Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayesian Tree Sampling
Greg Ewing
CIBIV
December 3, 2007
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Outline
1 Bayes Theorem
Bayes Theorem
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Outline
1 Bayes Theorem
Bayes Theorem
2 Markov Chains
definition
Properties
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Outline
1 Bayes Theorem
Bayes Theorem
2 Markov Chains
definition
Properties
3 MHMCMC
Algorithm
Examples
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Outline
1 Bayes Theorem
Bayes Theorem
2 Markov Chains
definition
Properties
3 MHMCMC
Algorithm
Examples
4 What is long enough
Its all about the die
Hats
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Outline
1 Bayes Theorem
Bayes Theorem
2 Markov Chains
definition
Properties
3 MHMCMC
Algorithm
Examples
4 What is long enough
Its all about the die
Hats
5 Phylogenetic Bayesian MCMC
In practice
Priors
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Outline
1 Bayes Theorem
Bayes Theorem
2 Markov Chains
definition
Properties
3 MHMCMC
Algorithm
Examples
4 What is long enough
Its all about the die
Hats
5 Phylogenetic Bayesian MCMC
In practice
Priors
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
The difference
The Bayesian approach asks the right question in a
hypothesis testing procedure, namely, "What is the
probability that this hypothesis is true, given the data?"
rather than the classical approach, which asks a
question like, "Assuming that this hypothesis is true,
what is the probability of the observed data?"
­Statistical Methods in Bioinformatics
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Derivation
We know that
Pr(A  B) = Pr(B|A) Pr(A),
from conditional probability.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Derivation
We know that
Pr(A  B) = Pr(B|A) Pr(A),
from conditional probability. Also
Pr(A  B) = Pr(B  A) = Pr(A|B) Pr(B).
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Derivation
We know that
Pr(A  B) = Pr(B|A) Pr(A),
from conditional probability. Also
Pr(A  B) = Pr(B  A) = Pr(A|B) Pr(B).
Therefore
Pr(A|B) Pr(B) = Pr(B|A) Pr(A)
Pr(A|B) =
Pr(B|A) Pr(A)
Pr(B)
.
This is Bayes formula or theorem.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Bayes Theorem
Pr(A|B) =
Pr(B|A) Pr(A)
Pr(B)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Bayes Theorem
Pr(A|B) =
Pr(B|A) Pr(A)
Pr(B)
Pr(A|B)
Posterior Density

Likelihood
L(A, B)
Prior
Pr(A)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Bayes Theorem
Pr(A|B) =
Pr(B|A) Pr(A)
Pr(B)
Bayesian, flips the probability around.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Bayes Theorem
Pr(A|B) =
Pr(B|A) Pr(A)
Pr(B)
Bayesian, flips the probability around.
It is easy to include prior information which is often
available.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Bayes Theorem
Pr(A|B) =
Pr(B|A) Pr(A)
Pr(B)
Bayesian, flips the probability around.
It is easy to include prior information which is often
available.
The Bayesian conditional probability is perhaps more
intuitive.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Making formulas tangible
Pr(T, M|D)  Pr(D|T, M) Pr(T, M)
The likelihood is L(T, D, M) = Pr(D|T, M)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Making formulas tangible
Pr(T, M|D)  Pr(D|T, M) Pr(T, M)
The likelihood is L(T, D, M) = Pr(D|T, M)
T is the tree.
D is the DNA/Protein etc sequence data.
M is the model parameters, like GTR.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Making formulas tangible
Pr(T, M|D)  Pr(D|T, M) Pr(T, M)
The likelihood is L(T, D, M) = Pr(D|T, M)
T is the tree.
D is the DNA/Protein etc sequence data.
M is the model parameters, like GTR.
In words: The likelihood is the probability of the DNA data
given the Tree and the model parameters.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Making formulas tangible
Pr(T, M|D)  Pr(D|T, M) Pr(T, M)
The likelihood is L(T, D, M) = Pr(D|T, M)
T is the tree.
D is the DNA/Protein etc sequence data.
M is the model parameters, like GTR.
In words: The likelihood is the probability of the DNA data
given the Tree and the model parameters.
The Prior is Pr(T, M) and indicates any information we
already know. i.e. The root is not older than 10 million
years.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
Making formulas tangible
Pr(T, M|D)  Pr(D|T, M) Pr(T, M)
The likelihood is L(T, D, M) = Pr(D|T, M)
T is the tree.
D is the DNA/Protein etc sequence data.
M is the model parameters, like GTR.
In words: The likelihood is the probability of the DNA data
given the Tree and the model parameters.
The Prior is Pr(T, M) and indicates any information we
already know. i.e. The root is not older than 10 million
years.
The Posterior density is Pr(T, M|D) the probability of the
tree and model parameters given the sequence data.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
The Bad News
We can't directly solve for the posterior distribution.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
The Bad News
We can't directly solve for the posterior distribution.
Therefore MHMCMC must be used, this means it will take
a lot of computer resources.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
The Bad News
We can't directly solve for the posterior distribution.
Therefore MHMCMC must be used, this means it will take
a lot of computer resources.
The "answer" is not a tree, but a distribution of trees/states.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Bayes Theorem
The Bad News
We can't directly solve for the posterior distribution.
Therefore MHMCMC must be used, this means it will take
a lot of computer resources.
The "answer" is not a tree, but a distribution of trees/states.
It will always be slower than ML.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Outline
1 Bayes Theorem
Bayes Theorem
2 Markov Chains
definition
Properties
3 MHMCMC
Algorithm
Examples
4 What is long enough
Its all about the die
Hats
5 Phylogenetic Bayesian MCMC
In practice
Priors
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chains
Assume that I have a machine that outputs random
numbers, ie a chain of numbers.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chains
Assume that I have a machine that outputs random
numbers, ie a chain of numbers.
If I can work out the probability of the next output by only
looking at the previous output, it is said to have the Markov
property.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chains
Assume that I have a machine that outputs random
numbers, ie a chain of numbers.
If I can work out the probability of the next output by only
looking at the previous output, it is said to have the Markov
property.
Example: Our machine flips a coin and either adds one to
the last output or subtracts one.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chains
Assume that I have a machine that outputs random
numbers, ie a chain of numbers.
If I can work out the probability of the next output by only
looking at the previous output, it is said to have the Markov
property.
Example: Our machine flips a coin and either adds one to
the last output or subtracts one.
Machine Output
1,2,1,0,1,0,-1,-2,-3,-2,-3,-4,-3,-2,-1,0,-1,0,1,2,1,2,3
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chains
Assume that I have a machine that outputs random
numbers, ie a chain of numbers.
If I can work out the probability of the next output by only
looking at the previous output, it is said to have the Markov
property.
Example: Our machine flips a coin and either adds one to
the last output or subtracts one.
Machine Output
1,2,1,0,1,0,-1,-2,-3,-2,-3,-4,-3,-2,-1,0,-1,0,1,2,1,2,3
We don't care about the whole sequence, just the last
output which is 3.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chains
Assume that I have a machine that outputs random
numbers, ie a chain of numbers.
If I can work out the probability of the next output by only
looking at the previous output, it is said to have the Markov
property.
Example: Our machine flips a coin and either adds one to
the last output or subtracts one.
Machine Output
1,2,1,0,1,0,-1,-2,-3,-2,-3,-4,-3,-2,-1,0,-1,0,1,2,1,2,3
We don't care about the whole sequence, just the last
output which is 3.
The next item has a 50% chance that it will be a 4, and a
50% chance that it will be a 2.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chains
Assume that I have a machine that outputs random
numbers, ie a chain of numbers.
If I can work out the probability of the next output by only
looking at the previous output, it is said to have the Markov
property.
Example: Our machine flips a coin and either adds one to
the last output or subtracts one.
Machine Output
1,2,1,0,1,0,-1,-2,-3,-2,-3,-4,-3,-2,-1,0,-1,0,1,2,1,2,3
We don't care about the whole sequence, just the last
output which is 3.
The next item has a 50% chance that it will be a 4, and a
50% chance that it will be a 2.
This is a Markov Chain.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Definition of a Markov Chain
Definition
A Markov Chain is a chain of randomly chosen values where
the probability of the next value is entirely determined by the
previous value.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Definition of a Markov Chain
Definition
A Markov Chain is a chain of randomly chosen values where
the probability of the next value is entirely determined by the
previous value.
Rough Math definition
Pr(Xn|Xn-1, Xn-2, . . .) = Pr(Xn|Xn-1)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
State Graph
A
B
c
Simple Markov Chains can be represented as a graph.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
State Graph
A
B
c
Simple Markov Chains can be represented as a graph.
Nodes or circles represent states (the last output).
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
State Graph
A
B
c
Simple Markov Chains can be represented as a graph.
Nodes or circles represent states (the last output).
Arrows are transitions between states.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
State Graph
A
B
c
Simple Markov Chains can be represented as a graph.
Nodes or circles represent states (the last output).
Arrows are transitions between states.
Transitions (Arrows) usually have probabilities on them.
That is the probability that this transition will be followed.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
State Graph
A
B
c
Simple Markov Chains can be represented as a graph.
Nodes or circles represent states (the last output).
Arrows are transitions between states.
Transitions (Arrows) usually have probabilities on them.
That is the probability that this transition will be followed.
For clarity, when transitions are equiprobable we omit the
transition probabilities.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C B
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C B A
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C B A B
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C B A B A
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C B A B A B
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C B A B A B A
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C B A B A B A C
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C B A B A B A C B
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C B A B A B A C B A
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C B A B A B A C B A C
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
definition
Markov Chain Graph
Example
State Graph
A
B
c
Output
A C B A B A B A C B A C
Note that the states can be anything. ie different trees
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Irreducibility
Reducible state diagram
A
B
c D E
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Irreducibility
Reducible state diagram
A
B
c D E
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Irreducibility
Reducible state diagram
A
B
c D E
Definition
A Markov Chain is Irreducible if and only if the chain can get
from any possible state to any other possible state eventually.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Irreducibility
Reducible state diagram
A
B
c D E
Definition
A Markov Chain is Irreducible if and only if the chain can get
from any possible state to any other possible state eventually.
The above state diagram is NOT irreducible.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Irreducibility
Reducible state diagram
A
B
c D E
Definition
A Markov Chain is Irreducible if and only if the chain can get
from any possible state to any other possible state eventually.
The above state diagram is NOT irreducible.
Adding a transition from D  C it would make this
irreducible
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Reversibility
A
B
c
Is this output reversed?
C A B C A B A B A B C
Note that there is no C  B transition or C  A transition.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Reversibility
A
B
c
Is this output reversed?
C A B C A B A B A B C
Note that there is no C  B transition or C  A transition.
Therefore we can tell that this output sequence is reversed.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Reversibility
Tricky Example
A
B
c
0. 5
0. 5
0. 1
0.5
0.9
0.5
Is this output reversed?
A B C A B C B C B C B C A B C B C A B A
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Reversibility
Is this output reversed?
A B C A B C B C B C B C A B C B C A B A
The transition B  A is much less likely than B  C in the
forward direction.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Reversibility
Is this output reversed?
A B C A B C B C B C B C A B C B C A B A
The transition B  A is much less likely than B  C in the
forward direction.
In this example there are 7 B  C transitions and only 1
B  A transition in the forward direction.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Reversibility
Is this output reversed?
A B C A B C B C B C B C A B C B C A B A
The transition B  A is much less likely than B  C in the
forward direction.
In this example there are 7 B  C transitions and only 1
B  A transition in the forward direction.
Conversely there are 4 B  C transitions and 4 B  A
transitions in the reverse direction.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Reversibility
Is this output reversed?
A B C A B C B C B C B C A B C B C A B A
The transition B  A is much less likely than B  C in the
forward direction.
In this example there are 7 B  C transitions and only 1
B  A transition in the forward direction.
Conversely there are 4 B  C transitions and 4 B  A
transitions in the reverse direction.
It seems we can guess that this output is not reversed.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Reversibility
Is this output reversed?
A B C A B C B C B C B C A B C B C A B A
The transition B  A is much less likely than B  C in the
forward direction.
In this example there are 7 B  C transitions and only 1
B  A transition in the forward direction.
Conversely there are 4 B  C transitions and 4 B  A
transitions in the reverse direction.
It seems we can guess that this output is not reversed.
But we stick to simple definitions for this course.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Reversibility
Definition
A Markov Chain is reversible if we cannot detect whether or not
the chain is running in "reverse". That is the output is
statistically identicle in both directions.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Aperiodic
Periodic-Aperiodic
1
3
2
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Aperiodic
Periodic-Aperiodic
1
3
2 1
3
2
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Aperiodic
Periodic-Aperiodic
1
3
2 1
3
2
Definition
A Markov Chain is periodic if there is some fixed "cycle" of
states, and it is aperiodic otherwise.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Why do we care?
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Why do we care?
If a MCMC chain has these 3 properties (reversible,
irreducible and aperiodic), then it is also ergodic.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Stationary distribution
1
3
2 4 -1
output
1 3 2 4 4 2 1 2 -1 -1 4 2 3 1 3 2 4 4 4 -1 -1 -1 4 2 3 1 2 3 -1
We can calculate statistics on the output, like mean and
standard deviation. Also we can plot histograms etc.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Stationary distribution
1
3
2 4 -1
output
1 3 2 4 4 2 1 2 -1 -1 4 2 3 1 3 2 4 4 4 -1 -1 -1 4 2 3 1 2 3 -1
We can calculate statistics on the output, like mean and
standard deviation. Also we can plot histograms etc.
Consider the distribution of the output.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Stationary distribution
1
3
2 4 -1
output
1 3 2 4 4 2 1 2 -1 -1 4 2 3 1 3 2 4 4 4 -1 -1 -1 4 2 3 1 2 3 -1
We can calculate statistics on the output, like mean and
standard deviation. Also we can plot histograms etc.
Consider the distribution of the output.
What about the start state. That is if the chain is started in
state 1, will the distribution be different from starting in 2.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Ergodic
Definition
If we can start from any state, and if we take samples for long
enough, and we end up with the same distribution, that
distribution is the stationary distribution of the Markov Chain,
and the Markov Chain is said to be ergodic
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Ergodic
Definition
If we can start from any state, and if we take samples for long
enough, and we end up with the same distribution, that
distribution is the stationary distribution of the Markov Chain,
and the Markov Chain is said to be ergodic
Definition
If a Markov Chain is reversible, irreducible and aperiodic then it
is also ergodic.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Ergodic
Definition
If we can start from any state, and if we take samples for long
enough, and we end up with the same distribution, that
distribution is the stationary distribution of the Markov Chain,
and the Markov Chain is said to be ergodic
Definition
If a Markov Chain is reversible, irreducible and aperiodic then it
is also ergodic.
So we can know that a chain will converge to the stationary
distribution without testing every state.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Ergodic
Definition
If we can start from any state, and if we take samples for long
enough, and we end up with the same distribution, that
distribution is the stationary distribution of the Markov Chain,
and the Markov Chain is said to be ergodic
Definition
If a Markov Chain is reversible, irreducible and aperiodic then it
is also ergodic.
So we can know that a chain will converge to the stationary
distribution without testing every state.
Usually the symbol  denotes the stationary distribution.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Properties
Extra Markov Chain Properties
Ergodic
Definition
If we can start from any state, and if we take samples for long
enough, and we end up with the same distribution, that
distribution is the stationary distribution of the Markov Chain,
and the Markov Chain is said to be ergodic
Definition
If a Markov Chain is reversible, irreducible and aperiodic then it
is also ergodic.
So we can know that a chain will converge to the stationary
distribution without testing every state.
Usually the symbol  denotes the stationary distribution.
Note that we have not said anything about how many
samples we need to get an accurate distribution.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Outline
1 Bayes Theorem
Bayes Theorem
2 Markov Chains
definition
Properties
3 MHMCMC
Algorithm
Examples
4 What is long enough
Its all about the die
Hats
5 Phylogenetic Bayesian MCMC
In practice
Priors
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Algorithm
Metropolis Hastings MCMC
Algorithm
Start in state Xn
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Algorithm
Metropolis Hastings MCMC
Algorithm
Start in state Xn
Randomly generate some new state X from X
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Algorithm
Metropolis Hastings MCMC
Algorithm
Start in state Xn
Randomly generate some new state X from X
Calculate the acceptance probability based on the
posterior density.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Algorithm
Metropolis Hastings MCMC
Algorithm
Start in state Xn
Randomly generate some new state X from X
Calculate the acceptance probability based on the
posterior density.
Accept the new state with that probability.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Algorithm
Metropolis Hastings MCMC
Algorithm
Start in state Xn
Randomly generate some new state X from X
Calculate the acceptance probability based on the
posterior density.
Accept the new state with that probability.
If we accept, then Xn+1 = X, otherwise Xn+1 = Xn.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Algorithm
If our new state generation step can get to any valid state
eventually (with non zero probability), then the chain is
irreducible.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Algorithm
If our new state generation step can get to any valid state
eventually (with non zero probability), then the chain is
irreducible.
If iťs possible to generate X from X and X from X then the
chain can be reversible.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Algorithm
If our new state generation step can get to any valid state
eventually (with non zero probability), then the chain is
irreducible.
If iťs possible to generate X from X and X from X then the
chain can be reversible.
The acceptance probability is chosen so that the chain will
be reversible and aperiodic.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Algorithm
If our new state generation step can get to any valid state
eventually (with non zero probability), then the chain is
irreducible.
If iťs possible to generate X from X and X from X then the
chain can be reversible.
The acceptance probability is chosen so that the chain will
be reversible and aperiodic.
Therefore the chain is ergodic with stationary distribution
.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Algorithm
If our new state generation step can get to any valid state
eventually (with non zero probability), then the chain is
irreducible.
If iťs possible to generate X from X and X from X then the
chain can be reversible.
The acceptance probability is chosen so that the chain will
be reversible and aperiodic.
Therefore the chain is ergodic with stationary distribution
.
The Key Idea
The stationary distribution is the posterior distribution of
interest. That is the MHMCMC chain is sampling the Bayesian
posterior distribution.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Example
Start with tree T = (a, b|c, d).
Output
(a, b|c, d)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Example
Start with tree T = (a, b|c, d).
Generate a new tree from T by a branch swap (b  c).
T = (a, c|b, d)
Output
(a, b|c, d)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Example
Start with tree T = (a, b|c, d).
Generate a new tree from T by a branch swap (b  c).
T = (a, c|b, d)
Calculate acceptance probability and then accept/reject.
We reject this time.
Output
(a, b|c, d)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Example
Start with tree T = (a, b|c, d).
Generate a new tree from T by a branch swap (b  c).
T = (a, c|b, d)
Calculate acceptance probability and then accept/reject.
We reject this time.
The new state is T = (a, b|c, d) which we output.
Output
(a, b|c, d) (a, b|c, d)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Example
Start with tree T = (a, b|c, d).
Generate a new tree from T by a branch swap (b  c).
T = (a, c|b, d)
Calculate acceptance probability and then accept/reject.
We reject this time.
The new state is T = (a, b|c, d) which we output.
The next generated state is T = (a, d|b, c) (b  d) and this
time we accept.
Output
(a, b|c, d) (a, b|c, d)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Example
Start with tree T = (a, b|c, d).
Generate a new tree from T by a branch swap (b  c).
T = (a, c|b, d)
Calculate acceptance probability and then accept/reject.
We reject this time.
The new state is T = (a, b|c, d) which we output.
The next generated state is T = (a, d|b, c) (b  d) and this
time we accept.
The new state is T = (a, d|b, c)
Output
(a, b|c, d) (a, b|c, d) (a, d|b, c)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Example
Start with tree T = (a, b|c, d).
Generate a new tree from T by a branch swap (b  c).
T = (a, c|b, d)
Calculate acceptance probability and then accept/reject.
We reject this time.
The new state is T = (a, b|c, d) which we output.
The next generated state is T = (a, d|b, c) (b  d) and this
time we accept.
The new state is T = (a, d|b, c)
We continue T = (a, c|b, d) (c  d), and accept.
Output
(a, b|c, d) (a, b|c, d) (a, d|b, c) (a, c|b, d)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
Wiki Formula
Pr(k|i, s) =
1
si
k-i
s

n=0
(-1)n i
n
k - sn - 1
i - 1
Die MHMCMC
Formula looks too complicated!
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
Wiki Formula
Pr(k|i, s) =
1
si
k-i
s

n=0
(-1)n i
n
k - sn - 1
i - 1
Die MHMCMC
Formula looks too complicated!
Use a simple MHMCMC instead.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
Wiki Formula
Pr(k|i, s) =
1
si
k-i
s

n=0
(-1)n i
n
k - sn - 1
i - 1
Die MHMCMC
Formula looks too complicated!
Use a simple MHMCMC instead.
Just pick one die at random and re-throw.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
Wiki Formula
Pr(k|i, s) =
1
si
k-i
s

n=0
(-1)n i
n
k - sn - 1
i - 1
Die MHMCMC
Formula looks too complicated!
Use a simple MHMCMC instead.
Just pick one die at random and re-throw.
This is reversible and the acceptance ratio is 1. i.e we
always accept.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
3 die
1 1 1
Output
3
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
3 die
1 1 1
4 1 1
Output
3 6
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
3 die
1 1 1
4 1 1
4 1 6
Output
3 6 11
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
3 die
1 1 1
4 1 1
4 1 6
2 1 6
Output
3 6 11 9
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
3 die
1 1 1
4 1 1
4 1 6
2 1 6
3 1 6
Output
3 6 11 9 10
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
3 die
1 1 1
4 1 1
4 1 6
2 1 6
3 1 6
3 1 4
Output
3 6 11 9 10 8
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
3 die
1 1 1
4 1 1
4 1 6
2 1 6
3 1 6
3 1 4
3 5 4
Output
3 6 11 9 10 8 12
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Examples
Die example
3 die
1 1 1
4 1 1
4 1 6
2 1 6
3 1 6
3 1 4
3 5 4
3 2 4
Output
3 6 11 9 10 8 12 9
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Outline
1 Bayes Theorem
Bayes Theorem
2 Markov Chains
definition
Properties
3 MHMCMC
Algorithm
Examples
4 What is long enough
Its all about the die
Hats
5 Phylogenetic Bayesian MCMC
In practice
Priors
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Its all about the die
More Die
By changing just one dice at each step, the sum can never
change by more than 5 from step to step.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Its all about the die
More Die
By changing just one dice at each step, the sum can never
change by more than 5 from step to step.
If we have 100 die and start at all ones, it will take a long
time to get to the "equilibrium".
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Its all about the die
More Die
By changing just one dice at each step, the sum can never
change by more than 5 from step to step.
If we have 100 die and start at all ones, it will take a long
time to get to the "equilibrium".
On the other hand we could roll every die at each step.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Its all about the die
More Die
By changing just one dice at each step, the sum can never
change by more than 5 from step to step.
If we have 100 die and start at all ones, it will take a long
time to get to the "equilibrium".
On the other hand we could roll every die at each step.
In this case we get to equilibrium in just a single step but
must generate 100 random numbers per step.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Its all about the die
More Die
100 die, rolling 1 dice per step
0 200 400 600 800 1000
100150200250300350400
Samples
DieSum
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Its all about the die
More Die
100 die, rolling 1 dice per step
0 200 400 600 800 1000
100150200250300350400
Samples
DieSum
100 die, rolling all per step
0 200 400 600 800 1000
100150200250300350400
Samples
DieSum
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Its all about the die
Effective Sample Size
Both chains were 1000 MCMC samples long, but each
sample is not independent of the other.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Its all about the die
Effective Sample Size
Both chains were 1000 MCMC samples long, but each
sample is not independent of the other.
Its clear that the second case gives better results.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Its all about the die
Effective Sample Size
Both chains were 1000 MCMC samples long, but each
sample is not independent of the other.
Its clear that the second case gives better results.
Effective sample size is the estimated number of
independent samples and is calculated with the Integrated
autocorrelation time. (in tracer for example)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Its all about the die
Effective Sample Size
Both chains were 1000 MCMC samples long, but each
sample is not independent of the other.
Its clear that the second case gives better results.
Effective sample size is the estimated number of
independent samples and is calculated with the Integrated
autocorrelation time. (in tracer for example)
Due to the correlations between samples we don't really
need every sample from the MCMC chain and instead only
collect every 100'th sample or so.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Its all about the die
Effective Sample Size
Both chains were 1000 MCMC samples long, but each
sample is not independent of the other.
Its clear that the second case gives better results.
Effective sample size is the estimated number of
independent samples and is calculated with the Integrated
autocorrelation time. (in tracer for example)
Due to the correlations between samples we don't really
need every sample from the MCMC chain and instead only
collect every 100'th sample or so.
Performance should be measured in the number of
effective samples per CPU cycle.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Hats
Witch's Hat
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Hats
Witch's Hat
Consider all non tree like signals.
Recombination, Horizontal Gene Transfer and other effects
could contribute to a lot of witch's hats.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Hats
Key Points for simple analysis
Check Effective Sample Size.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Hats
Key Points for simple analysis
Check Effective Sample Size.
Choose the correct sample intervals.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Hats
Key Points for simple analysis
Check Effective Sample Size.
Choose the correct sample intervals.
Check Burn in. It should be small enough that it does not
matter if you include it.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Hats
Key Points for simple analysis
Check Effective Sample Size.
Choose the correct sample intervals.
Check Burn in. It should be small enough that it does not
matter if you include it.
Not all moves are equal. How long depends on many
things
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Hats
Key Points for simple analysis
Check Effective Sample Size.
Choose the correct sample intervals.
Check Burn in. It should be small enough that it does not
matter if you include it.
Not all moves are equal. How long depends on many
things
Multiple runs from random starting locations
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Outline
1 Bayes Theorem
Bayes Theorem
2 Markov Chains
definition
Properties
3 MHMCMC
Algorithm
Examples
4 What is long enough
Its all about the die
Hats
5 Phylogenetic Bayesian MCMC
In practice
Priors
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Posterior
Pr(T, M|D)  Pr(D|T, M) Pr(T, M)
The likelihood is L(T, D, M) = Pr(D|T, M)
T is the tree.
D is the DNA/Protein etc sequence data.
M is the model parameters, like GTR.
Warning
Trees Make Life Difficult
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Moves and why you care about irreducibility
Many programs have a huge set of options.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Moves and why you care about irreducibility
Many programs have a huge set of options.
It is often possible to have moves that are not reversible or
irreducible.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Moves and why you care about irreducibility
Many programs have a huge set of options.
It is often possible to have moves that are not reversible or
irreducible.
Hence will not properly sample the posterior distribution.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Moves and why you care about irreducibility
Many programs have a huge set of options.
It is often possible to have moves that are not reversible or
irreducible.
Hence will not properly sample the posterior distribution.
It may not be possible to get to the parts of the state space
that are of interest.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Moves and why you care about irreducibility
Many programs have a huge set of options.
It is often possible to have moves that are not reversible or
irreducible.
Hence will not properly sample the posterior distribution.
It may not be possible to get to the parts of the state space
that are of interest.
The wrong choice of moves could make the chain run very
slowly.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Moves and why you care about irreducibility
Many programs have a huge set of options.
It is often possible to have moves that are not reversible or
irreducible.
Hence will not properly sample the posterior distribution.
It may not be possible to get to the parts of the state space
that are of interest.
The wrong choice of moves could make the chain run very
slowly.
Examples of real output.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Aside: Hot and Cold chains
Have more than one chain.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Aside: Hot and Cold chains
Have more than one chain.
Each extra chain is heated. With only one chain that is not.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Aside: Hot and Cold chains
Have more than one chain.
Each extra chain is heated. With only one chain that is not.
We swap states between chains at each step or as
frequently as desired.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Aside: Hot and Cold chains
Have more than one chain.
Each extra chain is heated. With only one chain that is not.
We swap states between chains at each step or as
frequently as desired.
Only collect samples from the cold chain. ie the only chain
with the correct distribution.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Aside: Hot and Cold chains
Have more than one chain.
Each extra chain is heated. With only one chain that is not.
We swap states between chains at each step or as
frequently as desired.
Only collect samples from the cold chain. ie the only chain
with the correct distribution.
The idea is that we won't get stuck.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
In practice
Aside: Hot and Cold chains
Have more than one chain.
Each extra chain is heated. With only one chain that is not.
We swap states between chains at each step or as
frequently as desired.
Only collect samples from the cold chain. ie the only chain
with the correct distribution.
The idea is that we won't get stuck.
Generally not as effective as just developing some better
moves.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Priors
Priors
Huge topic!
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Priors
Priors
Huge topic!
Without proper priors, the posterior density
may not even exist!
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Priors
Priors
Huge topic!
Without proper priors, the posterior density
may not even exist!
Priors do not need to be highly informed to be effective. e.g
root height.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Priors
Priors
Huge topic!
Without proper priors, the posterior density
may not even exist!
Priors do not need to be highly informed to be effective. e.g
root height.
Informative priors can make analysis possible by restricting
the state space
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Priors
Priors
Huge topic!
Without proper priors, the posterior density
may not even exist!
Priors do not need to be highly informed to be effective. e.g
root height.
Informative priors can make analysis possible by restricting
the state space
Priors should be considered with respect to the hypothesis
that will be tested.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Priors
Priors-Rules
Trees must have a prior.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Priors
Priors-Rules
Trees must have a prior.
Even if all the branch lengths in a topology are infinitely
long the likelihood is still finite.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Priors
Priors-Rules
Trees must have a prior.
Even if all the branch lengths in a topology are infinitely
long the likelihood is still finite.
Infinitely long branches do not make sense.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Priors
Priors-Rules
Trees must have a prior.
Even if all the branch lengths in a topology are infinitely
long the likelihood is still finite.
Infinitely long branches do not make sense.
Yule priors, coalescent priors and exponential priors are
common.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Priors
Priors-Rules
Trees must have a prior.
Even if all the branch lengths in a topology are infinitely
long the likelihood is still finite.
Infinitely long branches do not make sense.
Yule priors, coalescent priors and exponential priors are
common.
For rooted topologies, a simple bounded uniform prior is
sufficient.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Priors
Priors-Rules
Trees must have a prior.
Even if all the branch lengths in a topology are infinitely
long the likelihood is still finite.
Infinitely long branches do not make sense.
Yule priors, coalescent priors and exponential priors are
common.
For rooted topologies, a simple bounded uniform prior is
sufficient.
Even if the max root height is 100 expected substitutions
per site, the posterior can now be normalized.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Rules of thumb
Do more than one run. I recommend about 10 or so if
possible.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Rules of thumb
Do more than one run. I recommend about 10 or so if
possible.
Each run should always start from a random starting point.
Never use an NJ tree or any other "good" starting point.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Rules of thumb
Do more than one run. I recommend about 10 or so if
possible.
Each run should always start from a random starting point.
Never use an NJ tree or any other "good" starting point.
Burn in should be less than a tenth of the full run. In
general if the statistics are affected by the amount of burn
in, it wasn't run long enough.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Rules of thumb
Do more than one run. I recommend about 10 or so if
possible.
Each run should always start from a random starting point.
Never use an NJ tree or any other "good" starting point.
Burn in should be less than a tenth of the full run. In
general if the statistics are affected by the amount of burn
in, it wasn't run long enough.
More parameters will always take longer. Don't use more
parameters than are needed.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Rules of thumb
Do more than one run. I recommend about 10 or so if
possible.
Each run should always start from a random starting point.
Never use an NJ tree or any other "good" starting point.
Burn in should be less than a tenth of the full run. In
general if the statistics are affected by the amount of burn
in, it wasn't run long enough.
More parameters will always take longer. Don't use more
parameters than are needed.
Check your priors!
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Summary
Bayesian inference is not maximum likelihood.
Other points to consider.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Summary
Bayesian inference is not maximum likelihood.
It is not a black box. Care must be taken to get the chain
setup correctly, and when interpreting the results.
Other points to consider.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Summary
Bayesian inference is not maximum likelihood.
It is not a black box. Care must be taken to get the chain
setup correctly, and when interpreting the results.
Run the chain long enough! This is the most common
mistake.
Other points to consider.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Summary
Bayesian inference is not maximum likelihood.
It is not a black box. Care must be taken to get the chain
setup correctly, and when interpreting the results.
Run the chain long enough! This is the most common
mistake.
Other points to consider.
Generally slower than ML. (bootstrapped)
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Summary
Bayesian inference is not maximum likelihood.
It is not a black box. Care must be taken to get the chain
setup correctly, and when interpreting the results.
Run the chain long enough! This is the most common
mistake.
Other points to consider.
Generally slower than ML. (bootstrapped)
Support values are easier to interpret.
Bayes Theorem Markov Chains MHMCMC What is long enough Phylogenetic Bayesian MCMC Summary
Summary
Bayesian inference is not maximum likelihood.
It is not a black box. Care must be taken to get the chain
setup correctly, and when interpreting the results.
Run the chain long enough! This is the most common
mistake.
Other points to consider.
Generally slower than ML. (bootstrapped)
Support values are easier to interpret.
Can incorporate prior information easily.