Stat451 - Life Distribution
Upcoming SlideShare
Loading in...5
×
 

Stat451 - Life Distribution

on

  • 1,121 views

 

Statistics

Views

Total Views
1,121
Views on SlideShare
1,120
Embed Views
1

Actions

Likes
0
Downloads
6
Comments
0

1 Embed 1

http://www.slideshare.net 1

Accessibility

Upload Details

Uploaded via as Microsoft PowerPoint

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment
  • Only 1 example to demonstrate our group presentation
  • Continuous distribution
  • ore technically, although all posterior quantities are automatically defined as integrals with respect to the posterior distribution, it may be quite difficult to provide a numerical value in practice, and, in particular, an explicit form of the posterior distribution cannot always be derived.

Stat451 - Life Distribution Stat451 - Life Distribution Presentation Transcript

  • Bayesian Estimation
    Group H
    PhạmThiều Minh
    BùiLêQuýThái
    TrầnDiệpHuệMẫn
    NguyễnPhạmXuânQuỳnh
  • Content
    Background
    Bayesian estimation
    Credible interval
    Pros & Cons of Bayesian estimator
    References
  • Background
    View slide
  • Example
    View slide
  • Estimator
    Statistic used to estimate the value of an unknown parameter θ
  • Estimate
    Observed value of the estimator
  • Likelihood function
    We don’t know the parameters (for example mean μ or variance σ2)
    We have known data
     From known data, we can calculate missing parameter
  • Bayesian estimation
    What is Bayesian estimator?
    Terminology
    Squared error loss
    Absolute value loss
    Example
  • What is Bayesian estimator
    Bayesian estimator is an estimator that minimizes the expected loss (Bayes risk) of a given posterior distribution π(θ|D) over parameter θ.
  • Terminology
    Prior distributionπ(θ): initial beliefs about some unknown quantity
    Likelihood function p(x|θ): information in the data
    Given data D, the posterior densitywhere
  • Terminology - example
    Prior distribution: uniform distribution on (0,1)
    Likelihood function
    Data
  • Terminology
    The mean of discrete random variable:
    The mean of the prior distribution:
    The mean of the posterior distribution:
  • Terminology
    Bayesian estimator:
    True value: θ
    Loss function - to find a lower value that aindicate estimate is better estimate of θ
    Expected loss (Bayes risk):
  • How to minimize Bayes risk
  • Squared error loss (MSE)
    Other name is Minimum Squared Error (MSE)
    Loss function:= (true value – Bayesian estimator)2
    Bayes risk:
    Minimize the risk by taking the 1st derivation = 0
  • The Bayes estimator of a parameter θ ̂ with respect to squared loss is the mean of the posterior density
  • MSE - Example
  • MSE - Example
    Secondly, we calculate posterior density
  • Toss a coin 10 times, the number success (coin is head) is 6, then assuming a uniform (0,1) prior distribution on θ
    The posterior distribution is
  • MSE - Example
    Finally we evaluate Bayesian estimator
  • How to minimize Bayes risk
  • Absolute value loss
    Loss function:
    Bayes risk:
    Minimize the risk by taking the 1st derivation to be 0
  • The Bayes estimator of a parameter θ ̂ with respect to the absolute value loss is the median of the posterior density
  • Credible interval(Highest Density Regions )
  • What is HDR
    Highest Density Regions (HDR’s) are intervals containing a specified posterior probability. The figure below plots the 95% highest posterior density region.
    HDR
  • Pros & cons
  • Pros
    Incorporating prior knowledge into an analysis
    Loss functions allow a range of outcomes rather only 2 (the null & alternative hypothesis)
    Present data
    Past data
  • Cons
    Posterior
  • Reference
  • References
    Wikipedia (http://en.wikipedia.org/wiki/Bayes_estimator)
    FISH 497 course by Tim Esington (http://www.fish.washington.edu/classes/fish497/)
    Sheldon M. Ross – Probability and Statistics for Engineer and Scientists 3rd edition