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Example for Bayesian Analysis in UNIPASS

 

Consider the limit state function

G =  cond1- cond2 + 8

 

Where cond1 and cond2 are statistically independent random variables with distributions:

Cond1: Exponential(l = rand1, xl = 0)

Cond2: Normal(m = rand2, s = rand3)

 

Where rand1, rand2, and rand3 are distribution parameters with the following prior distributions:

Rand1: Gamma(m = 0.05,  s = 0.02)  

Rand2: Normal(m = 10, s  = 1)

Rand3: Uniform(xl=2, xu=4)

 

Five values of cond1 have been observed: 15, 17, 19, 23, 25.

Six values of cond2 have been observed: 12, 14, 15, 16, 18, 19.

 

The result of Bayesian Analysis was given below:

 

***** Bayesian Analysis *****

 

For the basic random variable with observed data:cond1  

prior mean of the basic random variable               = 2.1114816E+01

prior standard deviation of the basic random variable = 2.1114816E+01

 

Correction Factor for random variable cond1   = 6.0962821E+08

 

For the basic random variable with observed data:cond2  

prior mean of the basic random variable               = 1.0000000E+01

prior standard deviation of the basic random variable = 3.0000000E+00

 

Correction Factor for random variable cond2   = 7.2284221E+08

 

 

 

======= For the hyper parameter:rand1    =======

prior mean of the hyper parameter                   = 5.0000000E-02

prior standard deviation of the hyper parameter     = 2.0000000E-02

 

POSTERIOR DISTRIBUTION OF HYPER PARAMETER:

x= 2.0000000E-06 pdf= 1.6190875E-39

x= 3.8795102E-03 pdf= 3.3998853E-06

x= 7.7570204E-03 pdf= 1.7324564E-03

x= 1.1634531E-02 pdf= 4.6345047E-02

x= 1.5512041E-02 pdf= 3.7087047E-01

x= 1.9389551E-02 pdf= 1.5318749E+00

x= 2.3267061E-02 pdf= 4.1643069E+00

x= 2.7144571E-02 pdf= 8.4816598E+00

x= 3.1022082E-02 pdf= 1.3984453E+01

x= 3.4899592E-02 pdf= 1.9620730E+01

x= 3.8777102E-02 pdf= 2.4237740E+01

x= 4.2654612E-02 pdf= 2.7010614E+01

x= 4.6532122E-02 pdf= 2.7646363E+01

x= 5.0409633E-02 pdf= 2.6346509E+01

x= 5.4287143E-02 pdf= 2.3626079E+01

x= 5.8164653E-02 pdf= 2.0104439E+01

x= 6.2042163E-02 pdf= 1.6344250E+01

x= 6.5919673E-02 pdf= 1.2764869E+01

x= 6.9797184E-02 pdf= 9.6214115E+00

x= 7.3674694E-02 pdf= 7.0258925E+00

x= 7.7552204E-02 pdf= 4.9867397E+00

x= 8.1429714E-02 pdf= 3.4497668E+00

x= 8.5307224E-02 pdf= 2.3316194E+00

x= 8.9184735E-02 pdf= 1.5428339E+00

x= 9.3062245E-02 pdf= 1.0012859E+00

x= 9.6939755E-02 pdf= 6.3835509E-01

x= 1.0081727E-01 pdf= 4.0034897E-01

x= 1.0469478E-01 pdf= 2.4730073E-01

x= 1.0857229E-01 pdf= 1.5062664E-01

x= 1.1244980E-01 pdf= 9.0551744E-02

x= 1.1632731E-01 pdf= 5.3776899E-02

x= 1.2020482E-01 pdf= 3.1575190E-02

x= 1.2408233E-01 pdf= 1.8342637E-02

x= 1.2795984E-01 pdf= 1.0549415E-02

x= 1.3183735E-01 pdf= 6.0104302E-03

x= 1.3571486E-01 pdf= 3.3941449E-03

x= 1.3959237E-01 pdf= 1.9007278E-03

x= 1.4346988E-01 pdf= 1.0560231E-03

x= 1.4734739E-01 pdf= 5.8233673E-04

x= 1.5122490E-01 pdf= 3.1885350E-04

x= 1.5510241E-01 pdf= 1.7341258E-04

x= 1.5897992E-01 pdf= 9.3710230E-05

x= 1.6285743E-01 pdf= 5.0332043E-05

x= 1.6673494E-01 pdf= 2.6876822E-05

x= 1.7061245E-01 pdf= 1.4272602E-05

x= 1.7448996E-01 pdf= 7.5392567E-06

x= 1.7836747E-01 pdf= 3.9623709E-06

x= 1.8224498E-01 pdf= 2.0724188E-06

x= 1.8612249E-01 pdf= 1.0789078E-06

x= 1.9000000E-01 pdf= 5.5918909E-07

 

posterior mean of hyper parameter                 = 5.0223223E-02

posterior standard deviation of hyper parameter   = 1.4973643E-02

 

 

======= For the basic random variable:cond2    =======

prior mean of the basic random variable              = 1.0000000E+01

prior standard deviation of the basic random variable= 3.0000000E+00

 

POSTERIOR DISTRIBUTION OF THE BASIC RANDOM VARIABLE:

x=-1.1000000E+01 pdf= 3.5127121E-09

x=-1.0142857E+01 pdf= 1.1630726E-08

x=-9.2857143E+00 pdf= 3.6918178E-08

x=-8.4285714E+00 pdf= 1.1234966E-07

x=-7.5714286E+00 pdf= 3.2781591E-07

x=-6.7142857E+00 pdf= 9.1715559E-07

x=-5.8571429E+00 pdf= 2.4605787E-06

x=-5.0000000E+00 pdf= 6.3304830E-06

x=-4.1428571E+00 pdf= 1.5619326E-05

x=-3.2857143E+00 pdf= 3.6959728E-05

x=-2.4285714E+00 pdf= 8.3877236E-05

x=-1.5714286E+00 pdf= 1.8256019E-04

x=-7.1428571E-01 pdf= 3.8106486E-04

x= 1.4285714E-01 pdf= 7.6276167E-04

x= 1.0000000E+00 pdf= 1.4639249E-03

x= 1.8571429E+00 pdf= 2.6934024E-03

x= 2.7142857E+00 pdf= 4.7490831E-03

x= 3.5714286E+00 pdf= 8.0217425E-03

x= 4.4285714E+00 pdf= 1.2973007E-02

x= 5.2857143E+00 pdf= 2.0072970E-02

x= 6.1428571E+00 pdf= 2.9687530E-02

x= 7.0000000E+00 pdf= 4.1918700E-02

x= 7.8571429E+00 pdf= 5.6423618E-02

x= 8.7142857E+00 pdf= 7.2266113E-02

x= 9.5714286E+00 pdf= 8.7878201E-02

x= 1.0428571E+01 pdf= 1.0121068E-01

x= 1.1285714E+01 pdf= 1.1011127E-01

x= 1.2142857E+01 pdf= 1.1287550E-01

x= 1.3000000E+01 pdf= 1.0879579E-01

x= 1.3857143E+01 pdf= 9.8463870E-02

x= 1.4714286E+01 pdf= 8.3645780E-02

x= 1.5571429E+01 pdf= 6.6750805E-02

x= 1.6428571E+01 pdf= 5.0132231E-02

x= 1.7285714E+01 pdf= 3.5528265E-02

x= 1.8142857E+01 pdf= 2.3831729E-02

x= 1.9000000E+01 pdf= 1.5177363E-02

x= 1.9857143E+01 pdf= 9.2023707E-03

x= 2.0714286E+01 pdf= 5.3242719E-03

x= 2.1571429E+01 pdf= 2.9447245E-03

x= 2.2428571E+01 pdf= 1.5588847E-03

x= 2.3285714E+01 pdf= 7.9061031E-04

x= 2.4142857E+01 pdf= 3.8437787E-04

x= 2.5000000E+01 pdf= 1.7921641E-04

x= 2.5857143E+01 pdf= 8.0154979E-05

x= 2.6714286E+01 pdf= 3.4393712E-05

x= 2.7571429E+01 pdf= 1.4159643E-05

x= 2.8428571E+01 pdf= 5.5931375E-06

x= 2.9285714E+01 pdf= 2.1197148E-06

x= 3.0142857E+01 pdf= 7.7071836E-07

x= 3.1000000E+01 pdf= 2.6883172E-07

 

posterior mean of the basic random variable       = 1.1917890E+01

posterior stand. dev. of the basic random variable= 3.5993886E+00

 Finish Bayesian Analysis

 

 

It is noted that for the exponential distribution of cond1, the conjugate prior is gamma with parameters  and . Using known relations between these parameters and the mean and standard deviation, we obtain  = 125 and  = 6.25. The posterior is also gamma with the updated parameters:

 

The posterior mean and standard deviation of rand1 are

 

These two values verified the correct result from UNIPASS shown in the above table.

Last Updated 02/08/10

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