What is approximate inference? more
To me, approximate inference finds the best approximation to an integration problem confronting computational constraints.
Imagine you have a distribution and you want to compute the integral for some function of interest. We call the computation of this integral as inference. Examples include Bayesian inference where now is some posterior distribution and is the likelihood function of on unseen data. Or if is unnormalised, taking would return the integral as the normalising constant (or partition function) of . Unfortunately for many complicated models we are fancy on now (say neural networks) this integral is intractable, and here intractability means you can't compute the exact value of the integral due to computational constraints (say running time, memory usage, precision, etc). So instead we use approximate inference to approximate that integral.
Why I should care about it? more
Now you might ask: why I should care about this approximate integral in the first place? Couldn't we just forget about integrals and do point estimations instead? Or couldn't we just use models that can do exact inference (e.g. stacking invertible transformations, auto-regressive models, sum-product networks)? Well for the first question it's more like a debate between Frequentist and Bayesian which I don't think I should say too much about it here. But I shall emphasize that integration is almost everywhere in probability and statistics, say calculating expectations and computing marginal distributions. For the second question, those models are usually more computationally demanding (both in time and memory), which is exactly the point for not using them and do approximate inference instead if you don't have that much computational resource. Approximate inference only makes sense when you do have computational constraints, and the way you do approximate inference also depends on how much precision you want and computational resource you have.
What this list is about? more
There are mainly two ways to do approximate inference: directly approximating the integral you want, or, finding an accurate approximation to the target distribution and using it for integration later. The first approach is mainly dominated by sampling methods (and maybe adding student models to distill it) and quadrature. The second one, sometimes referred as the indirect approach, will be the focus of this list.
(Posts on specific topics coming 🙂 )
Theme I: algorithms for fitting approximate distributions:
- variational inference (VI);
- message passing, belief/expectation propagation;
- black-box/Monte Carlo approximation techniques;
- reparameterisation trick and control variates;
- lower-bounds and upper-bounds;
- optimisation meets VI (proximal gradient, trust-region method, etc);
- new methods using density ratio estimation, integral probability metrics, etc.
Theme II: approximate distribution design:
- invertible transform;
- non-parametric approximations (e.g. empirical distributions in sampling);
- latent variable models as approximate distributions;
- perturbations/corrections after fitting a simple approximation;
- bridging VI and MCMC.
Theme III: applications:
- learning generative models/latent variable models;
- Bayesian neural networks;
- Gaussian processes;
- probabilistic graphical models.