guenomu software -- model and agorithm in 2013

guenomu

Software and Model

Leonardo de O. Martins

University of Vigo

May, 16th 2013

Leo Martins (U Vigo) guenomu software 2013/5/16 1 / 15

Outline

1 The Model

2 The Sampling

3 The Code

Hierarchical Bayesian model

P(S ,Θ | D) ∝ P(θ0)P( ~λ0)P(α0)P(S)×

×N∏i=1

The mixture of distance distributions

P(G | ~λ, ~w , S) =

w1e−(dDUPS (G ,S)/λDUPS+dLOSS (G ,S)/λLOSS ) + w2e−(dILS (G ,S)/λILS ) + w3e−(dRF (G ,S)/λRF )

Z(~λ, ~w , S)

wi ∼ Gamma(αgene , 1)

λx ∼ Exp(Λx )

each gene has its own set of wi and λi

the distances dx (G , S) are scaled to account for different gene family sizes

P(G | ~λ, ~w , S) =

Z(~λ, ~w , S)

λx ∼ Exp(Λx )

P(G | ~λ, ~w , S) =

Z(~λ, ~w , S)

λx ∼ Exp(Λx )

P(G | ~λ, ~w , S) =

Z(~λ, ~w , S)

λx ∼ Exp(Λx )

P(G | ~λ, ~w , S) =

Z(~λ, ~w , S)

λx ∼ Exp(Λx )

Outline

1 The Model

2 The Sampling

3 The Code

Doubly-intractable distributions

π(y | θ) =qθ(y)

Z (θ)=

eθts(y)

Z (θ); Z (θ) =

eθts(y) (1)

augmented distribution: π(θ′, y ′, θ | y) ∝ π(y | θ)π(θ)h(θ′ | θ)π(y ′ | θ′)

Gibbs update of the auxiliary variables θ′,y ′:

I. draw θ′ ∼ h(· | θ)II. draw y ′ ∼ π(· | θ′)

exchange ratio from θ to θ′

qθ(y ′)π(θ′)h(θ | θ′)qθ′(y)

qθ(y)π(θ)h(θ′ | θ)qθ′(y ′)

We draw y ′ (the gene tree) through a secondary MCMC starting at itscurrent value

π(y | θ) =qθ(y)

Z (θ)=

eθts(y)

Z (θ); Z (θ) =

eθts(y) (1)

augmented distribution: π(θ′, y ′, θ | y) ∝ π(y | θ)π(θ)h(θ′ | θ)π(y ′ | θ′)Gibbs update of the auxiliary variables θ′,y ′:

π(y | θ) =qθ(y)

Z (θ)=

eθts(y)

Z (θ); Z (θ) =

eθts(y) (1)

I. draw θ′ ∼ h(· | θ)

II. draw y ′ ∼ π(· | θ′)exchange ratio from θ to θ′

π(y | θ) =qθ(y)

Z (θ)=

eθts(y)

Z (θ); Z (θ) =

eθts(y) (1)

π(y | θ) =qθ(y)

Z (θ)=

eθts(y)

Z (θ); Z (θ) =

eθts(y) (1)

π(y | θ) =qθ(y)

Z (θ)=

eθts(y)

Z (θ); Z (θ) =

eθts(y) (1)

Species tree proposal with the exchange algorithm

Generalized Multiple-Try Metropolis

MH: sample y , decide if accept it with probability r

r =π(y)

q(y , x)

q(x , y)=π(y)

p(x | y)

p(y | x)

MTM: choose y among several samples, according to their relative weights

r =w(y1, x) + · · ·+ w(yk , x)

w(x∗1 , y) + · · ·+ w(x∗k , y)

where w(x , y) = π(x)q(x , y)λ(x , y) = π(x)p(y | x)λ(x , y)

GMTM: weights w(.) do not need to represent probability distributions.

r =π(y)pk(x | y)

π(x)pk(y | x)

where Wy = wi (yi ,x)∑kj=1 wj (yj ,x)

for the chosen element i

r =π(y)

q(y , x)

q(x , y)=π(y)

p(x | y)

p(y | x)

r =w(y1, x) + · · ·+ w(yk , x)

w(x∗1 , y) + · · ·+ w(x∗k , y)

r =π(y)pk(x | y)

π(x)pk(y | x)

r =π(y)

q(y , x)

q(x , y)=π(y)

p(x | y)

p(y | x)

r =w(y1, x) + · · ·+ w(yk , x)

w(x∗1 , y) + · · ·+ w(x∗k , y)

r =π(y)pk(x | y)

π(x)pk(y | x)

gene tree proposal with GMTM or MTM

Outline

1 The Model

2 The Sampling

3 The Code

RF distance, Assignment cost (Hdist)

A parallel pseudo-random number generator (PRNG)

Given a seed and an algorithm, we have a stream of PRNs.

x11 x12

Given a seed and an algorithm, we have a stream of PRNs.

x11 x12

Using a second algorithm, the firststream will give us a sequence ofseeds. We use the 150 parametersets for the Tausworthe (LFSR)generators (L’ecuyer, Maths Comput1999, pp.261).Therefore, given the seed, we canpredict all states of all streams.

In our gene/species model:

x11 x12

we split gene families among jobs

all jobs receive seed (broadcast)and therefore can reproduce thesame x1. That’s cheaper thancommunicating the states.

each job uses its own x(i+1) forsampling new gene trees etc. andcan work in parallel. They use thecommon x1 for sampling e.g. newspecies tree, which needssynchronization.

the only thing that must be sharedis thus the proposal values(AllReduce) when updating”global” parameters”, so that alljobs can make the sameacceptance/rejection decision.

x11 x12

Each job looks like an independent analysis

https://bitbucket.org/leomrtns/guenomu

guenomu software -- model and agorithm in 2013

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