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# Stat451 - Life Distribution

## on Dec 18, 2009

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• 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 DistributionPresentation 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