Lesson 9.1 Exponential data

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From the course by University of California, Santa Cruz

Bayesian Statistics: From Concept to Data Analysis

1478 ratings

University of California, Santa Cruz

Bayesian Statistics: From Concept to Data Analysis

1478 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

Models for Continuous Data

This module covers conjugate and objective Bayesian analysis for continuous data. Lesson 9 presents the conjugate model for exponentially distributed data. Lesson 10 discusses models for normally distributed data, which play a central role in statistics. In Lesson 11, we return to prior selection and discuss ‘objective’ or ‘non-informative’ priors. Lesson 12 presents Bayesian linear regression with non-informative priors, which yield results comparable to those of classical regression.

Lesson 9.1 Exponential data4:03

Meet the Instructors

Herbert Lee
Professor
Applied Mathematics and Statistics

Let's consider the case of exponential data.

For example, suppose you're waiting for a bus that you think comes on average once

every ten minutes, but you're not sure exactly how often it comes.

We'll say Y falls in exponential distribution, with right lambda.

Your waiting time has a prior expectation of one over lambda.

It turns out, the gamma distribution is conjugate for an exponential likelihood.

Gammas actually are conjugate for a number of different things.

We need to specify a prior, so a particular gamma in this case.

If we think that the buses come on average every ten minutes,

that's a rate of one over ten.

So our prior mean should be one-tenth.

Thus, we'll want to specify a gamma distribution,

this first parameter divided by a second parameter is one-tenth.

We can think that about our variability.

Perhaps you want to specify a prior, which is a gamma, 100, 1000.

This will indeed have prior mean of one-tenth, and it'll have a prior standard

deviation, Of 1/100.

Thus if we think as a rough approximate of a mean plus or

minus two standard deviations as a rough interval for our prior, we would say we're

looking at 0.1 plus or minus 0.02 as a possible range for this rate parameter.

So, here's a particular prior we could specify.

Suppose that we wait for 12 minutes, and a bus arrives.

Now, you want to update you posterior for

lambda about how often this bus will arrive.

Posterior for lambda given y is proportional

to the likelihood times the prior.

In this case that's lambda e to the minus lambda y,

lambda to the alpha- 1, e to the minus beta lambda.

Doing a little bit of simplification, we get lambda to the alpha + 1- 1,

e to the beta plus y, lambda, negative.

I'm writing this as alpha + 1,- 1, so that it is in the form of a gamma distribution.

We can now say that lambda given y the posterior follows the gamma

distribution of parameters alpha + 1 and beta plus y.

Plugging in our particular prior thus gives

us a posterior for lambda which is a gamma

with parameters 101 and 1012.

Thus our posterior mean

is going to be 101/1012.

This is equal to 0.0998 or

approximately over 1/10.02.

This one observation doesn't contain a lot of data under this likelihood.

Our prior has a fair amount of information in it, and so

our posterior doesn't shift a whole lot.

When the bus comes and it takes 12 minutes instead of 10,

it barely shifts our posterior mean up.

It shifts it up a little bit, so we now think our expected waiting time

is slightly more than ten minutes, but not much more.

One data point doesn't have a big impact here.

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