Notes 20100610 CS 798 Course

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Inference with a Flat Prior back in Inference World

for all the topologies that include some particular subtree
a subquartet ((A,B),(C,D))
what is the average likelihood of the data?
likelihood of data is generally something unbelievably small
compare to ((A,C),(B,D)) or ((A,D),(B,C))
so we don't know what the right tree is-- but we can estimate the likelihood of a given quartet
if that number is >1, then a given quartet is likely-- if it is <1, then it is unlikely

If we assumed
Bayes' theorem
Pr(A | B) = Pr(B | A) Pr(A) ÷ Pr(B)
Data has been given Pr(A | Data) = Pr(Data | A) Pr(A) ÷ Pr(Data)
Data has been given Pr(A' | Data) = Pr(Data | A') Pr(A') ÷ Pr(Data)
Pr(A | Data) &divide Pr(A' | Data) = (Pr(Data | A)Pr(Data | A')) ÷ (Pr(A)Pr(A'))
- the factor Pr(Data | A)Pr(Data | A') is the likelihood ratio
- the factor Pr(A)Pr(A') is the prior ratio
note that we want to get rid of Pr(Data) where possible--
if we can't get rid of it, then we have to estimate
- Pr(Data) = ∑_{models of A}Pr(Data | A)Pr(A) -- we don't want to estimate this.
suppose we have a prior distribution over some family f of how likely we expect each entry is
- we could imagine doing Bayesian inference
the edge length of a given edge is normally distributed ...
- N(80, 10) -- truncated at 0 ⇐ N(real average, real standard deviation)
let's suppose we see an alignment that corresponds to a distance of .75
- now what do we think?
- what is the p that maximizes Pr(p | Data)?
Pr(p | .75) α Pr(.75 | p) ⋅ Pr(p | prior)
= N(p, σ)[74]⋅N(.8, .1)[p]

where the world is not so good
let's say that the true model f^ does not fall in F
- that is, reality is not a member of our set of models
trivial example
- suppose that every edge of a tree has two distances (parameters) attached to it
- which correspond to fast evolution and slow evolution
- one such model will generate a distribution over all of the choices of columns
- that is, we have two magical buttons in the generative model where fast produces long edge lengths, slow produces short.
suppose f^ is such a model, but F is a Juke-Cantor model of evolution
there exists a small (5 taxa) example where MLE in F is the wrong topology

as the number of columns goes to infinity, is the MLE the true topology
- MLE = maximum likelihood estimate
- this point of contention actually became motivation for the phylogeny wars
in verifying consistency, we say that there exists a family of generative models (F)
- which contains the one generating our data
each model f in F gives a distribution over all possible columns in the character matrix
- each column in the generative model is an independent event (assumption)
the generative model, f^ should be contained in F
as m (the number of columns) grows without bounds,
- LLN (law of large numbers) requires that frequency of each kind of column converges to its frequency f^
these are profile columns
every model f in F gives a distribution of columns, we just know that f^ is one of those models
- there is no guarantee that some f does not give the same distribution as another f
consider f in F,
- Pr(Data|f) = T_i=1...m Pr(col i|f) = T_{choice in column i} Pr(choice in column i | f)^{count of column of type i}
Pr(Data|f) -> T_{choice in column i}Pr(choice of column i | f)^{Pr(choice of column i | f^) * m}
logPr(Data|f) = (1/m)Σ_CiPr(Ci|f^)logPr(Ci|f) -- maximum if f^ is the same as f
- where Ci is the choice of type of column in column i (where a type is a distribution of alphabetical tokens)
Assume f^ in F
- then as m → ∞, Pr(data | f^) is the max of all the choices of f ∈ F

the object is to prove consistency
we have taken the statistician's method by looking at how a model behaves when the quantity of data it generates becomes large
the problem is that we can't guarantee that there aren't alternative generative models that are capable of generating the same data
we can't touch that problem when using only a consistency proof