Determining the Number of Non- Spurious Arcs in a Learned DAG Model: Investigation of a Bayesian and a Frequentist Approach Listgarten & Heckerman

2 Purpose  Design a vaccine for HIV By considering many patients and observing which HLA molekyles causes the T-killer cells of the imune system to react

3 Definitions  HLA = Human leukocyte antigen Each person usally has [3;6]  Epitopes = bits of protein Results of T-cell attacking HIV-peptide  Peptide = “small digestible” Link between amino acids

4 How?  Find out which HIV peptides interact with which HLA molekyles by using a graphical model.

5 Solution  A directed acyclic graph representing HLA and peptides HLA h 1 HLA h 2 HLA h 3 HLA h 4 peptide y 1 peptide y 2 peptide y 3 HLA h N peptide y M... Model for one patient. A design of a vaccine is to identify a set of peptide-HLA-pairs, which are epitopes for a large number of the population

6 Properties  Bi-partite model(2 levels)  HLA can have zero or several outgoing archs  Peptide can have zero or several ingoing archs  Each patient will have [3;6] HLA nodes that are “on”  Answers: which HLA molekyle(s) are(is) responsible for a given immune system reaction

7 Two approaches  Bayesian  Frequentist

8 Bayesian Approach cont. 1(2) true arch distribution bayesian expectation with given data D the number of archs both in G and G’ Ddata G’proposed model Gall possible graph structures

9 Bayesian Approach cont. 2(2)  Exponentional complexity…! Can be improved by limiting |Parent set| Limit=5, gives identical results

10 Frequentist Approach  FDR = False Discovery Rate  Given a set of hypotheses  Hypothesis i has a test score s: assumed to be independent in a given hypotheses

11 FDR cont. 1(4) Eexpected value Fnumber of false hypotheses Snumber of hypotheses with s i > t tthreshold

12 FDR cont. 2(4) Rewrite Where is a structure search algorithm

13 FDR cont. 3(4) – multiple data sets Q - – number of archs found by applying to real data, D

14 FDR cont. 4(4)  Standard FDR:  The average over multiple datasets  +1 – smooths the estimate

15 Results  PPV – positive predictive value Frequentist method: Bayesian method:

16 Results on non-HIV data

17 Results on non-HIV data

18 Results on synthetic HIV data

19 Results on real HIV data   8 results…. all matches