Bayesian inference

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Bayesian inference

• Based on Bayesian inference using Markov Chain
  Monte Carlo in phylogenetic studies by TorbojÖrn
  Karfunkel

Presented by Amir Hadadi Bioinformatics seminar,
spring 2005
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What is Bayesian inference ?

• Definition an approach to statistics in which
  all forms of uncertainty are expressed in terms
  of probability (Radford M. Neal)

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Probability reminder

• Conditional probability
• P(D?T) P(DT)?P(T)
• P(D?T) P(TD)?P(D)

Bayes theorem P(TD) P(DT)?P(T)/P(D)

• P(TD) is called the posterior probability of T
• P(T) is the prior probability, that is the
  probability assigned to T before seeing the data
• P(DT) is the likelihood of T, which is what we
  try to maximize in ML
• P(D) is the probability of observing the data D
  disregarding which tree is correct

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posterior vs. likelihood probabilitiesBayesian
inference vs. Maximum likelihood
Observation Fair Biased
1/6
1/21
1/6
2/21
1/6
3/21
1/6
4/21

• 100 dice
• some fair, some biased

1/6
5/21
1/6
6/21
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Example continued

• A die is drawn at random from the box
• Rolling the die twice gives us a and a
• Using the ML approach we get
• P( Fair) 1/6 ? 1/6 0.028
• P( Biased) 4/21 ? 6/21 0.054
• ML Conclusion the die is biased

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Example continued further

• Assume we have a prior knowledge about the dice
  distribution inside the box
• We know that in the box there are 90 fair dice
  and 10 biased dice

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Example conclusion

• Prior probability fair 0.1, biased 0.9
• Rolling the die twice gives us a and a
• Using the Bayesian approach we get
• P(Biased ) P(
  Biased)?P(Biased)/P( )0.179
• B.I. Conclusion the die is fair
• Conclusion ML and BI do not necessarily agree
• Resemblance of BI and ML results depends on the
  strength of prior assumptions we introduce

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Steps in B.I.

• formulate a model of the problem
• Formulate a prior distribution which captures
  your beliefs before seeing the data
• Obtain posterior distribution for the model
  parameters

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B.I. In phylogenetic reconstruction

• Phylogenetic reconstruction
• Finding an evolutionary tree which explains the
  Data (observed species)
• Methods of phylogenetic reconstruction
• Using a model of sequence evolution, e.g. maximum
  likelihood
• Not using sequence evolution, e.g. maximum
  parsimony, neighbor joining etc.
• Bayesian inference belongs to the first category

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Bayesian inference vs. Maximum likelihood

• The basic question in Bayesian inference
• What is the probability that this model (T) is
  correct, given the data (D) that we have observed
  ?
• Maximum likelihood asks a different question
• What is the probability of seeing the observed
  data (D) given that a certain model (T) is true
  ?
• B.I. seeks P(TD), while ML maximizes P(DT)

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Which priors should we assume ?

• Knowledge about a parameter can be used to
  approximate its prior distribution
• Usually we dont have prior knowledge about a
  parameters distribution. In this case a flat or
  vague prior is assumed.

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A flat prior
A vague prior
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How to find the posterior probability P(TD) ?

• P(T) is the assumed prior
• P(DT) is the likelihood
• Finding P(D) is infeasible we need to sum
  P(DT)P(T) over the entire tree space
• Markov Chain Monte Carlo (MCMC) gives us an
  indirect way of finding P(TD) without having to
  calculate P(D)

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MCMC Example
P1/2
P1/2
P1/2
P1/2
,
,
,
,
,
,
P(Palestine) 3/7, P(Tree) 4/7
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Symmetric simple random walk

• Definition A sequence of steps in ?, starting at
  0 and moving one step left or right with
  probability ½
• Properties
• After n steps the average distance from 0 is of
  magnitude ?n
• A random walk in one or two dimensions is
  recurrent
• A random walk in three dimensions or more is
  transient
• The Brownian motion is a limit of a random walk

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Definition of a markov chain

• A special type of stochastic process
• A sequence of random variables X0, X1, X2, such
  that
• Each Xi takes values in a state space S s1,
  s2,
• If x0, x1,, xn1 are elements of S, then
• P(Xn1 xn1Xn xn, Xn-1 xn-1,,X0 x0)
  
• P(Xn1 xn1Xn xn)

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Using MCMC to calculate posterior probabilities

• set S the set of parameters (e.g. tree
  topology, mutation probability, branch lengths
  etc.)
• Construct an MCMC with a stationary distribution
  equal to the posterior probability of the
  parameters
• Run the chain for a long time and sample from it
  regularly
• Use the samples to find the stationary
  distribution

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Constructing our MCMC

• The state space S is defined as the parameter
  space
• Start with random tree and parameters
• In each new generation, randomly propose either
• A new tree topology
• A new value for a model parameter
• If the proposed tree has higher posterior
  probability, ?proposed, than the current tree,
  ?current, the transition is accepted
• Otherwise the transition is accepted with
  probability ?proposed / ?current

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Algorithm visualization
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Convergence issues

• An MCMC might run for a long time until its
  sampled distribution is close to the stationary
  distribution
• The initial convergence phase is called the
  burn-in phase
• We wish to minimize burn-in time

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Avoiding getting stuck on local maxima

• Assume our landscape looks like this

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Avoiding local maxima (contd)

• descending a maximum can take a long time
• MCMCMC (Metropolis coupled MCMC) speeds the
  chains mixing rate
• Instead of running a single chain, multiple
  chains are run simultaneously
• The chains are heated to different degrees

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Chain heating
The cold chain has stationary distribution P(TD)
Heated chain number i has Stationary distribution
P(TD)1/i
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The MC3 algorithm

• Run multiple heated chains
• At each generation, attempt a swap between two
  chains
• If the swap is accepted, the hotter and cooler
  chains will swap states
• sample only from the cold chain

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Drawing conclusions

• To Decide the value of a parameter
• Draw a histogram showing the number of trees in
  each interval and calculate mean, mode,
  credibility intervals etc.
• To find the most likely tree topologies
• sort all sampled trees according to their
  posterior probabilities
• Pick the most probable trees until the cumulative
  probability is 0.95
• To Check whether a certain group of organisms is
  monophyletic
• Find the number of sampled trees in which it is
  monophyletic
• If it is monophyletic in 74 of the trees, it has
  a 74 probability of being monophyletic

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Summary

• Bayesian inference is very popular in many fields
  requiring statistical observations
• The advent of fast computers gave rise to the use
  of MCMC in B.I., enabling multi-parameter
  analysis
• Fields of genomics using Bayesian methods
• Identification of SNPs
• Inferring levels of gene expression and
  regulation
• Association mapping
• Etc.

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THE END
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A sample histogram