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Variational Inference

MCMC samples from the posterior and is asymptotically exact. It is also slow, and on large models it can be impractically so.

Variational inference takes a different route: pick a family of simple distributions, and find the member closest to the true posterior. Inference becomes optimisation.

The variational objective
q∗(θ)=arg⁡min⁡q∈QKL(q(θ) ∥ p(θ∣D))q^*(\theta) = \arg\min_{q \in \mathcal{Q}} \text{KL}\left(q(\theta)\,\|\,p(\theta \mid D)\right)

Take the closest tractable distribution to the true posterior, where closeness is measured by KL divergence.

The ELBO

Minimising that KL directly is impossible, since it contains the intractable posterior. The standard manipulation shows it is equivalent to maximising the evidence lower bound:

ELBO(q)=Eq[log⁡p(D,θ)]−Eq[log⁡q(θ)]\text{ELBO}(q) = E_q[\log p(D, \theta)] - E_q[\log q(\theta)]

which is computable. Maximising the ELBO minimises the KL, and now you have an ordinary optimisation problem that gradient methods can attack at scale.

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