{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# s4 - Stat 5102 Lecture Slides Deck 4 Charles J Geyer School...

This preview shows pages 1–8. Sign up to view the full content.

Stat 5102 Lecture Slides Deck 4 Charles J. Geyer School of Statistics University of Minnesota 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Bayesian Inference Now for something completely different. Everything we have done up to now is frequentist statistics. Bayesian statistics is very different. Bayesians don’t do confidence intervals and hypothesis tests. Bayesians don’t use sampling distributions of estimators. Modern Bayesians aren’t even interested in point estimators. So what do they do? Bayesians treat parameters as random variables. To a Bayesian probability is only way to describe uncertainty. Things not known for certain — like values of parameters — must be described by a probability distribution. 2
Bayesian Inference (cont.) Suppose you are uncertain about something. Then your uncer- tainty is described by a probability distribution called your prior distribution . Suppose you obtain some data relevant to that thing. The data changes your uncertainty, which is then described by a new prob- ability distribution called your posterior distribution . The posterior distribution reflects the information both in the prior distribution and the data. Most of Bayesian inference is about how to go from prior to posterior. 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Bayesian Inference (cont.) The way Bayesians go from prior to posterior is to use the laws of conditional probability, sometimes called in this context Bayes rule or Bayes theorem . Suppose we have a PDF g for the prior distribution of the pa- rameter θ , and suppose we obtain data x whose conditional PDF given θ is f . Then the joint distribution of data and parameters is conditional times marginal f ( x | θ ) g ( θ ) This may look strange because, up to this point in the course, you have been brainwashed in the frequentist paradigm. Here both x and θ are random variables. 4
Bayesian Inference (cont.) The correct posterior distribution, according to the Bayesian paradigm, is the conditional distribution of θ given x , which is joint divided by marginal h ( θ | x ) = f ( x | θ ) g ( θ ) f ( x | θ ) g ( θ ) Often we do not need to do the integral. If we recognize that θ f ( x | θ ) g ( θ ) is, except for constants, the PDF of a brand name distribution, then that distribution must be the posterior. 5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Binomial Data, Beta Prior Suppose the prior distribution for p is Beta( α 1 , α 2 ) and the con- ditional distribution of x given p is Bin( n, p ). Then f ( x | p ) = n x p x (1 - p ) n - x g ( p ) = Γ( α 1 + α 2 ) Γ( α 1 )Γ( α 2 ) p α 1 - 1 (1 - p ) α 2 - 1 and f ( x | p ) g ( p ) = n x Γ( α 1 + α 2 ) Γ( α 1 )Γ( α 2 ) · p x + α 1 - 1 (1 - p ) n - x + α 2 - 1 and this, considered as a function of p for fixed x is, except for constants, the PDF of a Beta( x + α 1 , n - x + α 2 ) distribution. So that is the posterior. 6
Binomial Data, Beta Prior (cont.) A bit slower, for those for whom that was too fast. If we look up the Beta( α 1 , α 2 ) distribution in the brand name distributions handout, we see the PDF f ( x ) = Γ( α 1 + α 2 ) Γ( α 1 )Γ( α 2 ) x α 1 - 1 (1 - x ) α 2 - 1 0 < x < 1 We want g ( p ). To get that we must change f to g , which is trivial, and x to p , which requires some care. That is how we got g ( p ) = Γ( α 1 + α 2 ) Γ( α 1 )Γ( α 2 ) p α 1 - 1 (1 - p ) α 2 - 1 on the preceding slide.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 127

s4 - Stat 5102 Lecture Slides Deck 4 Charles J Geyer School...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online