ベイズ統計と構造⼒学

Bayesian and Structure Mechanics

フロンティア⼯学研究グループ 01LAFEE

システム情報系 助教

⼭本亨輔

yamamoto_k@kz.tsukuba.ac.jp

講義︓2018年4⽉26⽇

Bayes Theorem
likelihood
尤度|prior distribution
事前分布: Parameters: Dataposterior distribution
事後分布

 If we know the relationship between “What we canobserve ( )” and “What we’d like to know ( )”, we canestimate “What we’d like to know ( )” from “What we canobserve ( )” by Bayes Theorem もし、「 ︓観測しやすいもの」と「 ︓知りたいもの」の関係が分かっているならば、ベイズ理論により「 ︓観測しやすいもの」から「 ︓知りたいもの」を推定できる

2Example

 Estimate “Young Modulus” of a Beam from observed
natural frequency.

 Material Parameters (Young Modulus:normal distributions.

and Density: ) follow logarithm Member Parameters (Length: , Width: , Height: ) follow logarithm normaldistributions.
 Observation Noise into the Natural Frequency follows a normal distribution.

3

Simulating DataI

A
5.08E-100.001508

 In this time, data is simulated instead of experiment.
 Let’s generate Observed Natural Frequency:

by Monte Carlo Simulation

 Setting Statistical characteristics of ParametersMaterial

Member[N/m2][kg/m3][m][m][m]

Average

2.05×1011

7850

0.75010.00201

1.829

S.D.

2.00×109

50.00150.00001

0.001

MemberAverage[m][m]

5.08E-100.001508

4

Basis of Statistics

follows Logarithm Normal Distribution

’s Average:, Standard Deviation:
1

exp

2
Average andStandard Deviation
exp
exp 2

ln2

Parameters of Log. Norm. Dist.
ln 12
exp

1ln

2

[N/m2][N/m2]

Average

2.05×1011

26.046

S.D.

2.00×1090.010
5Monte Carlo Simulation

 Generate

Frequency

Sample12345678910

2.10.E+117847.731

5.087E-10

1.509.E-03

7.502.E-01

2.011.E-03

1.829414

2.07.E+117856.437

5.151E-10

1.514.E-03

7.492.E-01

2.021.E-03

1.828602

2.04.E+117852.566

5.103E-10

1.515.E-03

7.536.E-01

2.010.E-03

1.828302

2.08.E+117846.214

4.969E-10

1.498.E-03

7.509.E-01

1.995.E-03

1.828626

2.08.E+117850.578

5.101E-10

1.511.E-03

7.504.E-01

2.013.E-03

1.829197

2.06.E+117852.279

5.114E-101.513.E-03

7.510.E-01

2.014.E-03

1.830040

2.05.E+117840.501

5.08E-10

1.509.E-03

7.509.E-01

2.010.E-03

1.827074

2.08.E+117851.889

5.013E-10

1.499.E-03

7.483.E-01

2.003.E-03

1.830499

2.05.E+117846.629

5.015E-10

1.501.E-03

7.498.E-01

2.002.E-03

1.829832

2.07.E+117853.577

5.083E-10

1.508.E-03

7.497.E-01

2.011.E-03

1.827193

1.41
1.41
1.39
1.39
1.40
1.40
1.40
1.40
1.39
1.40

 Natural Frequency

6

Observing Data Simulating Observed Data including Noise
Freq.

Noise

1.41
1.41
1.39
1.39
1.40
1.40
1.40
1.40
1.39
1.40

-0.000575-0.000463-0.001111-0.001155-0.000788-0.000414
0.000034-0.002033-0.000094-0.000160

Observed
Freq.
1.408
1.407
1.390
1.391
1.404
1.396
1.395
1.394
1.386
1.402

7

Bayes Theorem

 Bayes Theorem|

 Likelihood:|

 If
 The variance:

,

12

exp

Likelihood:2|

returns the maximum value.

can be set freely

89

3.00.E+11

2.90.E+112.80.E+112.70.E+112.60.E+112.50.E+112.40.E+112.30.E+110.000
2.20.E+110.200

2.10.E+11|

2.00.E+110.400

1.90.E+110.800

1.80.E+111.000

1.70.E+111.200

1.60.E+113.00.E+112.90.E+112.80.E+112.70.E+112.60.E+112.50.E+112.40.E+112.30.E+112.20.E+112.10.E+112.00.E+111.90.E+111.80.E+111.70.E+111.60.E+11

Estimation of Parameter Step One From the Prior Distribution, we can estimate the distribution
First Observed data:=1.4082.50.E+00

2.00.E+00

0.600
1.50.E+00

1.00.E+00
5.00.E-01

0.00.E+00

Estimation of Parameter Step: 2
 From the Prior Distribution, we can estimate the distribution|

The last posterior dist.
is taken as the new prior dist

2nd Observed data:=1.4074.50.E+00

2.50.E+00

4.00.E+00
2.00.E+00

3.50.E+00
3.00.E+00

1.50.E+00

2.50.E+00
2.00.E+00

1.00.E+00

1.50.E+00

5.00.E-01

1.00.E+00

0.00.E+00

0.00.E+00

10

3.00.E+11

2.90.E+11

2.80.E+11

2.70.E+11

2.60.E+112.50.E+112.40.E+112.30.E+112.20.E+112.10.E+112.00.E+111.90.E+111.80.E+111.70.E+111.60.E+113.00.E+112.90.E+112.80.E+112.70.E+112.60.E+112.50.E+112.40.E+112.30.E+112.20.E+112.10.E+112.00.E+111.90.E+111.80.E+111.70.E+111.60.E+115.00.E-01

2.00.E+02

0.000
0.00.E+00

113.00.E+110.200

2.90.E+114.00.E+02

2.80.E+110.400

2.70.E+116.00.E+022.60.E+110.600

2.50.E+118.00.E+022.40.E+110.800

2.30.E+111.00.E+03

2.20.E+111.000

2.10.E+11 By the production of Likelihoods into which data:

2.00.E+111.20.E+03

1.90.E+111.200

1.80.E+11|

1.70.E+11⋅⋅⋅

1.60.E+11

3.00.E+11|

2.90.E+112.80.E+112.70.E+112.60.E+112.50.E+11

2.40.E+112.30.E+112.20.E+112.10.E+112.00.E+11

1.90.E+111.80.E+111.70.E+111.60.E+11

Estimation of Parameter Repeat the procedure
are substituted|

12