# Tutorial 2 – Bayesian inference

In this tutorial we use Fesslix for

We investigate the two-degree of freedom problem discussed in [1] (It is a two-story frame structure that is represented as a two-degrees-of-freedom shear building model). The two stiffness coefficients of the model are considered uncertain. The selected two-dimensional prior distribution represents the normalized stiffness coefficients (stiffness coefficients divided by 29.7e6 N/m): The prior consists of two independent log-Normal distributions with modes 1.3 and 0.8 and standard deviation 1.0. The lumped story masses are 16.5e3 kg 16.1e3 kg; and are treated as deterministic. The influence of damping is neglected.

Bayesian updating is performed based on the measured first two eigen-frequencies of the system: and . The likelihood function of the problem is expressed as:

*Bayesian inference*. As method to perform the inference we employ*BUS-SuS*(BUS: Bayesian Updating with Structural relaibility methods [1]; SuS: Subset Simulation [2] ).We investigate the two-degree of freedom problem discussed in [1] (It is a two-story frame structure that is represented as a two-degrees-of-freedom shear building model). The two stiffness coefficients of the model are considered uncertain. The selected two-dimensional prior distribution represents the normalized stiffness coefficients (stiffness coefficients divided by 29.7e6 N/m): The prior consists of two independent log-Normal distributions with modes 1.3 and 0.8 and standard deviation 1.0. The lumped story masses are 16.5e3 kg 16.1e3 kg; and are treated as deterministic. The influence of damping is neglected.

Bayesian updating is performed based on the measured first two eigen-frequencies of the system: and . The likelihood function of the problem is expressed as:

where and

with as the th eigen-frequency predicted by the model.
The example problem has a bimodal posterior distribution.

The problem is solved using different numcerical formulations of the likelihood function:

The problem is solved using different numcerical formulations of the likelihood function:

Overview

- Tutorial 2a: Likelihood function formulated directly in Fesslix
- Tutorial 2b:
*Matlab/Octave*likelihood function - Tutorial 2c:
*Python*likelihood function - Tutorial 2d: Likelihood function evaluated with
*external*application

Note: Even more interfaces to Fesslix are discussed in Tutorial 1.
The interfaces discussed in Tutorial 1 but not in this tutorial can be transferred
with only minor modifications from Tutorial 1.

## References

**[1]**Straub D., Papaioannou I. (2015): Bayesian Updating with Structural Reliability Methods.*Journal of Engineering Mechanics*, Trans. ASCE, 141(3) . dx.doi.org/10.1061/(ASCE)EM.1943-7889.0000839**[2]**Au, S. K., & Beck, J. L. (2001). Estimation of small failure probabilities in high dimensions by subset simulation.*Probabilistic Engineering Mechanics*, 16(4), 263-277. doi:10.1016/S0266-8920(01)00019-4