3_MLE_printable.pdf

Week 3
Maximum Likelihood Estimation
Applied Statistical Analysis II
Jeffrey Ziegler, PhD
Assistant Professor in Political Science & Data Science
Trinity College Dublin
Spring 2023

Road map for today
Maximum Likelihood Estimation (MLE)
I Why do we need to think like this?
I Getting our parameters and estimates
I Computational difficulties
Next time: Binary outcomes (logit)
By next week, please...
I Problem set #1 DUE
I Read assigned chapters
1 49

Overview: Motivation for MLE
Previously we have seen “closed form” estimators for
quantities of interest, such as b = (X0X)−1X0y
Moving to nonlinear models for categorical and limited
support outcomes requires a more flexible process
Maximum Likelihood Estimation (Fisher 1922, 1925) is a
classic method that finds the value of the estimator “most
likely to have generated the observed data, assuming the
model specification is correct”
There is both an abstract idea to absorb and a mechanical
process to master
2 49

Background: GLM review
Suppose we care about some social phenomenon Y, and
determine that it has distribution f()
The stochastic component is:
Y ∼ f(µ, π)
The systematic component is:
µ = g−1
(Xβ)
This setup is very general and covers all of non-linear regression
models we will cover
3 49

Background: GLM review
You have seen linear model in this form before:
Yi = Xiβ + i
i = N(0, σ2
)
But now we are going to think of it in this more general way:
Yi ∼ N(µi, σ2
)
µi = Xiβ
We typically write this in expected value terms:
E(Y|X, β) = µ
4 49

Background: Likelihood Function
Assume that:
x1, x2, ..., xn ∼ iidf(x|β),
where θ is a parameter critical to data generation process
Since these values are independent, joint distribution of
observed data is just product of their individual PDF/PMFs:
f(x|θ) = f(x1|θ), f(x2|θ), f(xn|θ) =
n
Y
i=1
f(xi|θ)
But once we observe data, x is fixed
It is θ that is unknown, so re-write joint distribution function
according to:
f(x|θ) = L(θ|x)
Note: Only notation change, maths aren’t different
5 49

Background: Likelihood Function
Fisher (1922) justifies this because at this point we know x
A semi-Bayesian justification works as follows, we want to
perform:
p(x|θ) =
p(x)
p(θ)
p(θ|x)
but p(x) = 1 since data has already occured, and if we put a
finite uniform prior on θ over its finite allowable range
(support), then p(θ) = 1
Therefore:
p(x|θ) = 1p(θ|x) = p(θ|x)
Only caveat here is finite support of θ
6 49

Background: Poisson MLE
Start with Poisson PMF for xi:
p(X = xi) = f(xi|θ) =
e−θθxi
xi!
which requires assumptions:
I Non-concurrance of arrivals
I Number of arrivals is proportion to time of study
I This rate is constant over the time
I There is no serial correlation of arrivals
7 49

Likelihood function is created from joint distribution:
L(θ|x) =
n
Y
i=1
e−θθxi
xi!
=
e−θθx1
x1!
e−θθx2
x2!
...
e−θθxn
xn!
= e−nθ
θ
P
xi
n
Y
i=1
xi!
!−1
Suppose we have data: x = {5, 1, 1, 1, 0, 0, 3, 2, 3, 4}, then
likelihood function is:
L(θ|x) =
e−10θθ20
207360
which is probability of observing this exact sample
8 49

Remember, it’s often easier to deal logarithm of MLE:
log L(θ|x) = `(θ|x) = log e−nθ
θ
P
xi
n
Y
i=1
xi!
!−1!
= −nθ +
n
X
i=1
xi log(θ) − log
n
Y
i=1
xi!
!
For our small example this is:
`(θ|x) = −10θ + 20log(θ) − log(207360)
9 49

Importantly, for family of functions that we use (exponential
family), likelihood function and log-likelihood function have
same mode (max of function) for θ
They are both guaranteed to be concave to x-axis
10 49

Grid Search vs. Analytic solutions
Most crude approach to ML estimation is a “grid search”
I While seldom used in practice, it illustrates very well that
numerical methods can find ML estimators to any level of
precision desired
“Likelihood in this sense is not a synonym for probability,
and is a quantity which does not obey laws of probability
I Property of values of parameters, which can be determined
from observations without antecedent knowledge”
In R we can do a grid search very easily (I’ll show you soon)
11 49

Obtaining Poisson MLE
1st year calculus: where is maximum of function?
I At point when first derivative of function equals zero
So take first derivative, set it equal to zero, and solve
∂
∂θ `(θ|x) ≡ 0 is called likelihood equation
For the example:
`(θ|x) = −10θ + 20log(θ) − log(207360)
Taking derivative, and setting equal to zero:
∂
∂θ
`(θ|x) = −10 + 20θ−1
= 0
so that 20θ−1 = 10, and therefore θ̂ = 2 (note hat)
12 49

Obtaining Poisson MLE
More generally...
`(θ|x) = −nθ +
n
X
i=1
log(θ) − log
n
Y
i=1
xi!
!
∂
∂θ
`(θ|x) = −n +
1
θ
n
X
i=1
xi ≡ 0
θ̂ =
1
n
n
X
i=1
xi = x̄
It’s not true that MLE is always data mean
13 49

General Steps of MLE
1. Identify PMF or PDF
2. Create the likelihood function from joint distribution of
observed data
3. Change to log for convenience
4. Take first derivative with respect to parameter of interest
5. Set equal to zero
6. Solve for MLE
14 49

Poisson Example in R
1 # POISSON LIKELIHOOD AND LOG−LIKELIHOOD FUNCTION
2 llhfunc − function ( input_data , input_params , do . log =TRUE ) {
3 # data
4 d − rep ( input_data , length ( input_params ) )
5 # 10 x values for given parameters ( lambda for Poisson )
6 q . vec − rep ( length ( y . vals ) , length ( input_params ) )
7 print (q . vec )
8 # create vector to feed into dpois
9 p . vec − rep ( input_params , q . vec )
10 d . mat − matrix ( dpois (d , p . vec , log=do . log ) , ncol= length ( input_params ) )
11 print (d . mat)
12 i f (do . log ==TRUE ) {
13 # function along columns ( so , add )
14 apply (d . mat , 2 , sum)
15 }
16 else {
17 # or multiply
18 apply (d . mat , 2 , prod )
19 }
20 }
15 49

1 # TEST OUR FUNCTION
2 y . vals − c ( 1 , 3 , 1 , 5 , 2 , 6 , 8 , 11 , 0 , 0)
3 llhfunc ( y . vals , c (4 , 30) )
[1] 10 10
[,1] [,2]
[1,] -2.613706 -26.59880
[2,] -1.632876 -21.58817
[3,] -2.613706 -26.59880
[4,] -1.856020 -17.78150
[5,] -1.920558 -23.89075
[6,] -2.261485 -16.17207
[7,] -3.514248 -13.39502
[8,] -6.253070 -10.08914
[9,] -4.000000 -30.00000
[10,] -4.000000 -30.00000
[1] -30.66567 -216.11426
16 49

1 # USE R CORE FUNCTION FOR OPTIMIZING , par=STARTING VALUES ,
2 # control = l i s t ( fnscale = −1) INDICATES A MAZIMIZATION , bfgs=
QUASI−NEWTON ALGORITHM
3 mle − optim ( par =1 , fn= llhfunc , X=y . vals , control = l i s t (
fnscale = −1) , method=BFGS )
4 # MAKE A PRETTY GRAPH OF LOG AND NON−LOG VERSIONS
5 ruler − seq ( from =.01 , to =20 , by= . 0 1 )
6 poison . l l − llhfunc ( y . vals , ruler )
7 poison . l − llhfunc ( y . vals , ruler , do . log =FALSE )
1 plot ( ruler , poison . l , col = purple , type= l , xaxt = n )
2 text (mean( ruler ) ,mean( poison . l ) , Poisson Likelihood
Function )
3 plot ( ruler , poison . l l , col = red , type= l )
4 text (mean( ruler ) ,mean( poison . l l ) / 2 , Poisson Log− Likelihood
Function )
17 49

0e+00
2e−14
4e−14
ruler
poison.l
Poisson Likelihood Function
0 5 10 15 20
−200
−150
−100
−50
ruler
poison.ll
Poisson Log−Likelihood Function
18 49

Measuring Uncertainty of MLE
First derivative measures slope and second derivative
measures “curvature” of function at a given point
I More peaked function is at MLE, more “certain” data are
about this estimator
I Square root of negative inverse of expected value of second
derivative is SE of MLE
I In multivariate terms for vector θ, we take negative inverse of
expected Hessian
19 49

Measuring Uncertainty of MLE
Poisson example:
∂
∂θ
`(θ|x) = −n +
1
θ
n
X
i=1
xi
∂2
∂2θ
`(θ|x) =
∂
∂θ
(
∂
∂θ
`(θ|x))
= −θ−2
+
n
X
i=1
xi
20 49

Uncertainty of Multivariable MLE
Now θ is a vector of coefficients to be estimated (eg. regression)
Score function is:
∂
∂θ
`(θ|x)
which we use to get MLE θ̂
Information matrix is:
I = −Ef

∂2
`(θ|x)
∂θ∂θ0
θ̂

≡ Eb

∂`(θ|x)
∂θ
∂`(θ|x)
∂θ0
θ̂

where equivalence of these forms is called information equality
Variance-covariance of θ̂ is produced by:
E = E(I)−1
Expected value (estimate) of θ is MLE, so:
SE(θ̂) = [I(θ)]−1 θ̂2
Pn
i=1 xi
=
x̄2
nx̄
=
x̄
n
21 49

Properties of the MLE (Birnbaum 1962)
Convergence in probability: plimθ̂ = θ
Asymptotic normality:
θ̂ ∼a N(θ, I(θ)−1
) where I(θ) = −E
∂2`(θ)
∂θ∂θ0
Asymptotic efficiency: No other estimator has lower
variance, variance of MLE meets Cramer-Rao Lower Bound
22 49

MLEs Are Not Guaranteed To Exist
Let X1, X2, ..., Xn be iid from B(1, p), where p ∈ (0, 1)
If we get sample (0, 0, ..., 0), then MLE is obviously X̄ = 0
But this is not an admissible value, so MLE does not exist
23 49

Likelihood Problem #1
What if likelihood function is flat around mode?
I This is an indeterminant (and fortunately rare) occurrence
I We say that model is “non-identified” because likelihood
function cannot discriminate between alternative MLE values
I Usually this comes from a model specification that has
ambiguities
24 49

Likelihood Problem #2
What if likelihood function has more than one mode?
I Then it’s difficult to choose one, even if we had perfect
knowledge about shape of function
I This model is identified provided that there is some criteria
for picking a mode
I Usually this comes from complex model specifications, like
non-parametrics
25 49

Root Mode ﬁnding with Newton-Raphson
Newton’s method exploits properties of a Taylor series
expansion around some given point
General form (to be derived):
x(1)
= x(0)
−
f(x(0))
f0(x(0))
Taylor series expansion gives relationship between value of
a mathematical function at point, x0, and function value at
another point, x1, given (with continuous derivatives over
relevant support) as:
f(x1) = f(x0) + (x1 − x0)f0
(x0) +
1
2!
(x1 − x0)2
f00
(x0)
+
1
3!
(x1 − x0)3
f000
(x0) + ...,
where f0 is 1st derivative with respect to x, f00 is 2nd derivative
with respect to x, and so on
26 49

Note that it is required that f() have continuous derivatives
over relevant support
I Infinite precision is achieved with infinite extending of series
into higher order derivatives and higher order polynomials
I Of course, factorial component in denominator means that
these are rapidly decreasing increments
This process is both unobtainable and unnecessary, and only
first two terms are required as a step in an iterative process
Point of interest is x1 such that f(x1) = 0
I This value is a root of function, f() in that it provides a
solution to polynomial expressed by function
27 49

It is also point where function crosses x-axis in a graph of x
versus f(x)
This point could be found in one step with an infinite Taylor
series:
0 = f(x0) + (x1 − x0)f0
(x0) +
1
2!
(x1 − x0)2
f00
(x0)
+
1
∞!
(x1 − x0)∞
f∞
(x0) + ...,
While this is impossible, we can use first two terms to get
closer to desired point:
0 u f(x0) + (x1 − x0)f0
(x0)
28 49

Now rearrange to produce at (j + 1)th step:
x(j+1)
u x(j)
−
f(x(j))
f0(x(j))
so that progressively improved estimates are produced until
f(x(j+1)) is sufficiently close to zero
Method converges quadratically to a solution provided that
selected starting point is reasonably close to solution,
although results can be very bad if this condition is not met
29 49

Newton-Raphson for Statistical Problems
Newton-Raphson algorithm, when applied to mode finding
in an MLE statistical setting, substitutes β(j+1) for x(j+1) and
β(j) for x(j)
I Where β values are iterative estimates of parameter vector
and f() is score function
For a likelihood function, L(β|X), score function is first
derivative with respect to parameters of interest:
`(β|X) =
∂
∂β
`(β|X)
Setting `(β|X) equal to zero and solving gives maximum
likelihood estimate, β̂
30 49

Goal is to estimate a k-dimensional β̂ estimate, given data
and a model
Applicable multivariate likelihood updating equation is now
provided by:
β(j+1)
= β(j)
−
∂
∂β
`(βj
|X)

∂2
∂β∂β0
`(βj
|X)
−1
31 49

It is computationally convenient to solve on each iteration
by least squares
I Problem of mode finding reduces to a repeated weighted
least squares application in which inverse of diagonal values
of second derivative matrix in denominator are appropriate
weights
I This is a diagonal matrix by iid assumption
So we now need to use weighted least squares...
32 49

Intro to Weighted Least Squares
A standard technique for compensating for non-constant
error variance in LMs is to insert a diagonal matrix of
weights, Ω, into calculation of β̂ = (X0X)−1X0Y such that
heteroscedasticity is mitigated
Ω matrix is created by taking error variance of the ith case
(estimated or known), vi, and assigning inverse to ith
diagonal: Ωii = 1
v1
Idea is that large error variances are reduced by
multiplication of reciprocal
Starting with Yi = Xiβ + i, observe that there is
heteroscedasticity in error term so: i =
vi
, where shared
(minimum) variance is (i.e. non-indexed), and differences
are reflected in vi term
33 49

Let’s collect our ideas
Partial derivatives take the derivative wrt each parameter,
treating the other parameters as fixed
Gradient vector is vector of first partial derivatives wrt θ
Second partial derivatives are partial derivatives of first
partials, include all cross partial
Hessian is symmetric matrix of second partials created by
differentiating gradiant with regard to θ0
Variance-covariance matrix of our estimators is inverse of
negative of expected value of Hessian
34 49

Ex: OLS and MLE
We want to know function parameters β and σ that most
likely fit data
I We know function because we created this example, but bear
with me
1 beta − 2 . 7
2 sigma − sqrt ( 1 . 3 )
3 ex_data − data . frame ( x = runif (200 , 1 , 10) )
4 ex_data$y − 0 + beta *ex_data$x + rnorm(200 , 0 , sigma )
1 beta − 2 . 7
2 sigma − sqrt ( 1 . 3 )
3 ex_data − data . frame ( x = runif (200 , 1 , 10) )
4 ex_data$y − 0 + beta *ex_data$x + rnorm(200 , 0 , sigma )
5 pdf ( . . /graphics/normal_mle_ex . pdf , width =9)
6 plot ( ex_data$x , ex_data$y , ylab= ’Y ’ , xlab= ’ X ’ )
35 49

Ex: OLS and MLE
2 4 6 8 10
0
5
10
15
20
25
X
Y
36 49

Ex: OLS and MLE
We assume that points follow a normal probability
distribution, with mean xβ and variance σ2, y ∼ N(xβ, σ2)
Equation of this probability density function is:
1
√
2πσ2
e

−
(yi−xiβ)2
2σ2

What we want to find is the parameters β and σ that
maximize this probability for all points (xi, yi)
Use log of likelihood function:
log(L) =
n
X
i=1
−
n
2
log(2π) −
n
2
log(σ2
) −
1
2σ2
(yi − xiβ)2
37 49

Ex: OLS and MLE
We can code this as a function in R with θ = (β, σ)
1 linear . l i k − function ( theta , y , X ) {
2 n − nrow ( X )
3 k − ncol ( X )
4 beta − theta [ 1 : k ]
5 sigma2 − theta [ k +1]^2
6 e − y − X%*%beta
7 logl − −.5 *n* log (2 * pi ) −.5 *n* log ( sigma2 ) − ( ( t ( e ) %*%
e ) / (2 *sigma2 ) )
8 return ( − logl )
9 }
38 49

Ex: Normal MLE
This function, at different values of β and σ, creates a surface
1 surface − l i s t ( )
2 k − 0
3 for ( beta in seq (0 , 5 , 0 . 1 ) ) {
4 for ( sigma in seq ( 0 . 1 , 5 , 0 . 1 ) ) {
5 k − k + 1
6 logL − linear . l i k ( theta = c (0 , beta , sigma ) , y = ex_
data$y , X = cbind ( 1 , ex_data$x ) )
7 surface [ [ k ] ] − data . frame ( beta = beta , sigma = sigma ,
logL = −logL )
8 }
9 }
10 surface − do . c a l l ( rbind , surface )
11 l i b r a r y ( l a t t i c e )
1 wireframe ( logL ~ beta *sigma , surface , shade = TRUE )
39 49

Ex: Normal MLE
As you can see, there is a maximum point somewhere on this
surface
beta
sigma
logL
40 49

Ex: Normal MLE in R
We can find parameters that specify this point with R’s
built-in optimization commands
1 linear . MLE − optim ( fn= linear . lik , par=c ( 1 , 1 , 1 ) , hessian=
TRUE , y=ex_data$y , X=cbind ( 1 , ex_data$x ) , method =
BFGS )
2
3 linear . MLE$par
[1] -0.01307278 2.70435684 1.05418169
This comes reasonably close to uncovering true parameters,
β = 2.7, σ =
√
1.3
41 49

Ex: Compare MLE to lm in R
Ordinary least squares is equivalent to maximum likelihood
for a linear model, so it makes sense that lm would give us
same answers
I Note that σ2
is used in determining the standard errors
1 summary( lm ( y ~ x , ex_data ) )
Coefficients:
Estimate Std. Error t value Pr(|t|)
(Intercept) -0.01307 0.17228 -0.076 0.94
x 2.70436 0.02819 95.950 2e-16 ***
42 49

Recap
We can write normal regression model, which we usually
estimate via OLS, as a ML model
Write down log-likelihood
Take derivatives w.r.t. parameters
Set derivatives to zero
Solve for parameters
43 49

Recap
We get an estimator of the coefficient vector which is
identical to that from OLS
ML estimator of the variance, however, is different from least
squares estimator
I Reason for difference is that OLS estimator of variance is
unbiased, while the ML estimator is biased but consistent
I In large samples, as assumed by ML, difference is insignificant
44 49

Inference
We can construct a simple z-score to test null hypothesis
concerning any individual parameter, just as in OLS, but
using normal instead of t-distribution
Though we have not yet developed it, we can also construct
a likelihood ratio test for null hypothesis that all elements
of β except first are zero
I Corresponds to F-test in a least squares model, a test that
none of independent variables have an effect on y
45 49

Using ML Normal Regression
So should you now use this to estimate normal regression
models?
I No, of course not!
Because OLS is unbiased regardless of sample size
Enormous amount of software available to do OLS
Since the ML and OLS estimates are asymptotically identical,
there is no gain in switching to ML for this standard problem
46 49

Not a waste of time!
Now seen a fully worked, non-trivial, application of ML to a
model you are already familiar with
But there is a much better reason to understand the ML
model for normal regression
I Once usual assumptions no longer hold, ML is easily adapted
to new case, something that cannot be said of OLS
47 49

Next week
Specific GLMs: Binary outcomes
48 49

Class business
Read required (and suggested) online materials
Turn in Problem set # 1 before next class!
49 / 49

3_MLE_printable.pdf

Recomendados

Recomendados

Mais conteúdo relacionado

Semelhante a 3_MLE_printable.pdf

Semelhante a 3_MLE_printable.pdf (20)

Último

Último (20)

3_MLE_printable.pdf