4. Basic use of Probeye - Crash course Bayesian system identification

This package provides a transparent and easy-to-use framework for solving parameter estimation problems (i.e., inverse problems) primarily via sampling methods in a characteristic two-step approach:

In the first step, the problem at hand is defined in a solver-independent fashion, i.e., without specifying which computational means are supposed to be utilized for finding a solution.
In the second step, the problem definition is handed over to a user-selected solver, that finds a solution to the problem. The currently supported solvers focus on Bayesian methods for posterior sampling.

Some of the features include:

Multiple prior types accepted
Measurement data bookkeeping
Various methods supported
Postprocessing functionalities

You can install it by running pip from the command line:

pip install probeye

More information can be found the Probeye documentation page

Optional 💽

🖱️Right-click and select “Save link as” to download the exercise HERE 👈 so you can work on it locally on your computer

Hint

Steps to follow:

Define ForwardModelBase child class
Define InverseProblem
Add Latent parameters
Add Fixed parameters
Add Experiment
Add Forward model
Add Likelihood model
Solve with emcee

See Probeye simple linear regresion example

Import necessary libraries¶

# Problem definition
from probeye.definition.inverse_problem import InverseProblem
from probeye.definition.forward_model import ForwardModelBase
from probeye.definition.distribution import Normal
from probeye.definition.sensor import Sensor
from probeye.definition.likelihood_model import GaussianLikelihoodModel

# Inference
from probeye.inference.emcee.solver import EmceeSolver

# Postprocessing
from probeye.postprocessing.sampling_plots import create_posterior_plot

Problem setup¶

# Fixed parameters
I = 1e9  # mm^4
L = 10_000  # mm
Q = 100  # kN

# Measurements
x_meas = 5000  # mm
d_meas = 50  # mm
sigma_meas = 10  # mm

# Prior
E_mean = 60  # GPa
E_std = 20  # GPa

Define forward model¶

## Write your code where the three dots are

def midspan_deflection(E):
    return Q * L ** 3 / (48 * E * I)

class BeamModel(ForwardModelBase):
    def interface(self):
        self.parameters = ...
        self.input_sensors = ...
        self.output_sensors = ...

    def response(self, inp: dict) -> dict:
        E = ...
        return {...}

Define inverse problem¶

## Write your code where the three dots are

# Instantiate the inverse problem
problem = ...

# Add latent parameters
problem.add_parameter(...)

# Add fixed parameters
problem.add_parameter(...)

# Add measurement data
problem.add_experiment(...)

# Add forward model
problem.add_forward_model(...)

# Add likelihood model
problem.add_likelihood_model(...)

Solve with MCMC¶

## Write your code where the three dots are

emcee_solver = EmceeSolver(...)
inference_data = emcee_solver.run(...)

Plot¶

post_plot_array = create_posterior_plot(
    inference_data,
    emcee_solver.problem,
    kind="kde",
    title="Kernel density estimate of the posterior distribution",
)

Complete Code! 📃💻

Here’s the complete code that you would run in your PC:

# Problem definition
from probeye.definition.inverse_problem import InverseProblem
from probeye.definition.forward_model import ForwardModelBase
from probeye.definition.distribution import Normal
from probeye.definition.sensor import Sensor
from probeye.definition.likelihood_model import GaussianLikelihoodModel

# Inference
from probeye.inference.emcee.solver import EmceeSolver

# Postprocessing
from probeye.postprocessing.sampling_plots import create_posterior_plot

# Fixed parameters
I = 1e9  # mm^4
L = 10_000  # mm
Q = 100  # kN

# Measurements
x_meas = 5000  # mm
d_meas = 50  # mm
sigma_meas = 10  # mm

# Prior
E_mean = 60  # GPa
E_std = 20  # GPa


#Define the forward model
def midspan_deflection(E):
    return Q * L ** 3 / (48 * E * I)

class BeamModel(ForwardModelBase):
    def interface(self):
        self.parameters = ["E"]
        self.input_sensors = Sensor("x")
        self.output_sensors = Sensor("y", std_model="sigma")

    def response(self, inp: dict) -> dict:
        E = inp["E"]
        return {"y": midspan_deflection(E)}
    

#Define the inverse problem
# Instantiate the inverse problem
problem = InverseProblem("Beam model with one sensor", print_header=False)

# Add latent parameters
problem.add_parameter(
    "E",
    tex="$E$",
    info="Elastic modulus of the beam (GPa)",
    prior=Normal(mean=E_mean, std=E_std),
)

# Add fixed parameters
problem.add_parameter(
    "sigma",
    tex="$\sigma$",
    info="Standard deviation of the measurement error (mm)",
    value=sigma_meas,
)

# Add measurement data
problem.add_experiment(
    name="TestSeries_1",
    sensor_data={"x": x_meas, "y": d_meas}
)

# Add forward model
problem.add_forward_model(BeamModel("BeamModel"), experiments="TestSeries_1")

# Add likelihood model
problem.add_likelihood_model(
    GaussianLikelihoodModel(experiment_name="TestSeries_1", model_error="additive")
)

#solve with mcmc
emcee_solver = EmceeSolver(problem, show_progress=True)
inference_data = emcee_solver.run(n_steps=2000, n_initial_steps=2000)

#plot
post_plot_array = create_posterior_plot(
    inference_data,
    emcee_solver.problem,
    kind="kde",
    title="Kernel density estimate of the posterior distribution",
)

Exercises

3. Numerical integration

Exercises

5. Multiple sensors and correlations