Lesson 4.2 Likelihood function and maximum likelihood

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From the course by University of California, Santa Cruz

Bayesian Statistics: From Concept to Data Analysis

1238 ratings

University of California, Santa Cruz

Bayesian Statistics: From Concept to Data Analysis

1238 ratings

This course introduces the Bayesian approach to statistics, starting with the concept of probability and moving to the analysis of data. We will learn about the philosophy of the Bayesian approach as well as how to implement it for common types of data. We will compare the Bayesian approach to the more commonly-taught Frequentist approach, and see some of the benefits of the Bayesian approach. In particular, the Bayesian approach allows for better accounting of uncertainty, results that have more intuitive and interpretable meaning, and more explicit statements of assumptions. This course combines lecture videos, computer demonstrations, readings, exercises, and discussion boards to create an active learning experience. For computing, you have the choice of using Microsoft Excel or the open-source, freely available statistical package R, with equivalent content for both options. The lectures provide some of the basic mathematical development as well as explanations of philosophy and interpretation. Completion of this course will give you an understanding of the concepts of the Bayesian approach, understanding the key differences between Bayesian and Frequentist approaches, and the ability to do basic data analyses.

From the lesson

Statistical Inference

This module introduces concepts of statistical inference from both frequentist and Bayesian perspectives. Lesson 4 takes the frequentist view, demonstrating maximum likelihood estimation and confidence intervals for binomial data. Lesson 5 introduces the fundamentals of Bayesian inference. Beginning with a binomial likelihood and prior probabilities for simple hypotheses, you will learn how to use Bayes’ theorem to update the prior with data to obtain posterior probabilities. This framework is extended with the continuous version of Bayes theorem to estimate continuous model parameters, and calculate posterior probabilities and credible intervals.

Lesson 4.1 Confidence intervals4:57

Lesson 4.2 Likelihood function and maximum likelihood7:06

Lesson 4.3 Computing the MLE3:02

Lesson 4.4 Computing the MLE: examples4:21

Introduction to R6:57

Plotting the likelihood in R4:38

Plotting the likelihood in Excel4:48

Meet the Instructors

Herbert Lee
Professor
Applied Mathematics and Statistics

Let's do another example.

Consider a hospital where 400 patients are admitted over a month for heart attacks,

and a month later 72 of them have died and 328 of them have survived.

We can ask, what's our estimate of the mortality rate?

Under the frequentist paradigm, we must first establish our reference population.

What do we think our reference population is here?

One possibility is we could think about heart attack patients in the region.

Another is we could think about heart attack patients that are admitted to

this hospital, but over a longer period of time.

Both of these might be reasonable attempts, but in this case

our actual data are not a random sample from either of those populations.

We could sort of pretend they are and move on, or

we could also try to think harder about what a random sample situation might be.

One would be, is we could think about all people in the region who might possibly

have a heart attack and might possibly get admitted to this hospital.

It's a bit of an odd hypothetical situation, and

so there are some philosophical issues with the setup of this whole

problem with the frequentist paradigm.

In any case, let's forge forward and think about how we might do some estimation.

We can say each patient comes from a Bernoulli distribution

with unknown parameter theta.

I'm now going to use the Greek letter theta because this is an unknown

we're really trying to estimate.

So the probability that Y i = 1 = theta for

all the individuals admitted.

In this case we're going to say that a success is mortality.

The probability density function for

the entire set of data we can write in vector form.

Probability of all the Y's take some value of little y given a value of theta.

So this is the probability Y 1 takes some value little y 1,

Y 2 takes some value little y 2, 0 or 1, and so on up to Y n.

Because we're viewing each of these as independent,

then this is just the probability of each of these individual ones.

So we can write this in product notation.

And applying what we know about a Bernoulli, this then works out

to be theta to the y sub i times 1 minus theta to the 1 minus y sub i.

This is the probability of observing the actual data that we collected,

conditioned on a value of the parameter theta.

We can now think about this expression as a function of theta.

This is a concept of a likelihood.

The likelihood function is this density function

thought of as a function of theta.

So we can write this L of theta given y.

It looks like the same function, but up here this is a function of y given theta.

And now we're thinking of it as a function of theta given y.

This is not a probability distribution anymore, but

it is still a function for theta.

One way to estimate theta is that we choose the theta that gives us the largest

value of the likelihood.

It makes the data the most likely to occur for the particular data we observed.

This is referred to as the maximum likelihood estimate, or

MLE, maximum likelihood estimate.

It's the theta that maximizes the likelihood.

In practice, it's usually easier to maximize the natural logarithm of

the likelihood, referred to as the log likelihood.

At this point we usually drop the conditional notation on y as well.

Since the logarithm is a monotone function, if we maximize the logarithm of

the function, we also maximize the original function.

In this case we're dealing with data that are independent and

identically distributed.

Sometimes we write that down as Y sub i, independent and

identically distributed, iid, Bernoulli's theta.

With independent identically distributed data,

the likelihood is a product because they're independent.

And therefore, when we take the log of a product we get a sum, and

sums are much easier to work with.

So in this case, the log likelihood is the log

of the product of theta to the y sub i,

1 minus theta to the 1 minus y sub i.

The log of our product is the sum of the logs.

And when we take the log of the inside, the log of theta to the y sub i,

we'll get simplification there as well.

So that becomes y sub i times the log

of theta plus 1 minus y sub i times the log of 1 minus theta.

Now we can pull the thetas out because they don't change

with i as we take the sum.

And so this turns into the sum of the y sub i's times the log of theta,

plus the sum of 1 minus y sub i times the log of 1 minus theta.

How do we find the theta that maximizes this function?

If you recall from calculus,

we can maximize a function by taking the derivative and setting it equal to 0.

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