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T. Bayes, Phil. Trans. Roy. Soc., 330 (1763).
Bayesian Inference of Phylogeny
T
TipTiSp
TipTiSpSTip
)()(
)()()(
Ι
ΙΙ
p(Ti|S) probability of the tree Ti given the sequence data Sp(S|Ti) probability or likelihood of the data S given tree Ti
p(Ti) prior probability of Ti
“The denominator sums the probabilities over all possible trees”
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ESTIMACION BAYESIANA• Inferencias están basadas en la
probabilidad de distribución posterior de un parámetro.
• La unión de las probabilidades de todos los parámetros son calculados.
• Las probabilidades están basadas en algún modelo (esperado a priori), luego de aprender algo de los datos.
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ESTIMACION BAYESIANA
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DADOS
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ESTIMACION BAYESIANA
• ¿Cuál es la probabilidad de tomar un dado trucado?
• Respuesta :1/10.
• Esta número representa la probabilidad a priori de tomar un dado sesgado.
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ESTIMACION BAYESIANA
Supongamos ahora que otra persona toma un par de dados de la caja y los tira.
Resultando:
¿Podemos creer que este resultado esta sesgado?
Dos aproximaciones: Maximum Likelihood e Inferencia Bayesiana.
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PROBABILIDADES
OBSERVACION NORMALES SESGADOS
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PR
PR
NORM
SESG
PROBABILIDADES
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ESTIMACION BAYESIANA
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Pr [Sesgados
INFERENCIA BAYESIANA
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ESTIMACION BAYESIANA
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11 44
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posterior
a priori
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Probabilidad a posteriori
Likelihood Probabilidad a priori
Σ de todas las probabilidades a posteriori
Integración de todas las posibles combinaciones de largo de ramas y modelos de sustitución nucleotídica.
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INFERIR UNA FILOGENIA
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POSIBLES FILOGENIAS
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Arboles equiprobables
Proporcional a observaciones: supuestos ej. alineamiento
Combinación: probabilidades a priori y Likelihood
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ALINEAMIENTO
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Estimación de las probabilidades a posteriori : ¿Cómo aproximarse?
• Calcular esta probabilidad implica: involucrar todos los árboles posibles….para cada árbol se debe integrar sobre todas las combinaciones de largo de rama y modelos de sustitución nucleotídica.
(IMPOSIBLE ANALÍTICAMENTE!!!) • Por necesidad la solución debe ser aproximada
• Método de Montecarlo
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Monte Carlo y cadenas Markovianas (MCMC)
• MCMC trabaja del siguiente modo:• a) Comienza una cadena markoviana con un
árbol ya sea 1) elegido al azar o 2) elegido por el investigador.
• b) Un nuevo árbol es propuesto….el proceso de cambio del arbol 1 al 2 debe satisfacer las siguientes condiciones:
1) El mecanismo debe ser estocástico; 2) cada arbol posible debe ser obtenido por aplicaciones repetidas del mismo mecanismo y 3) la cadena debe ser aperiodica.
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MARKOV CHAIN MONTE CARLO (MCMC)
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At each step in the chain a new tree is proposed by altering the At each step in the chain a new tree is proposed by altering the topology, or by changing branch lengths or the parameters of the topology, or by changing branch lengths or the parameters of the
model of sequence evolution.model of sequence evolution.
The Metropolis-Hastings algorithm is then used to accept or reject The Metropolis-Hastings algorithm is then used to accept or reject the new tree.the new tree.
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• Involucra correr algunas cadenas independientemente.
• La primera cadena que se cuenta (cold chain) el resto se denomina cadenas accesorias (heated chain).
• Saltos son intentados al azar entre dos cadenas distintas.
• Se necesita correr varios análisis independientes para confirmar convergencias.
METROPOLIS-COUPLED MARKOV CHAIN MONTE CARLO (MCMCMC o MC3)
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Resultado de esta búsqueda se obtiene un tercer término para la estimación de las probabilidades a posteriori (Proposal Ratio o Término de Hasting)
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INFERENCIA FILOGENÉTICA BAYESIANA
Phylogenetic tree
DNA Data
Evolutionary modelLikelihood
Prior probability
Posterior prob.
MCMC
Starting treeProposal
A sequence of Samples
inferencia
Approximate the distribution
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MrBayes: Bayesian Inference of Phylogeny
MrBayes is a program for Bayesian inference of phylogeny using Markov chain Monte Carlo methods. Avaialble for Mac, PC, and Unix.
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Métodos filogenéticos más usados
Data set
Algorithm
Algorithmicmethod
Optimization method
Distance matrix Character data
UPGMA
Neighbor-join
Fitch-Margolish
StatisticalSupported
Maximum Parsimony
MaximumLikelihood
Bayesian Methods
Search Strategy
Greedy search
Divide &Conquer
Stochastic search
DCM, HGT, Quartet
GA, SAMCMC
ExhaustiveBranch & Bound
Exact search
Stepwise additionGlobal arrangementStar decomposition
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Mapping characters onto phylogenies
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Mapping Uncertainty
parsimony ML
Bayesian
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Phylogenetic and Mapping Uncertainty
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