lunes, 3 de marzo de 2008

On IB

Bayesian methods deal with the notion of a probability distribution for the parameter; the distribution of the parameter before the data are analyzed is called the prior distribution. The Bayes theorem is used in Bayesian inference (BI) to calculate the posterior distribution of the parameter, that is, the conditional distribution of the parameter given the data (Holder and Lewis, 2003).

The general idea beyond the “tree inference” done by IB is to construct a Markov Chain that has as its state space the parameters of the statistical model, and a stationary distribution that is the posterior probability distribution of the parameters and run a sampling chain for long enough time; then sort sampled trees in probability order and pick trees until cumulative probability is reached! (Yang, 2006). Clearly it is not conceived as a search mechanism, but instead as a sampling mechanism and therefore it probably won’t find the individual trees of maximum a posteriori probability. There is still the difficulty of when to know whether the chain has run long enough and when the method converges, facts that are ignore in many publications.

There is an idea that IB provides measures of support faster than ML bootstrapping. Bayesian inference produces both a tree estimate and measures of uncertainty for the groups on the tree (Holder and Lewis, 2003). But the problem is that it attribute a high probability to false groups that should at least be recognized as ambiguous (Albert, 2005) and, indeed, when recognizing monophyletic group IB does it a frequentist way: check how many sampled trees claim a particular group is monophyletic and this is the probability of our group of being monophyletic!.

The optimal hypothesis under BI is the one that maximizes the posterior probability (Holder and Lewis, 2003). The posterior probability for a hypothesis is proportional to the likelihood multiplied by the prior probability of that hypothesis. In many publications Prior probabilities are ignored or a uniform distribution over the range of the parameters (flat priors) is used. It have been suggested that priors can be specified by using either an objective assessment of prior evidence concerning the parameter or the researcher's subjective opinion! (Yang, 2006), but when no available information about the parameter exists, it is unclear which prior is more reasonable. In the other hand it has been shown that uniform priors are not non informative -no prior represent total ignorance- and is generally accepted that personal prejudices influence statistical inference.

Note on Support:

If a data set contains homoplasy then different characters support different trees, hence which tree (or trees) a given data supports will depend on which characters have been sampled (Page and Holmes, 1998). I consider that support is a measure of how perturbation in the data gives a different result given that repeated sampling from the population is difficult and sometimes we are interested in what we call repeatability: the probability that another such sample shares the groups with the original one.

Estimates of phylogeny based on samples will be accompanied by sampling error. One way to measure sampling error is to take multiple resamples (pseudoreplicates) from our sample and build a tree. The variation among estimates derived from each pseudoreplicate is a measure of the sampling error associated with our sample. The simple bootstrap can be applied therefore as a perturbation tool to asses the stability (in the sense of continuity, a small perturbation in the data that produces only a small perturbation in the data that produces only small perturbation in the estimate) of the estimator (Holmes, 2003). Bootstrapping and jackknife (Bayesian methods based on Markov chain Monte Carlo as well) essentially make confidence statements for the trees. The other approach, Bremer support, examines how many extra steps are needed to lose a branch in the consensus tree of near-most-parsimonious trees. This method explores suboptimal solutions and determines how much worse a solution must be for a hypothesized group not to be recovered -the amount of contradictory evidence required to refute a group (Bremer, 1994)-.

1 comentario:

Salva dijo...

Hola Christian

Uno de los problemas lógicos de los 'flat priors' es que en realidad no son tales, después de todo en una árbol, la presencia de un grupo influye en la presencia de un grupo que en el que este anidado, o incluso en la presencia (o ausencia) de grupos en otras regiones del cladograma. Al final, por motivos mismos de la topología ciertos grupos (eso depende de la cantidad de terminales en el grupo) tienen mayor probabilidad que otros, como es imposible enumerar todos los arboles, conocer los priores para cada posible grupo es imposible, y su influencia en los resultados es todo un misterio. En realidad, la gente no sabe que priores esta usando! Pero para salir del asunto les llaman flat priors...

No se si la idea de apoyo que yo tengo se parezca a lo que describes, yo no creo que dependa mucho del 'muestreo', después de todo el muestreo en filogenia no es nada al azar! Aún así, es valido por ejemplo realizar boots, o jacks en datos no producto de muestreos (como datos morfológicos), si lo que intentas es descubrir la sensibilidad de los resultados a la interacción entre los datos. Es por eso, que metodos de remuestreo, así como los basados en árboles suboptimos miden precisamente el apoyo ;) [mira a Goloboff et al., 2003]