6_Variational Inference

Question 1

ELBO (evidence lower bound; marginal likelihood lower bound)

Answer 1

F(v) := E_q[log( p(x,z) / q(z) )]

F(v) = E_q[log( p(x|y) )] - KL(q(z) || p(y))

Question 2

Score Function

Answer 2

score function (log derivative trick)

\nabla_v log(q(z|v)) = \nabla_v q(z|v) / q(z|v)

\Leftrightarrow \nabla_v q(z|v) = \nabla_v log(q(z|v)) * q(z|v)

Question 3

Natural Gradient

Answer 3

\tilde{\nabla}_v F(v) = F^{-1} \nabla_v F(v)

F^{-1} = (Fisher information matrix)^{-1}

Question 4

Noisy Updates of Variational Parameters

Answer 4

v_{t+1} = v_t + \rho_t \hat{\nabla}_v F(v)

Question 5

\nabla_v ELBO using the score function

Answer 5

\nabla_v F(v) = E_q[\nabla_v log(q(z|v)) * ( log(p(x,z) - log(q(z|v) )]

Use Monte Carlo to compute this

Question 6

Change of variables

Answer 6

|q(z|v) dz| = |p(\varepsilon) d\varepsilon|

Question 7

Reparametrisation trick

Answer 7

Base distribution p(\varepsilon) [normal or uniform] and a deterministic transformation z = t(\varepsilon, v) s.t. z~q(z|v). Then:

\nabla_v E_{q(z|v)}[f(z)] = E_{p(\varepsilon)}[\nabla_v f(t(\varepsilon, v))]

Note, we take the expectation w.r.t. base distribution now.

Question 8

Reparametrisation ELBO gradient

Answer 8

\nabla_v F(v) = E_{p(\varepsilon)}[\nabla_v * ( log(p(x,t(\varepsilon, v)) - log(q(t(\varepsilon, v)|v) )]

\nabla_v F(v) = E_{p(\varepsilon)}[\nabla_z * ( log(p(x,z) - log(q(z|v) ) * \nabla_v t(\varepsilon, v)]

where z = t(\varepsilon, v)

Question 9

What are the score function ELBO gradient properties

Answer 9

\+ Works for all models (continuous and discrete)
\+ Works for a large class of variational approximations
- Variance can be high, thus, slow convergence

Question 10

What are the path wise gradient estimator ELBO gradient properties

Answer 10

• Requires differentiable models
• Requires the variational approximation to be expressed as a deterministic transformation z = t(\varepsilon, v)
  + Generally lower variance

Question 11

Amortised variational inference in hierarchical Bayesian models

Answer 11

F(v) = E_q[log(p(x,\beta,z_{1:N}) - log(q(\beta, z_{1:N} | \lambda, \phi_{1:N})],
where v = {\lambda, \phi_{1:N}}

F(v) = E_q[log(p(x,\beta,z_{1:N})]
- E_q[log(q(\beta|\lambda) + \sum_n log(q(z_n| f(x_n, \theta) )],
where \phi_n = f(x_n, \theta), f is a deep neural network

Question 12

Amortised SVI (Algorithm)

Answer 12

1. Input: data x, model p(\beta, z, x)
2. Initialise global variational parameters \lambda randomly
3. Repeat:
   3.1 Sample \beta ~ q(\beta | \labmda)
   3.2 Sample data point x_n uniformly at random
   3.3 Compute stochastic natural gradients
   »> \tilde\nabla_\lambda ELBO
   »> \tilde\nabla_\theta ELBO
   3.4 Update global parameters
   »> \lambda += \rho_t \tilde\nabla_\lambda ELBO [global variational parameters]
   »> \theta += \rho_t \tilde\nabla_\theta ELBO [inference network parameters]

Question 13

BBVI (Algorithm) [black box variational inference]

Answer 13

1. Input: model p(x, z), variational approximation q(z|v)
2. Repeat:
   2.1 Draw S samples z^{(S)} ~q(z|v)
   2.2 Update variational parameters [MC estimate of the score-function gradient of the ELBO]
   »> v += \rho_t * 1/S \sum_s \nabla_v log(q(z^{(s)} | v) * (log(p(x, z^{(s)}) - log(q(z^{(s)} | v)) )
   2.3 t += 1

Question 14

SVI (Algorithm) [stochastic variational inference]

Answer 14

1. Input: data x, model p(\beta, z, x)
2. Initialise global variational parameters \lambda randomly
3. Repeat:
   3.1 Sample data point x_n uniformly at random
   3.2 Update local parameter \phi_n
   3.3 Compute intermediate global parameter \hat\lambda based on noisy natural gradient
   3.4 Set global parameter
   »> \lambda = (1-\rho_t) * \lambda + \rho_t * \hat\lambda

Question 15

Mean-Field Approximation (Algorithm) [CW3]

Answer 15

1. Input: data x, model p(\beta, z, x)
2. Initialise global variational parameters \lambda randomly
3. While ELBO has not converged, repeat:
4. 1 For each data point x_n
5. 1.1 Update local variational parameters \phi_n
6. 2 Update global variational parameters \lambda