additional slides on bayesian statistics for sta 101 l.
Download
Skip this Video
Download Presentation
Additional Slides on Bayesian Statistics for STA 101

Loading in 2 Seconds...

play fullscreen
1 / 30

Additional Slides on Bayesian Statistics for STA 101 - PowerPoint PPT Presentation


  • 112 Views
  • Uploaded on

Additional Slides on Bayesian Statistics for STA 101. Prof. Jerry Reiter Fall 2008. Can we use this method to learn about means and percentages?.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Additional Slides on Bayesian Statistics for STA 101' - roxy


Download Now An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
can we use this method to learn about means and percentages
Can we use this method to learn about means and percentages?
  • To learn about population averages and percentages, we’ve used data (like the DNA test results), but not prior information (like the list of suspects).
  • We show how to combine data and prior information in class.
combining the prior beliefs and the data using bayes rule
Combining the prior beliefs and the data using Bayes Rule
  • In Bayes rule problem before break, we combine the prior beliefs and the data using Bayes rule.
  • Pr(p|X=1) represents our posterior beliefs about µ .
estimation of unknown parameters in statistical models bayesian and non bayesian
Estimation of unknown parameters in statistical models (Bayesian and non-Bayesian)
  • Suppose we posit a probability distribution to model data. How do we estimate its unknown parameters?
  • Example: assume data follow regression model. Where do the estimates of the regression coefficients come from?
  • Classical statistics: maximum likelihood estimation.
  • Bayesian statistics: Bayes rule.
estimating percentage of dukies who plan to get advanced degree
Estimating percentage of Dukies who plan to get advanced degree
  • Suppose we want to estimate the percentage of Duke students who plan to get an advanced degree (MBA, JD, MD, PhD, etc.). Call this percentage p.
  • We sample 20 people at random, and 8 of them say they plan to get an advanced degree.
  • What should be our estimate of p?
estimating the average iq of duke professors
Estimating the average IQ of Duke professors
  • Let µ be the population average IQ of Duke profs.
  • Suppose we randomly sample 25 Duke profs and record their IQs.
  • What should be our estimate of µ?
maximum likelihood estimation a principled approach to estimation
Maximum likelihood estimation: Aprincipled approach to estimation
  • Usually we can use subject-matter knowledge to specify a distribution for the data. But, we don’t know the parameters of that distribution.1) Number out of 20 who want advanced degree: binomial distribution.2) Profs’ IQs: normal distribution.
maximum likelihood estimation
Maximum likelihood estimation
  • We need to estimate the parameters of the distribution.Why do we care? A) So we can make probability statements about future events.B) The parameters themselves may be important.
maximum likelihood estimation9
Maximum likelihood estimation
  • The maximum likelihood estimate of the unknown parameter is the value for which the data were most likely to have occurred.
  • Let’s see how this works in the examples.
advanced degree example
Advanced degree example
  • Let Y be the random variable for the number of people out of 20 that plan to get an advanced degree.
  • Y has a binomial distribution with n = 20, and unknown probability p.
  • In the data, Y= 8 . If we knew p, the value of the probability distribution function at Y= 8 would be:
mle for degree example
MLE for degree example
  • Let’s graph Pr(Y = 8) as a function of the unknown p.
  • Label the function L(p). L(p) is called the likelihood function for p.
maximum likelihood
Maximum likelihood
  • The maximum likelihood estimate of p is the value of p that maximizes L(p).
  • This is a reasonable estimate because it is the value of p for which the observed data (y= 8 ) had the greatest chance of occurring.
finding the mle for degree example
Finding the MLE for degree example
  • To maximize the likelihood function, we need to take the derivative of

with respect to p, set it equal to zero, and finally solve for p. You get the sample percentage!

estimating the average iq of duke professors14
Estimating the average IQ of Duke professors
  • Let µ be the population average IQ of Duke profs.
  • Suppose we randomly sample 25 Duke profs and record their IQs.
  • What should be our estimate of µ?
model for professors iqs
Model for Professors’ IQs
  • The mathematical function for a normal curve for any prof’s IQ, which we label Y, is:
  • All normal curves have this form, with different means and SDs. Here, we’ll assume the σ = 15. We don’t know µ, which is what we’re after.
model for all 25 iqs
Model for all 25 IQs
  • We need the function for all 25 IQs.
  • Assuming each prof’s IQ is independent of other profs’ IQs, we have
model for all 25 iqs17
Model for all 25 IQs
  • With some algebra and simplifications, the likelihood function is:
likelihood function and maximum likelihood estimates
Likelihood function and maximum likelihood estimates
  • A graph of the likelihood function looks something like this:
  • The function is maximized when µ is the sample average. So, we use 132.16 as our estimate of the average Duke prof’s IQ.
  • This sample average is the MLE for µ in any normal curve.
the bayesian approach to estimation of means
The Bayesian approach to estimation of means
  • Let’s show how to combine data and prior information to address the following motivating question: What is a likely range for the average IQ of Duke professors?
combining the prior beliefs and the data using bayes rule20
Combining the prior beliefs and the data using Bayes Rule
  • We combine our prior beliefs and the data using Bayes rule.
  • f(µ|data) represents our posterior beliefs about µ .
formalizing a model for prior information
Formalizing a model for prior information
  • Let’s assign a distribution for µ that reflects our a priori beliefs about its likely range. Label this f(µ).
  • Using the data you supplied in class, the curve describing our beliefs about µ is the normal curve with mean = 128 SD = 15
mathematical equation for normal curve
Mathematical equation for normal curve
  • We can write down the equation for this normal curve.
model for the data 25 iqs
Model for the data (25 IQs)
  • If we knew µ, the model for the data (the professors’ IQs) is
estimating the average iq of duke professors24
Estimating the average IQ of Duke professors
  • Let µ be the population average IQ of Duke profs.
  • Suppose we randomly sample 25 Duke profs and record their IQs.
  • What should be our estimate of µ?
combining the prior beliefs and the data using bayes rule25
Combining the prior beliefs and the data using Bayes Rule
  • We combine the model for the prior beliefs and the model for the data using Bayes rule.
  • f(µ|data) represents our posterior beliefs about µ .
posterior distribution
Posterior distribution
  • Using calculus, one can show that f(µ|data) is a normal curve with

mean =

SD =

posterior distribution27
Posterior distribution
  • For our data and prior beliefs, the posterior beliefs, f(µ|data), is a normal curve with

mean =

SD =

using the posterior distribution to summarize beliefs about
Using the posterior distribution to summarize beliefs about µ
  • Because f(µ|data) describes beliefs about µ, we can make probability statements about µ.
  • For example, using a normal curve with mean equal to 132.06 and SD equal to 2.314, Pr(µ > 130 | data) = .813
  • A 95% posterior interval for µ stretches from127.52 to 136.59.
bayesian statistics in general
Bayesian statistics in general
  • Bayesian methods exist for any population parameter, including percentiles, maxima and minima, ratios, etc.
  • The method is general: 1) specify a mathematical curve that reflects prior beliefs about the population parameter.2) specify a mathematical curve that describes the distribution of the data, given a value of the population parameter.3) combine the curves from 1 and 2 mathematically to get posterior beliefs for the parameter, updated for the data.
differences between frequentist and bayesian
FREQUENTIST

Parameters are not random.

Confidence intervals.

BAYESIAN

Parameters are random.

Posterior distributions.

Differences between frequentist and Bayesian