exponential

# exponential - One Parameter Models September 22, 2010...

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One Parameter Models September 22, 2010 Reading: Hoff Chapter 3 One Parameter Models – p. 1/2 1

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Highest Posterior Density Regions Find Θ 1 - α = { θ : p ( θ | Y ) h α } such that P ( θ Θ 1 - α | Y ) = 1 - α All points in Θ 1 - α have a higher density than any point outside the regions. Often requires iterative solution: Find points such that p ( θ | Y ) > h Find probability of that set Adjust h until reach desired coverage may not have symmetric tail areas multimodal posterior may not have an interval One Parameter Models – p. 2/2 1
solve.HPD.beta Key ideas: use relative posterior by dividing posterior density by the density at the mode (range is 0 to 1) for a height h in (0,1) find points where p ( θ | Y ) = h on either side of the mode, using the uniroot function use pbeta function to caclulate area move h up or down so that area is closer to coverage 1 - α One Parameter Models – p. 3/2 1

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Exponential Example: Rate β The time between accidents modeled with an exponential distribution with a rate of β accidents per day. n = 10 observations: 1.5 15.0 60.3 30.5 2.8 56.4 27.0 6.4 110.7 25.4 f ( y | β ) = β exp( - ) y > 0 L ( β ; y 1 , . . . , y n ) = Y i β exp ( - y i β ) ( Likelihood ) = β n exp ( - X i y i β ) Look at plot of L ( β ) for y i = 336 . One Parameter Models – p. 4/2 1
Likelihood function l.exp = function(theta, y) { n = length(y) sumy = sum(y) l = thetaˆn * exp(-sumy * theta) return(l) } Vectorized: can be used to evaluate L ( θ ) at multiple values ot θ rather than using a loop beta = seq(.00001, .25, length=1000) l.exp(beta, y) One Parameter Models – p. 5/2 1

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Plot of Likelihood 0.00 0.05 0.10 0.15 0.20 0.25 0.0e+00 5.0e-21 1.0e-20 1.5e-20 2.0e-20 2.5e-20 β L ( β 29 Exponential Likelihood One Parameter Models – p. 6/2 1
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exponential - One Parameter Models September 22, 2010...

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