1. Parameter Estimation using likelihood functions Tutorial #1
This class has been cut and slightly edited from Nir Friedman’s full course of 12 lectures which is available at Changes made by Dan Geiger and Ydo Wexler. .

2. Example: Binomial Experiment
Head
Tail
When tossed, it can land in one of two positions: Head or Tail
We denote by  the (unknown) probability P(H).
Estimation task:
Given a sequence of toss samples x[1], x[2], …, x[M] we want to estimate the probabilities P(H)=  and P(T) = 1 - 

3. Statistical Parameter Fitting
Consider instances x[1], x[2], …, x[M] such that
The set of values that x can take is known
Each is sampled from the same distribution
Each sampled independently of the rest
i.i.d.
samples
The task is to find a vector of parameters  that have generated the given data. This vector parameter  can be used to predict future data.
i.I.d - idependent, identically distributed

4. The Likelihood Function
How good is a particular ? It depends on how likely it is to generate the observed data
The likelihood for the sequence H,T, T, H, H is
0.2
0.4
0.6
0.8 1 
L()

5. Sufficient Statistics
To compute the likelihood in the thumbtack example we only require NH and NT (the number of heads and the number of tails)
NH and NT are sufficient statistics for the binomial distribution
For example: H,T,T,H,H is just like T,H,H,H,T

6. Sufficient Statistics
A sufficient statistic is a function of the data that summarizes the relevant information for the likelihood
Datasets
Statistics
Formally, s(D) is a sufficient statistics if for any two datasets D and D’
s(D) = s(D’ )  LD() = LD’ ()

7. Maximum Likelihood Estimation
MLE Principle:
Choose parameters that maximize the likelihood function
This is one of the most commonly used estimators in statistics
Intuitively appealing
One usually maximizes the log-likelihood function defined as lD() = logeLD()

8. Example: MLE in Binomial Data
Applying the MLE principle we get
(Which coincides with what one would expect)
0.2
0.4
0.6
0.8 1
L()
Example:
(NH,NT ) = (3,2)
MLE estimate is 3/5 = 0.6

9. From Binomial to Multinomial
For example, suppose X can have the values 1,2,…,K (For example a die has 6 sides)
We want to learn the parameters 1, 2. …, K
Sufficient statistics:
N1, N2, …, NK - the number of times each outcome is observed
Likelihood function:
MLE:
A case of a die where 1….6 are the possible values

10. Example: Multinomial Let be a protein sequence
We want to learn the parameters q1, q2,…,q20
corresponding to the frequencies of the 20 amino acids
N1, N2, …, N20 - the number of times each amino acid is observed in the sequence
Likelihood function:
MLE:
A case of a die where 1….6 are the possible values

11. Is MLE all we need? Suppose that after 10 observations,
ML estimates P(H) = 0.7 for the thumbtack
Would you bet on heads for the next toss?
Suppose now that after 10 observations,
ML estimates P(H) = 0.7 for a coin
Would you place the same bet?
Solution: The Bayesian approach which incorporates your subjective prior knowledge. E.g., you may know a priori that some amino acids have the same frequencies and some have low frequencies. How would one use this information ?

12. Bayes’ rule
Bayes’ rule:
Where
It hold because:

13. Example: Dishonest Casino
A casino uses 2 kind of dice:
99% are fair
1% is loaded: 6 comes up 50% of the times
We pick a die at random and roll it 3 times
We get 3 consecutive sixes
What is the probability the die is loaded?

14. Dishonest Casino (cont.)
The solution is based on using Bayes rule and the fact that while P(loaded | 3sixes) is not known, the other three terms in Bayes rule are known, namely:
P(3sixes | loaded)=(0.5)3
P(loaded)=0.01
P(3sixes) =
P(3sixes | loaded) P(loaded)+P(3sixes | not loaded) (1-P(loaded))

15. Dishonest Casino (cont.)

16. Biological Example: Proteins
Extracellular proteins have a slightly different amino acid composition than intracellular proteins.
From a large enough protein database (SWISS-PROT), we can get the following:
p(ai|int) - the frequency of amino acid ai for intracellular proteins
p(ai|ext) - the frequency of amino acid ai for extracellular proteins
p(int) - the probability that any new sequence is intracellular
p(ext) - the probability that any new sequence is extracellular

17. Biological Example: Proteins (cont.)
What is the probability that a given new protein sequence x=x1x2….xn is extracellular?
Assuming that every sequence is either extracellular or intracellular (but not both)
we can write
Thus,

18. Biological Example: Proteins (cont.)
Using conditional probability we get ,
By Bayes’ theorem
The probabilities p(int), p(ext) are called the prior probabilities.
The probability P(ext|x) is called the posterior probability.

19. Bayesian Parameter Estimation (step by step)
View the unknown parameter  as a random variable.
Use probability to quantify the uncertainty about the unknown parameter.
Update of the parameter follows from the rules of probability using Bayes rule
The updated parameter is set to be the expected value of the posterior
posterior of
likelihood
prior of

20. Example: Binomial Data Revisited
Prior: say, a uniform distribution for  in [0,1]
P( ) = 1 (say due to seeing one head h = 1 and one tail t = 1 before seeing the data)
Then for data (NH,NT ) = (4,1)
MLE for is 4/5 = 0.8
Bayesian estimation gives
0.2
0.4
0.6
0.8 1
Dirichlet
integral

21. The multinomial case
The hyper-parameters 1,…,K can be thought of as “imaginary” counts from our prior experience
Imaginary sample size = 1+…+K
The larger the imaginary sample size the more confident we are in our prior. The more it influences the outcome.
Bayesian Estimation vs. MLE

22. Protein Example Continued (estimation of ext)
Let’s assume that from biological literature we know that 40% of the proteins are extracellular and 60% are intracellular
ext= p(ext)=0.4
Recall that we had
Now, given a protein sequence x we have all the information we need to calculate p(ext|x).

23. Protein Example (cont.) (updating ext)
To update ext after seeing the sequence x, we need to decide what weight should we give our prior knowledge.
If our prior knowledge is worth seeing M sequences
then the update will be as follow: