Chapter 9

Chapter 9: Variational Inference

MCMC can be slow for very large models. Variational Inference (VI) is a much faster alternative based on optimization, not sampling.

9.1 The Basic Idea

VI finds a simpler, tractable distribution (e.g., a Normal distribution) that is the "closest" possible approximation to the true posterior. It's a trade-off: VI is incredibly fast but can be less accurate than MCMC, often underestimating the variance of the posterior.

9.2 Code Example: VI in PyMC

PyMC makes it easy to switch to VI using pm.fit().
Python

# Using the same non_conjugate_model definition...
with non_conjugate_model:
    # Instead of pm.sample(), use pm.fit() for VI
    # This returns an approximation object
    approx = pm.fit(method='advi', n=30000)

# We can then draw samples from this approximation to analyze it
trace_vi = approx.sample(1000)

# Plot the VI posterior and compare to the MCMC one
az.plot_posterior(trace_vi, color='blue', hdi_prob=0.94)
az.plot_posterior(trace_mcmc, color='red', hdi_prob=0.94) # from previous chapter
plt.suptitle("VI Approximation (Blue) vs. MCMC 'Truth' (Red)")
plt.show()
The resulting posterior plots will be very similar, but you might notice the blue VI distributions are slightly narrower than the red MCMC ones.