Deep Bayes: Variational inference

Introduction

这篇笔记比较系统地阐述了一些变分推断的方法，欢迎食用。转载请注明：
https://blog.nowcoder.net/n/c0e560ac1acb42f9a88b525da39b9189
Reference: Deep Bayes

Full Bayesian inference

Training Stage

Testing Stage

Comment: The denominator in training stage sometimes may be intractable. Posterior distributions can be calculated analytically only for simple conjugate models.

Approximate inference

Probabilistic model:

Variational Inference:
Approximate
Biased but fast and more scalable
MCMC:
samples from unnormalized
unbiased but need a lot of samples
Some mathematic magic:

The first item is ELBO, evidence lower bound
THe second item is KL divergence, Kullback-Leibler divergence
Variational Inference: ELBO interpretation
Final optimisation problem

The first item is data item, the second item is regularizer.
Mean field approximation


then we could use the following replacement to reformulate the equation:

So the above equation can become:

Algorithm
Initialize

Iterations:
\ Update each factor
Parametric optimization

Inference Summary

Statistical Inference

continuous latent variables can be regarded as a mixture of a continuum of distributions
E-step can be done in closed form only in case of contiguous distributions, otherwise the true posterior is intractable.

Typically continuous latent variables are used for dimension reduction also known as representation learning.
Example: PCA model
Consider ,such that D>>d
Joint distribution:

consists of matrix V, D-dimensional vector and scalar
EM-PCA and Mixture of PCA
joint distribution:

Variational autoencoder

EM for VAE

However, the denominator is still intractable.
Variational inference
parametric variational inference
Instead of direct infering of p(z_i | x_i,\theta) let us define flexible variational approximation

This additional Neural Network ensures tractability of the distribution while being very flexible.

Stochastic optimization

Problem 1: The training data is assumed to be large which means iterations might be expensive
Problem 2: The integral in ELBO is still intractable
Solution: Compute stochastic gradients by using mini-batching and Monto-Carlo estimation
Optimization w.r.t.

Mini-batching

However, if we use Monte-Carlo estimation:

However, when it comes to , it is another case:
Can no longer move gradient inside integral

Log-derivative trick

if we apply the trick, it yields to:

Then the expectations can be estimated using monte carlo methods.
Log-derivative trick for ELBO

Now consider its first term and apply mini-batching and log-derivative trick

We can prove that the score function: is zero mean.
REINFORCE

However, the term can be arbitrary large negative that leads to very unstable stochastic gradients
A partial solution is to use baselines
Consider a function , such that:

Remember that the so-called score function can meet the requirements.

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I am a lazy man.

Reparameterization trick

Consider differentiation of complex expectation

Express as a deterministic function g(.) of random and and perform change-of-variables rule

Then stochastic differentiation is simply

我又懒了～～

Conclusion

Good Good Study，Day Day Up