Prediction by Bayesian linear regression

Questions
8

I’m not sure I understand what this is supposed to mean.

Here is a model:

with pm.Model() as linear_model:
    # Intercept
    intercept = pm.Normal('Intercept', mu = 0, sd = 10)
    
    # Slope 
    slope1 = pm.Normal('slope1', mu = 0, sd = 10)
    slope2 = pm.Normal('slope2', mu = 0, sd = 10)
    slope3 = pm.Normal('slope3', mu = 0, sd = 10)
    
    # Standard deviation
    sigma = pm.HalfNormal('sigma', sd = 10)
    
    # Estimate of mean
    mean = (intercept + 
            slope1 * X_train[:,0] +
            slope2 * X_train[:,1] +
            slope3 * X_train[:,2])
    
    # Observed values
    Y_obs = pm.Normal('Y_obs',
                      mu = mean,
                      sd = sigma,
                      observed = y_train)

    # Posterior distribution
    linear_trace = pm.sample()

Now you can now use X_test make predictions about y_test. But how you do this will strongly depend on what exactly you are looking for. Presumably, you are interested in the predictions associated with the individual observations in the test set.

# which observation from the test
# set do we want to inspect?
obs_index = 0

ppc = (linear_trace['Intercept'] + 
      (linear_trace['slope1'] * X_test[obs_index,0]) + 
      (linear_trace['slope2'] * X_test[obs_index,1]) + 
      (linear_trace['slope3'] * X_test[obs_index,2]) )

The variable ppc will contain a full posterior distribution over values of y for the relevant observation (i.e., the value found in y_test[obs_index]). What you do from there depends on your needs.

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