BTRY 7210: Topics in Quantitative Genomics and Genetics

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1 BTR 70: Topics in Quantitative Genomics and Genetics Jason Mezey Biological Statistics and Computational Biology (BSCB Department of Genetic Medicine March 9 05

2 Lecture 5: Basics of eqtl network modeling = the foundation: intro to robabilistic Graphical Models

3 Motivation: network analysis leveraging eqtl to find novel regulatory relationships No cis-trans regulation Battle et al. A B C Unidenti ed or equivalent models D Resolvable in theory (not in practice Resolvable in theory and in practice Downloaded from genome.cshlp.org on March ublished by Cold Spring Harbor Laboratory ress from Schadt et al. 005 Nature Genetics from Battle et al. 04 Genome Research Figure 3 A portion of the core subnetwork derived from the liver transcriptional subnetworks representative of gene expression signatures of the mouse models of the candidate genes. The liver

4 Using eqtl to discover regulatory relationships between genes What we would like to discover: The core of this modeling / discover technique are robabilistic Graphical Models Understanding what these are from a global level will allow you to start digging into a diverse and confusing literature + will provide a basis for the specific models and approaches we will use

5 robabilistic Graphical Models This is a core and diverse set of models that play a central role in computational statistics / statistics machine learning and computational / systems biology They are used for: prediction discovery and modeling in many different ways They include: Hidden Markov Models Markov Random Fields Gaussian Graphical Models ARACNE Bayesian Networks Cyclic graphs Ancestral graphs... even linear regression(!! The literature is very confusing when first jumping in due to both the sophistication of how these are investigated but even more critically the use and explanation of the models conflate the model the algorithm used for analyzing data and the statistics employed A great way to approach this field is to focus on what these models are representing and build from there (this is what we will do

6 n (xi µ cn( X00 Bern(0.4 X00 Bern(0.4 or H0 : = c L( x = p e i= r(t... r(t (X H : (X H = c X:00 ern(0.45 X(X Overview: Bern( r(t(x (X r(t Bern(0.4 0 r(t = c 0 GMs n!n r(n (x n 4 (xi µ 3µ i r(t (X HA0GM : is=a picture c i= represents of ai= multivariate = the L( x =thatl( x e pstructure e r(t (X H : = c 0 probability distribution variables random!n for several r( under a set of assumptions that!restrict the possible model: (x µ n i n i= r( L( x = p e nr((x i µ 3 r( L( x = p e i= (4 a GM makes assumptions r( More specifically converning the marginal r( r( the conditional probability distributions of random variables and r( 33random 44 variables: r( distributions r(of these (5 334 r( r( 3 4 r( r( r( (6 r( 3 r( 3 r( 3 3 r( 33 (7 r( r( 4 4 r( 3 4 r( 4 r( r( (8 r( 3 random variables r( probability 3 measures We therefore need toreview: 4 probability distribution functions marginal r(distributions 3 4and conditional (9 r( = { ossible Individuals } 3 4 r( 3 4 probability e Individuals } = { g \ } r( r( 3 = { (0 3 4 }! g

7 robability + Random variables X = x r(x X E Random Variable X( r(f (Sample Space F (Sigma Algebra

8 robability + Random variables II F! Random variable - a real valued function on the sample space:! X( :! R A critical point to note: because we have defined a probability function on the sample space this induces a probability function on the random variable X: r(f! r(x

9 robability and random vectors With more than one random variable we have a probability distribution on a random vector 8 < Consider the two coin flips and assume a probability function for a fair coin as a model: r(hh=r(ht=r(th=r(tt=0.5 : Let s define two random variables: number of Tails and first flip is Heads X ( = 8 < : X (HH=0 X (HT=X (TH= X (TT= The probability function induces the following pmf for the random vector X=[X X] where we use bold X do indicate a vector (or matrix: X ( = r(x =r(x = x X = x = X (x = X X (x x r(x =0X = 0 = 0.0r(X =0X = = 0.5 r(x =X = 0 = 0.5r(X =X = = 0.5 r(x =X = 0 = 0.5r(X =X = = 0.0 X (TH=X (TT=0 X (HH=X (HT=

10 Marginal distributions marginal distributions of random vectors are the probability of a r.v. of a random vector after summing (discrete or integrating (continuous over all the values of the other random variables: X (x = max(x X X =min(x Z f X (x = Z r(x = x \X = x = X r(x = x X = x r(x = x r(x = x \X = x dx = Again as a simple illustration consider our two coin flip example: Z Z X =0 X = X = X = X = (8 r(x = x X = x r(x = x dx

11 Random vectors: conditional probability and independence We can define conditional probability for random vectors: As a simple example (discrete in this case - but continuous is the same! consider the two flip sample space fair coin probability model random variables: number of tails and first flip is heads : X =0 X = X = X = X = We can similarly consider whether r.v. s of a random vector are independent e.g. r(x X = r(x X r(x r(x =0 X = = r(x =0 X = r(x = = =0.5 r(x =0 X = = 0.5 = r(x = 0r(X = = =0.5

12 n! r(t (X H : 00 =r(t 0X r(t... (X (X r(t (X H :n = c 0Bern(0.4 Bern(0.45 cbern(0.4 X00 r( nn (xi(xiµ µ!n= = p e ei=i=!n L( x L( x n = c (xi µ (x µ n i er( i= 3 L( x = p i= p L( x = e!for r(t (X H = c(multivariate probability distribution: n0 :a joint n (xi µ r( r(! 3 4 n i= 3 r(f 4! r(x p e (4 n r( (xi µ = r( i= p L( x e (4 r( joint 3 4 r( 3 4 A complete characterization would require information on this r( marginal :! R and X( distributions distribution and each of the possible Bringing this back to GMs r( r( conditional 3 4 distributions: (5 r( r( r( 3 3 r(f! r(x r( (5 r( X( r( R 3 r( r( r( (6 3 4 :! (6 r( 3 For the moment we don t need to worry about the specific r( 3(7 r( 3r( r( r( 3 4 ( r( parameterized probability distribution(s i.e. it could be anything justr( 3 about 4 r(the marginals think admissible (!! we are going to and r( (8 r( r( (8 r( 3 = { ossible Individuals } 3 4 conditionals 4 3r( 6= r( r( 3 r( (9 } g \ = { r( r( More 3 specifically (9 about 3 which 4 4 r( we will3 be concerned of these r( 3 = r( (0 }r(are independent relationships and non-independent: e Individuals r( 3 r( 3 4 r( 3 4 = (4 (5 (6 (7 (8 (9 (0! p ( (0 d + ( (p } 3 6=a r( r( 6= r( = { g r( ossible Individuals } r( 3 4r( 3 4 ( ( r( 3 4 = r(( r( 3 = r( = { dg \ = } {A A A A A A } g

13 Looking ahead: connecting this to the graph (picture... We assume a graphical model G=(V E describes the joint distribution where each random variable is represented as a vertex and each edge represents a conditional relationship: r( =r(... k V =( =... k E( i j = r( i j = r( i yj yk yj yk yj yk yj yk yj yk yj yk z

14 That s it for today!

Probability Theory and Simulation Methods

Probability Theory and Simulation Methods Feb 28th, 2018 Lecture 10: Random variables Countdown to midterm (March 21st): 28 days Week 1 Chapter 1: Axioms of probability Week 2 Chapter 3: Conditional probability and independence Week 4 Chapters