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Stat451 - Life Distribution

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Stat451 - Life Distribution

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

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