Leaving the topology and fixed adjust the tree small region

Bayesian Phylogenetics

Bret Larget

The Reverand Thomas Bayes was born in London in 1702.

He was the son of one of the first Noncomformist ministers to be ordained in England.

Bayes’ Theorem explains how to calculate inverse probabilities. For example, suppose that Box B1 contains four balls, three of which are black and one of which is white.

Box B2 has four balls, two of which are black and two of which are white.

If a ball is chosen uniformly at random from Box B1, there is a 3/4 chance that it is black.

But if a black ball is drawn, how likely is it that it came from Box B1?

Mathematical Background

Bayes’ Theorem

Things are further complicated in that additional parameters such as branch lengths and likelihood model parameters affect the likelihood, but are also unknown.

A posterior distribution is a probability distribution on parameters after data is observed.

Maximum Likelihood	Bayesian

	Only defined	Describes everything

Parameters		Random
Nuisance	Optimize them	Average over them

Model

Likelihood

Mathematical Background

Likelihood

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Then we are interested in computing

P(clade \| data)	=	tree with clade�
P(clade \| data)	=	tree with clade�	P(data \| tree)P(tree) P(data)

Methods

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Bayesian Phylogenetic Methods

P tree with cladeP(tree)	R

Metropolis-Hastings Example

L(d) =	�1	�50	×	�1 4− 1 4e− 4 3d	�9	×	�1 4 + 3 4e− 4 3d

p(d) =	λ (1 + λd)2 ,	d > 0

	Example	11 / 27

Example

�

Bayesian Phylogenetics

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density	10	0.0	0.2	0.4	prior (lambda = 10) posterior (lambda = 10) prior (lambda = 1) posterior (lambda = 1) likelihood
	8
	6				0.6
	4
	2
	0

	Example	13 / 27

What is Markov Chain Monte Carlo?

Metropolis-Hastings is a form of MCMC that works using any Markov chain to propose the next item to sample, but rejecting proposals with specified probability.

5	1	If accepted, set xi+1 = x∗.
5	2	If rejected, set xi+1 = xi.

We have a function h(θ) from which we want to sample.

We only need to know h up to a normalizing constant.

We begin the Markov chain at a single point.

We evaluate the value of h at this point.

Current state θ; Proposed state θ∗

This proposal is accepted.

Bayesian Phylogenetics

Computation

Example

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Second Proposal

Accept with probability 0.153

Example

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Third Proposal

The proposal was rejected, so proposed state is sampled again and remains current.

θ θ*

Bayesian Phylogenetics		Example	21 / 27

Sample So Far

G	G

Bayesian Phylogenetics

Example

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Repeat this for 10,000 proposals and show the sample.

Large Sample

Bayesian Phylogenetics

Example

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Bayesian Phylogenetics		Example	24 / 27

distribution at all: almost any type of proposal method would have

worked.

▶ The sample mean converges to the mean of the target.

▶ The sample median converges to the median of the target.

The model parameters for a Bayesian phylogenetics analysis typically includes:

▶ a tree (topology and branch lengths);
▶ substitution process parameters.

Cautions