Probabilistic Inversion with PyMC3: A Step-by-Step Guide

Sure, here's a step-by-step illustration of how to use PyMC3 for probabilistic inversion, using a linear regression example:

Example Problem: Imagine you're trying to estimate the parameters of a linear regression model with two predictor variables, X1 and X2, and one response variable, Y. The model is represented by:

Y = β0 + β1X1 + β2X2 + ε

where β0, β1, and β2 are the parameters to be estimated, and ε is the error term.

Step 1: Define Prior Distributions for the Parameters

We'll assume that the prior distributions for the parameters are normally distributed with a mean of 0 and a standard deviation of 10:

beta0 = pm.Normal('beta0', mu=0, sd=10) beta1 = pm.Normal('beta1', mu=0, sd=10) beta2 = pm.Normal('beta2', mu=0, sd=10)

Step 2: Define the Likelihood Function

We'll assume that the errors are normally distributed with a mean of 0 and a standard deviation of σ. The likelihood function is given by:

Y ~ Normal(μ, σ)

where μ is the mean of the model, calculated as:

μ = beta0 + beta1X1 + beta2X2

In PyMC3, the likelihood function is defined like this:

likelihood = pm.Normal('likelihood', mu=mu, sd=sigma, observed=y)

where 'y' is the vector of observed response variable values, and 'sigma' is the standard deviation of the errors.

Step 3: Define the Model and Run Inference

The model can be defined as follows:

with pm.Model() as model: beta0 = pm.Normal('beta0', mu=0, sd=10) beta1 = pm.Normal('beta1', mu=0, sd=10) beta2 = pm.Normal('beta2', mu=0, sd=10)

mu = beta0 + beta1*X1 + beta2*X2

likelihood = pm.Normal('likelihood', mu=mu, sd=sigma, observed=y)

Inference is then carried out using the NUTS (No-U-Turn Sampler) as shown:

with model: trace = pm.sample(1000, tune=1000)

This generates 1000 samples from the posterior distribution using NUTS, discarding the first 1000 samples as burn-in.

Step 4: Analyze the Results

Analyze the results by examining the posterior distributions of the parameters and calculating the posterior mean and credible intervals:

beta0_mean = np.mean(trace['beta0']) beta1_mean = np.mean(trace['beta1']) beta2_mean = np.mean(trace['beta2'])

beta0_ci = pm.stats.hpd(trace['beta0'], alpha=0.05) beta1_ci = pm.stats.hpd(trace['beta1'], alpha=0.05) beta2_ci = pm.stats.hpd(trace['beta2'], alpha=0.05)

print('beta0 mean:', beta0_mean) print('beta0 95% CI:', beta0_ci) print('beta1 mean:', beta1_mean) print('beta1 95% CI:', beta1_ci) print('beta2 mean:', beta2_mean) print('beta2 95% CI:', beta2_ci)

This will output the posterior mean and credible intervals for each parameter.

Important Note: This is a simplified example. Real-world applications might involve more complex models and inferences. However, the core steps of defining prior distributions, the likelihood function, the model, and running inference remain the same.