1. Lecture 6: Mutations and variational inference
Genome evolution
Lecture 6: Mutations and variational inference

2. Bayesian inference vs. Maximum likelihood
Maximum likelihood estimator
Introducing prior beliefs on the process
(Alternatively: think of virtual evidence)
Computing posterior probabilities on the parameters
No prior beliefs
Parameter Space
PME
Beliefs
MAP
MLE
Parameter Space

3. KL-divergence Entropy (Shannon) Kullback-leibler divergence
Not a metric!! KL

4. Expectation-Maximization
Relative entropy>=0
EM maximization
Dempster

5. Expectation-Maximization
Decompose over alignment positions
Group terms with the same free parameter
(weights are essentially the posterior of the parent child – prove it!

6. Terminology: Do you know how to define these by now?
Inference Parameter learning Likelihood Total probability/Marginal probability Exact inference/Approximate inference
Sampling is the a natural way to do approximate inference
Marginal
Probability
(integration over
A sample)
Marginal
Probability
(integration over
all space)

7. Sources of mutations Mistakes Endogenous DNA Damage
Replication errors (point mutations)
Recombination errors (mainly indels)
Endogenous DNA Damage
Spontaneous base damage: Deaminations, depurinations
Byproducts of metabolism: Oxygen radicals that damage DNA
Exogenous DNA Damage UV
Chemicals
All of these mechanisms cross talk with the surrounding sequence

8. DNA polymerases
replicating DNA
A good polymerase domain has a misincorporation rate of 10-5 (1/100,000)
Any misincorps are clipped off with 99% efficiency by the “proofreading” activity of the polymerase
Further mismatch repair that works in 99.9% of the case bring the fidelity of the main Polymerases to 10-10
Some dedicated polymerases are not as accurate!

9. Recombination errors
A consequence of partial homology between different chromosomal loci
Can introduce translocations if the matching sequences are on different chromsomes
Can introduce inversion or deletion if the matching sequences are on the same chromsome
Can generate duplication or deletions if the matching sequences are in tandem

10. Endogenous DNA damage: Deamination of CytosinesNH H O N 2 H*
deNHn
Cytosine
Uracil
*Thymine has CH3 here

11. Deamination of Cytosine creates a G-U mismatch
Easy to tell that U is wrong
Deamination of Cytosine creates a G-T mismatch
Not easy to tell which base is the mutation.
About 50% of the time the G is “corrected” to A
resulting in a mutation

12. Exogenous DNA damage UV irradiation generate primarily Thymine dimers:
Chemicals -
Food
Benzopyrene – smoke
UV radiations (Sunlight)
Ionizing raidation
radon
Cosmic rays
X rays

13. Repairing DNA damage
Direct repair

14. Thymine Dimers can be corrected by a direct repair mechanism
Photon

15. BER
Deaminated bases
are repaired by a
base excision mechanism.

16. BER Spontaneously occuring abasic sites are repaired
by the same mechanism

17. NER Dimeric bases and bulky lesions, e.g., large chemical adducts
are repaired by
Nucleotide excision repair

18. Adaptive mutations: Cairns et al. 88
Experimental system: lacz frameshift
The experiment suggests adaptive mutations
Luria-Delbruk’s observation

19. The “Mutator” paradigm:
Ability to switch to the mutator phenotype depends on particular DNA repair mechanisms (Double Strand Break repair in E. Coli)
Mutator phenotype is suggested to be important in pathogenesis, antibiotic resistance, and in cancer
Species occasionally change (adaptively or even by drift) their repair policy/efficiency
The resulted substitution landscape must be very complex

20. Dynamic Bayesian Networks
Conditional probabilities 1 3 2 4
T=1
T=2
T=3
T=4
T=5 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4
Synchronous discrete time process

21. Context dependent Markov Processes1 2 3 4
Context determines A markov process rate matrix
Any dependency structure make sense, including loops A A A C A A G A A
When context is changing, computing probabilities is difficult.
Think of the hidden variables as the trajectories A A A C
Continuous time Bayesian Networks
Koller-Noodleman 2002

22. Modeling simple context in the tree: PhyloHMM
hpaij
Heuristically approximating the Markov process?
Where exactly it fails?
hij-1
hij
hpaij-1
hpaij
hpaij+!
hkj-1
hkj
hkj+1
hij-1
hij
hij+!
Siepel-Haussler 2003

23. Log-likelihood to Free Energy
We have so far worked on computing the likelihood:
Computing likelihood is hard. We can reformulate the problem by adding parameters and transforming it into an optimization problem. Given a trial function q, define the free energy of the model as:
The free energy is exactly the likelihood when q is the posterior:
Better: when q a distribution, the free energy bounds the likelihood:
D(q || p(h|s))
Likelihood

24. Energy?? What energy?
In statistical mechanics, a system at temperature T with states x and an energy function E(x) is characterized by Boltzman’s law:
Z is the partition function:
Given a model p(h,s|T) (a BN), we can define the energy using Boltzman’s law
If we think of P(h|s,q):

25. Free Energy and Variational Free Energy
The Helmoholtz free energy is defined in physics as:
This free energy is important in statistical mechanics, but it is difficult to compute, as our probabilistic Z (= p(s))
The variational transformation introduce trial functions q(h), and set the variational free energy (or Gibbs free energy) to:
The average energy is:
The variational entropy is:
And as before:

26. Solving the variational optimization problem
Maxmizing U?
Maxmizing H?
Focus on max configurations
Spread out the distribution
So instead of computing p(s), we can search for q that optimizes the free energy
This is still hard as before, but we can simplify the problem by restricting q
(this is where the additional degrees of freedom become important)

27. Simplest variational approximation: Mean Field
Maxmizing U?
Maxmizing H?
Focus on max configurations
Spread out the distribution
Let’s assume complete independence among r.v.’s posteriors:
Under this assumption we can try optimizing the qi – (looking for minimal energy!)

28. Mean Field Inference We optimize iteratively:
Select i (sequentially, or using any method)
Optimize qi to minimize FMF(q1,..,qi,…,qn) while fixing all other qs
Terminate when FMF cannot be improved further
Remember: FMF always bound the likelihood
qi optimization can usually be done efficiently

29. Mean field for a simple-tree model
Just for illustration, since we know how solve this one exactly:
We select a node and optimize its qi while making sure it is a distribution:
The energy decomposes, and only few terms are affected:
To ease notation, assume the left (l) and right (r) children are hidden

30. Mean field for a simple-tree model
Just for illustration, since we know how solve this one exactly:
We select a node and optimize its qi while making sure it is a distribution:

31. Mean field for a phylo-hmm model
Now we don’t know how to solve this exactly, but MF is still simple:
hj-1pai
hjpai
hjpai
hj-1i
hji
hj+1i
hj-1r
hjl
hj+1l
hj-1r
hjr
hj+1r

32. Mean field for a phylo-hmm model
Now we don’t know how to solve this exactly, but MF is still simple:
hj-1pai
hjpai
hjpai
hj-1i
hji
hj+1i
As before, the optimal solution is derived by making logqi equals the sum of affected terms:
hj-1r
hjl
hj+1l
hj-1r
hjr
hj+1r

33. Simple Mean Field is usually not a good idea
Why?
Because the MF trial function is very crude
For example, we said before that the joint posteriors cannot be approximated by independent product of the hidden variables posteriors
A/C
A/C
A/C A C A C

34. Exploiting additional structure
We can greatly improve accuracy by generalizing the mean field algorithm using larger building blocks
The approximation specify independent distributions for each loci, but maintain the tree dependencies.
We now optimize each tree q separately, given the current other tree potentials.
The key point is that optimizing for any given tree is efficient: we just use a modified up-down algorithm

35. Tree based variational inference
Each tree is only affected by the tree before and the tree after:

36. Tree based variational inference
We got the same functional form as we had for the simple tree, so we can use the up-down algorithm to optimize qj.

37. Chain cluster variational inference
We can use any partition of a BN to trees and derive a similar MF algorithm
For example, instead of trees we can use the Markov chains in each species
What will work better for us?
Depends on the strength of dependencies at each dimension – we should try to capture as much “dependency” as possible