Lecture 06
Hidden Variables and Missing Data
o Say I have a data point x i
I can imagine an imaginary binary indicator
from distribution/cluster 0 or 1 if
Then, the PDF of X is actually
I can just check what
z i that is 0 if
x i comes
x i comes from distr
Lecture 07
Application Example of Markov Chains: Representing a Genome
o Say we have a genome of length 100 megabases
o Say we divide it into blocks of length 200
100,000,000
=500,000
We start at block 1 and end at block
200
o Say there are 3 types of fe
Lecture 04
Data Likelihood
o Say my dataset is x, where
x
and
d
x
h
is the set of data from the healthy individuals
is the set of data from the diseased individuals, and
expression data from healthy individual i, and
d
xj
h
xi
is the gene
is the gene expr
Lecture 12
Review
o A model of sequence evolution is defined by its rate matrix
JC69: Non-diagonal entries are and diagonal are
P (t )=eQt P ( 0 ) where Pi ( t ) is the probability of being in state i after time t
o
o Model-based calculation of distance
Lecture 05
Likelihood Function
n
o
L ( data, p )= P ( x i , p )=L ( p )
i=1
p is the parameter
o Given the Likelihood function, I can deterministically find the best parameter p
that matches our data by choosing a value of p that maximizes the Likelihood
Lecture 13
Our Models So Far
o Assumed all sites are i.i.d. all sites evolve under identical phylogeny
Interestingly, not all parts of genome share same evolutionary history
o Captured sequence evolution as series of substitutions
There are more types o
Lecture 01
Time-variant clustering model for understanding cell fate decisions. Wei Huang et al.,
PNAS, 2014
Cell Fate Decision in Mammals
o Differentiation: The process by which more general cell-types (e.g. stem cells)
transition into more specific cell
Lecture 02
Assumptions of Modern Computations
o Suppose we have some function f ( x 1 , , x 10 )
x i has 104
If each
4
f will have ( 10 )
10
possible values
possible inputs
'
o If f was continuous, we could maximize by finding f ( x 1 , , x 10) =0 and
ver
Lecture 08
Calculate Area of Circle Using Monte Carlo
o Say we have a square with side-length 2r and we have a shape (circle) inside the
square
o Pick 2 random numbers from uniform distribution from 0 to 2r, which will
represent a random 2D point (x, y)
o
Lecture 14
Pairwise Alignment
o For one branch of the true
o True Alignment: Two letters are in the same column only if they are homologous
Homologous: Related to each other through common ancestry
Cannot be results of insertions, but substitutions are
Lecture 03
Distribution Function
o Cumulative Distribution Function:
F X ( x )=P ( X x )
This function maps a set of outcomes ( X x ) to a real number
( P ( X x)
CDF has a max value of 1
o Probability Density Function: f X ( x ) =P ( X=x )
This function m
Lecture 16
Soft Computing
o Many problems are NP-Hard, necessitating heuristics
o Many optimization problems have diminishing returns: small improvements in
optimization score come at a large cost (running time)
o Remember, all models are wrong to begin w
Lecture 15
Causality
o Correlation is not causation
o Association is not causation
We can see associations that are not causal if there are other variables that
we dont take into account
Say people who are healthy in general tend to take vitamins
We wou
Lecture 10
Frequent Universe
o Results of an experiment (e.g.
X =cfw_ x 1 , , x n ) are drawn from a larger
population of all results in repetitions of identical experiments
o We dont see those repetitions
But if we had access to them, we could draw new
Lecture 09
Statistical Models
o Mathematical construct that ideally captures mechanisms or behavior of a system
How is the data generated?
o Models are always wrong, but sometimes useful (George E. P. Box)
Occams razor, robust, tractable, interpretable,
Lecture 11
Phylogeny
o Phylogenies are trees that tell us information about evolution
Can be based on whole genome, based on sequence, languages, anything
o The lengths can be arbitrary, or they can tell us some information about time
o Given sequences f