Probabilistic Inversion with PyMC: A Simple Forward Model Example

Sure! Here's an example of using PyMC to perform probabilistic inversion on a simple forward model:

Suppose we have a simple forward model that takes in a parameter 'x' and returns an output 'y'. We also have some measured data 'y_obs' that we want to use to infer the value of 'x'. Our forward model is given by:

def forward_model(x):
    return x**2

We can use PyMC to infer the value of 'x' by defining a prior distribution for 'x' and a likelihood function that compares the output of the forward model to the measured data. Here's how we can set up the model:

import pymc3 as pm

# Define the forward model
def forward_model(x):
    return x**2

# Define the measured data
y_obs = 4.0

# Define the prior distribution for x
with pm.Model() as model:
    x = pm.Normal('x', mu=0, sd=1)
    
    # Define the likelihood function
    y = pm.Deterministic('y', forward_model(x))
    likelihood = pm.Normal('likelihood', mu=y, sd=0.1, observed=y_obs)

In this model, we define a normal prior distribution for 'x' with mean 0 and standard deviation 1. We also define a deterministic variable 'y' that represents the output of the forward model given the value of 'x'. We then define a normal likelihood function that compares the value of 'y' to the measured data 'y_obs' with a standard deviation of 0.1.

We can then use PyMC to sample from the posterior distribution of 'x' given the measured data 'y_obs':

# Perform the MCMC sampling
with model:
    trace = pm.sample(10000)
    
# Print the posterior distribution of x
pm.plot_posterior(trace, var_names=['x'])

This will give us a posterior distribution of 'x' based on the measured data 'y_obs'. We can use this posterior distribution to make probabilistic statements about the value of 'x'.

Note that this is a very simple example, but the same basic approach can be used for more complex forward models and data sets. The key is to define a prior distribution for the parameters of interest, a likelihood function that compares the output of the forward model to the measured data, and then use PyMC to sample from the posterior distribution given the measured data.