NGA172 MSAT4-39 F21M12 MSAT4-8 MSAT2-38 ATHCHIB MSAT11-0 NGA248 T27K12 NGA128 F5I14 MSAT1-13 MSAT-36 MSAT2-41 MSAT2-7 MSAT2-10 MSAT2-22 MSAT3-19 MSAT3-32 MSAT3-21 MSAT3-18 NGA8 MSAT4-35 MSAT4-15 MSAT4-18 MSAT4-9 MSAT4-37 MSAT5-14 NGA139 MSAT5-22 MSAT5-9 MSAT5-12 MSAT1-5 MSAT cm Genetic Map BAY-0 x SHADARA
8 Linear Model Quantitative Traits Genetic Map The model To determine the benchmark dose for a predefined endpoint we can use: p Y i = β 0 + β j X ij I Xij M k + ɛ ij j=1 where I Xij M k = 1 if X ij is in model M k and zero otherwise and { 1 if Locus j comes from Parent A X ij = 0 if Locus j comes from Parent B There are 2 p = 2 38 possible first order linear models.
9 Quantitative Traits Genetic Map The model Typical approaches to determining the locations may be: ANOVA Step-wise Selection Forward Selection Mallows C p
10 Quantitative Traits Genetic Map The model Typical approaches to determining the locations may be: ANOVA Step-wise Selection Forward Selection Mallows C p
11 Quantitative Traits Genetic Map The model Typical approaches to determining the locations may be: ANOVA Step-wise Selection Forward Selection Mallows C p
12 Quantitative Traits Genetic Map The model Typical approaches to determining the locations may be: ANOVA Step-wise Selection Forward Selection Mallows C p
13 Inclusion Probabilities Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space An alternative to standard likelihood or Bayes factor inferences can be achieved via Inclusion Probabilities: P(β j 0 D) = P(Locus j D) These give us the probability that Locus j is important regardless of the model. This only depends on the data.
14 Bayesian Model Averaging Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space For a model space, M with M models then for each model M c M the posterior model probability given a set of data D is given by: P(D M c )P(M c ) P(M c D) = M k=1 P(D M k)p(m k ) where P(D M c ) = P(D θ c, M c )P(θ c M c )dθ c. where θ c is the parameter vector for model M c.
15 Determining the Marginals Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space Determining P(D M c ) can be done various ways: AIC, BIC or DIC based approximations Laplace approximation Numerical Integration Exact Solution
16 Determining the Marginals Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space Determining P(D M c ) can be done various ways: AIC, BIC or DIC based approximations Laplace approximation Numerical Integration Exact Solution
17 Determining the Marginals Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space Determining P(D M c ) can be done various ways: AIC, BIC or DIC based approximations Laplace approximation Numerical Integration Exact Solution
18 Determining the Marginals Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space Determining P(D M c ) can be done various ways: AIC, BIC or DIC based approximations Laplace approximation Numerical Integration Exact Solution
19 Determining the Marginals Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space For the linear model with Normal-Inv-χ 2 prior distribution the exact marginal is given by: Γ ( ) ν ν+n (νλ) 2 P(D µ c, V c, ν, X c, M c ) = π n 2 Γ ( ν 2 [λν + (Y X c µ c ) (I + X c V c X c) 1 (Y X c µ c )] ν+n 2, (1) 2 ) I + Xc V c X c 1/2
20 Stochastic Search Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space To search through the model space a stochastic search can be employed. To the probability of moving from model M c to model M t is given by the Metropolis-Hastings algorithm. α = min { 1, P(M t)p(d M t ) P(M c )P(D M c ) where q is a proposal distribution q(m t M c ) q(m c M t ) }, (2)
21 Inclusion Probabilities: Part Duex Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space In this framework the Inclusion Probabilities can be calculated via: P(β j 0 D) = = M P(β j 0 D, M k )P(M k D) k=1 M I Xj M k (M k )P(M k D) k=1
22 Restricted Model Space Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space In the case where p > n a restricted model space can be employed be restricting the models to have only r loci at a time. This allows: All loci to be considered. Enough degrees of freedom for model fitting. Smaller model space.
23 Restricted Model Space Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space In the case where p > n a restricted model space can be employed be restricting the models to have only r loci at a time. This allows: All loci to be considered. Enough degrees of freedom for model fitting. Smaller model space.
24 Restricted Model Space Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space In the case where p > n a restricted model space can be employed be restricting the models to have only r loci at a time. This allows: All loci to be considered. Enough degrees of freedom for model fitting. Smaller model space.
25 Effect of Restriction Bayesian Model Averaging Determining the Marginals Inclusion Probabilities Restricted Model Space For the Arabidopsis thaliana. Table: Inclusion probabilities P(β j 0 D) for highest probability loci with restrictions r = 5, 10, 15 and r = p = 38. Locus r = 5 r = 10 r = 15 r = p = 38 ATHCHIB F21M MSAT MSAT MSAT The restriction suppresses the larger inclusion probabilities.
26 Theorem Theorem Performance Theorem Under H o :β j = 0 for all j = 1,..., p with restriction r < p and P(M k ) is uniform for all M k M then P(β j 0 D) r 1 i=1 ( ) r 1 i ). r i=1 ( p i (3)
27 Corollary Theorem Performance Corrollary Under H o :β j = 0 for all j = 1,..., p, r = p and P(M k ) is uniform for all M k M then P(β j 0 D) 1 2. This is consistent with expectations.
28 Upper Cut-off Values Theorem Performance Let q = P(β j 0 D) under H 0 : β j = 0 then a simple upper cut-off value for a single locus is: q(1 q) P(β j 0 D) q + z 1 α, n s or for p loci the upper cut off can be found as an order statistic q (max) g(q (max)) = Φ(q (max)) p 1 φ(q (max)), where g is the sampling distribution for q under H 0. For simplicity the normal distribution can be used as an approximation. Note: a Bayesian approach using a Beta distribution could be employed.
29 Simulation Study Theorem Performance To evaluate the performance of the algorithm a simulation study was conducted. Theoretical vs Empirical Cut off Values Power for effect sizes 0, 1, 2,4 and 8 Restriction sizes 5, 10 and 15
30 Simulation Study Theorem Performance To generate the simulation data the loci data from the Arabidopsis thaliana was used for X i and the response was generated using: Y i = β 0 + δx ij + ɛ i ɛ i N(0, 2) δ = 0, 1, 2,4 and 8 X j was randomly selected from the 38 loci P(β j 0 D) based on 100,000 stochastic steps
31 Theoretical vs. Empirical Theorem Performance Table: Empirical and theoretical α = 0.05 upper cut-off values for a single locus. Empirical values are based on 300 simulations. Restriction Empirical Theoretical r = r = r = Note: at r = 15 the normal distribution assumption is invalid.
32 Power Theorem Performance Power study for various effect sizes. Table: Power of theoretical upper cut-off values for a single locus. Effect Size Restriction r = r = r = Note: In simulated data σ = 2.
33 Arabidopsis thaliana data Table: Inclusion probabilities for highest probability loci with significance using proposed individual cut-off value of for r = 5 and for r = 10. r = 5 r = 10 Locus P(β j 0 D) Significant P(β j 0 D) Significant ATHCHIB Y Y F21M Y Y MSAT Y Y MSAT N N MSAT Y N
34 Conclusions This gives frequentists a method to evaluate Inclusion Probabilities Shows how to deal with the effect of restricted model spaces Consistent with expectations This shows the power of the method Shows how to apply the method
35 Conclusions This gives frequentists a method to evaluate Inclusion Probabilities Shows how to deal with the effect of restricted model spaces Consistent with expectations This shows the power of the method Shows how to apply the method
36 Conclusions This gives frequentists a method to evaluate Inclusion Probabilities Shows how to deal with the effect of restricted model spaces Consistent with expectations This shows the power of the method Shows how to apply the method
37 Conclusions This gives frequentists a method to evaluate Inclusion Probabilities Shows how to deal with the effect of restricted model spaces Consistent with expectations This shows the power of the method Shows how to apply the method
38 Conclusions This gives frequentists a method to evaluate Inclusion Probabilities Shows how to deal with the effect of restricted model spaces Consistent with expectations This shows the power of the method Shows how to apply the method
39 Future Work How does restricted model spaces affect epistasis models? Can we determine a better sampling distribution for P(β j 0 D)? A blind stochastic search is expensive. Can we form a more intelligent search method? Can we form a more intelligent restriction method?
40 Future Work How does restricted model spaces affect epistasis models? Can we determine a better sampling distribution for P(β j 0 D)? A blind stochastic search is expensive. Can we form a more intelligent search method? Can we form a more intelligent restriction method?
41 Future Work How does restricted model spaces affect epistasis models? Can we determine a better sampling distribution for P(β j 0 D)? A blind stochastic search is expensive. Can we form a more intelligent search method? Can we form a more intelligent restriction method?
42 Future Work How does restricted model spaces affect epistasis models? Can we determine a better sampling distribution for P(β j 0 D)? A blind stochastic search is expensive. Can we form a more intelligent search method? Can we form a more intelligent restriction method?
43 Contact Info Thank You. Edward L. Boone Department of Statistical Sciences and Operations Research Richmond, Virginia
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