Lecture10.pdf - Stat210A Theoretical Statistics Lecture Date Sept 30 2014 Bayesian Inference Lecturer Tamara Broderick 1 Scribe Ross Boczar Being

Lecture10.pdf - Stat210A Theoretical Statistics Lecture...

This preview shows page 1 - 3 out of 4 pages.

Stat210A: Theoretical Statistics Lecture Date: Sept. 30, 2014 Bayesian Inference Lecturer: Tamara Broderick Scribe: Ross Boczar 1 Being Bayesian Bayes’ Theorem (for densities): X, Θ are random variables p ( θ | x ) = p ( x | θ ) p ( θ ) p ( x ) Joint p ( x, θ ) Marginal p ( θ ) Conditionals p ( x | θ ) , p ( θ | x ) Want to use these to do inference One school of thought: statistics ⇐⇒ “inverse probability” Probabilistic model/simulation: fixed θ random X Statistical inference: observe x knowledge of Θ Bayesian thinking: Represent knowledge of Θ with a distribution: p ( θ ) (prior); p ( θ | x ) (posterior) Example Θ: true proportion of U.S. residents that watch Game of Thrones X : # who watch out of n residents Bayesian model: p ( x | θ ) = Bin( x | n, θ ) = n x θ x (1 - θ ) n - x p ( θ ) = Beta( θ | a, b ) = Γ( a + b ) Γ( a )Γ( b ) θ a - 1 (1 - θ ) b - 1 1
Image of page 1

Subscribe to view the full document.

2 Bayesian Inference For the prior distribution, a, b are known as hyperparameters . Note that when a = b = 1, we have a uniform distribution. In general, a, b changing can represent us having more information about θ . Some useful identities: E [Θ] = a a + b Var(Θ) =
Image of page 2
Image of page 3
  • Fall '17

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes