Week 8 : Bayesian modeling
Introduction to Bioinformatics (LF:DSIB01)


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Bayes’ theorem
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Adobe Systems
Bayes’ theorem
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Adobe Systems
Bayes’ theorem
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Adobe Systems
Bayes’ theorem
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Bayes’ theorem - example
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•You might be interested in finding out a patient’s probability of having liver disease if they are
an alcoholic.
‒Past data tells you that 10% of patients entering your clinic have liver disease. P(A) = 0.10.
‒Five percent of the clinic’s patients are alcoholics.
‒Among those patients diagnosed with liver disease, 7% are alcoholics.

Adobe Systems
Bayes’ theorem - example
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•You might be interested in finding out a patient’s probability of having liver disease if they are
an alcoholic.
‒Past data tells you that 10% of patients entering your clinic have liver disease. P(A) = 0.10.
‒Five percent of the clinic’s patients are alcoholics.
‒Among those patients diagnosed with liver disease, 7% are alcoholics.
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•P(A|B) = (0.07 * 0.1) / 0.05 = 0.14
•If the patient is an alcoholic, their chances of having liver disease is 0.14 (14%). This is a
large increase from the 10% suggested by past data.

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Bayes’ theorem - example
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•What is the probability that a woman has cancer if she has a positive mammogram result?
‒One in 1000 of women have breast cancer.
‒98 percent of women who have breast cancer test positive on mammograms.
‒1 percent of women without breast cancer have a positive mammogram.
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•

Adobe Systems
Bayes’ theorem - example
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•What is the probability that a woman has cancer if she has a positive mammogram result?
‒One in 1000 of women have breast cancer.
‒98 percent of women who have breast cancer test positive on mammograms.
‒1 percent of women without breast cancer have a positive mammogram.
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•P(A)=0.001, P(-A)=0.999, P(B|A)=0.98,P(B|-A)=0.01
•(0.98 * 0.001) / ((0.98 * 0.001) + (0.01 * 0.999)) = 0.0893
•The probability of a woman having cancer, given a positive test result, is ~9%.

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Bayes’ theorem - applications
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•Bayesian statistics
‒Data modeling
‒Parameter estimation
•Bayesian networks
‒Naïve Bayesian classifier
‒Dynamic Bayesian networks
•Hidden markov models
•Can be used in many different context
‒Neural networks
‒

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Bayesian statistics
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•Bayesian statistical methods use Bayes' theorem to compute and update probabilities after
obtaining new data.
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•

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Bayesian vs. frequentists statistics
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•Bayesian interpretation of probability where probability expresses a degree of belief in an event
•In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist
inference, a hypothesis is typically tested without being assigned a probability.
•The frequentist interpretation that views probability as the limit of the relative frequency of an
event after many trials
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Bayesian vs. frequentists statistics
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•Two main sticking points
‒Prior believe
‒Small amount of data situations
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Bayesian statistics – parameter estimation
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•Data from some probability distribution function (PDF)
•The goal is to get PDF of the parameter of this function
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PDF estimation with sampling
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•Impossible (very hard) to compute analytically
•Markov chain Monte Carlo (MCMC)
‒This allowed usage of Bayesian statistics
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PDF estimation with sampling
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•Impossible (very hard) to compute analytically
•Markov chain Monte Carlo (MCMC)
‒This allowed usage of Bayesian statistics
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•
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Bayesian statistics - example
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•Student t-test

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Bayesian statistics – Generalized linear model
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y = sig(β0 + β1x1 + β2x2 )
y = β0 + β1x1 + β2x2
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Bayesian statistics – Generalized linear model
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y Scale Type
Link Function
pdf
metric
identity
normal
dichotomous
logistic
Bernoulli
ordinal
thresholded cumulative normal
categorical
count
exponential
Poisson
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Adobe Systems
Bayesian statistics – Generalized linear model
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Response variable type
Explenatory variable type
Example test type
Categorical
Categorical
Fisher test
Categorical (two groups)
Continuous
t-test
Categorical (multiple groups)
Continuous
ANOVA
Continuous
Continuous
Linear regression
Continuous
Categorical (two groups)
Logistic regression

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Bayesian statistics – hierarchical models
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Bayesian networks
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•Diracted acyclic probabilistic graph
•Represents a set of variables and their conditional dependencies

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Bayesian networks - example
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E1:
symptom1
E2:
lab result
E4:
lab result2
C:
diagnosis
E3:
symptom2
CPT(E1|C,E6)
CPT(E2|C)
CPT(E3|C,E7)
CPT(E4|C,E5,E7)
E5:
Family genentics
E6:
Smoker
E7:
Hot weather
P(E6)
CPT(C|E5,E6)
P(E5)
P(E7)

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Dynamic Bayesian networks
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•Hidden markov models (hmm) are a (simple) special case of DBN

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Naive Bayesian Classifier
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Naive Bayesian Classifier – simple example
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E1:
symptom1
E2:
lab result
E4:
lab result2
C:
diagnosis
E3:
symptom2
CPT(E1|C)
CPT(E2|C)
CPT(E3|C)
CPT(E4|C)

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Naive Bayes classifiers - properties
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•highly scalable, requiring number of parameters linear in the number of variables
‒curse of dimensionality
•training can be done by evaluating a closed-form expression which takes linear time
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•Assumption of independence
‒Simple model
•Successfully being used
•Best if you have poor understanding of the problem.
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Naive Bayesian Classifier - example
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•Pancreatic tumor classification based on miRNA levels
•miRNA sequencing from plasma samples
‒~300 miRNAs
•Select ~20 miRNAs to classify tumor types
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•Bayesian model - 24 miRNAs -> 85% success rate
Log expression ratio
(miRNA12,miRNA9)
tumor classification
…
Log expression ratio
(miRNA25,miRNA60)
Log expression ratio
(miRNA24,miRNA51)

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60 minutes lunch break.

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