# Variational Inference and Learning ... Recap I Learningandinferenceofteninvolvesintractableintegrals

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Variational Inference and Learning

Michael Gutmann

Probabilistic Modelling and Reasoning (INFR11134) School of Informatics, University of Edinburgh

Spring semester 2018

Recap

I Learning and inference often involves intractable integrals I For example: marginalisation

p(x) = ∫

y p(x, y)dy

I For example: likelihood in case of unobserved variables

L(θ) = p(D; θ) = ∫

u p(u,D; θ)du

I We can use Monte Carlo integration and sampling to approximate the integrals.

I Alternative: variational approach to (approximate) inference and learning.

Michael Gutmann Variational Inference and Learning 2 / 36

History

Variational methods have a long history, in particular in physics. For example:

I Fermat’s principle (1650) to explain the path of light: “light travels between two given points along the path of shortest time” (see e.g. http://www.feynmanlectures.caltech.edu/I_26.html)

I Principle of least action in classical mechanics and beyond (see e.g. http://www.feynmanlectures.caltech.edu/II_19.html)

I Finite elements methods to solve problems in fluid dynamics or civil engineering.

Michael Gutmann Variational Inference and Learning 3 / 36

http://www.feynmanlectures.caltech.edu/I_26.html) http://www.feynmanlectures.caltech.edu/II_19.html

Program

1. Preparations

2. The variational principle

3. Application to inference and learning

Michael Gutmann Variational Inference and Learning 4 / 36

Program

1. Preparations Concavity of the logarithm and Jensen’s inequality Kullback-Leibler divergence and its properties

2. The variational principle

3. Application to inference and learning

Michael Gutmann Variational Inference and Learning 5 / 36

log is concave

I log(u) is concave

log(au1 +(1−a)u2) ≥ a log(u1)+(1−a) log(u2) a ∈ [0, 1]

I log(average) ≥ average (log)

I Generalisation

logE[g(x)] ≥ E[log g(x)]

with g(x) > 0

log(u)

u

log(u)

I Jensen’s inequality for concave functions.

Michael Gutmann Variational Inference and Learning 6 / 36

Kullback-Leibler divergence

I Kullback Leibler divergence KL(p||q)

KL(p||q) = ∫

p(x) log p(x)q(x)dx = Ep(x) [ log p(x)q(x)

] I Properties

I KL(p||q) = 0 if and only if (iff) p = q (they may be different on sets of probability zero)

I KL(p||q) 6= KL(q||p) I KL(p||q) ≥ 0

I Non-negativity follows from the concavity of the logarithm.

Michael Gutmann Variational Inference and Learning 7 / 36

Non-negativity of the KL divergence Non-negativity follows from the concavity of the logarithm.

Ep(x) [ log q(x)p(x)

] ≤ logEp(x)

[q(x) p(x)

] = log

∫ p(x)q(x)p(x)dx

= log ∫

q(x)dx

= log 1 = 0.

From Ep(x)

[ log q(x)p(x)

] ≤ 0

it follows that

KL(p||q) = Ep(x) [ log p(x)q(x)

] = −Ep(x)

[ log q(x)p(x)

] ≥ 0

Michael Gutmann Variational Inference and Learning 8 / 36

Asymmetry of the KL divergence

Blue: mixture of Gaussians p(x) (fixed)

Green: (unimodal) Gaussian q that minimises KL(q||p)

Red: (unimodal) Gaussian q that minimises KL(p||q)

−30 −20 −10 0 10 20 30 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Barber Figure 28.1, Section 28.3.4

Michael Gutmann Variational Inference and Learning 9 / 36

Asymmetry of the KL divergence argminq KL(q||p) = argminq

∫ q(x) log q(x)p(x) dx

I Optimal q avoids regions where p is small. I Produces good local fit, “mode seeking”

argminq KL(p||q) = argminq ∫

p(x) log p(x)q(x) dx I Optimal q is nonzero where p is nonzero

(and does not care about regions where p is small) I Corresponds to MLE; produces global fit/moment matching

−30 −20 −10 0 10 20 30 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Michael Gutmann Variational Inference and Learning 10 / 36

Asymmetry of the KL divergence

Blue: mixture of Gaussians p(x) (fixed)

Red: optimal (unimodal) Gaussians q(x)

Global moment matching (left) versus mode seeking (middle and right). (two local minima are shown)

minq KL( p || q) minq KL( q || p) minq KL( q || p)

Bishop Figure 10.3

Michael Gutmann Variational Inference and Learning 11 / 36

Program

1. Preparations Concavity of the logarithm and Jensen’s inequality Kullback-Leibler divergence and its properties

2. The variational principle

3. Application to inference and learning

Michael Gutmann Variational Inference and Learning 12 / 36

Program

1. Preparations

2. The variational principle Variational lower bound Free energy and the decomposition of the log marginal Free energy maximisation to compute the marginal and

conditional from the joint

3. Application to inference and learning

Michael Gutmann Variational Inference and Learning 13 / 36

Variational lower bound: auxiliary distribution

Consider joint pdf /pmf p(x, y) with marginal p(x) = ∫

p(x, y)dy I Like for importance sampling, we can write

p(x) = ∫

p(x, y)dy = ∫ p(x, y)

q(y) q(y)dy = Eq(y) [p(x, y)

q(y)

] where q(y) is an auxiliary distribution (called the variational distribution in the context of variational inference/learning)

I Log marginal is

log p(x) = logEq(y) [p(x, y)

q(y)

] I Instead of approximating the expectation with a sample

average, use now the concavity of the logarithm.

Michael Gutmann Variational Inference and Learning 14 / 36

Variational lower bound: concavity of the logarithm

I Concavity of the log gives

log p(x) = logEq(y) [p(x, y)

q(y)

] ≥ Eq(y)

[ log p(x, y)q(y)

] This is the variational lower bound for log p(x).

I Right-hand side is called the (variational) free energy

F(x, q) = Eq(y) [ log p(x, y)q(y)

] It depends on x through the joint p(x, y), and on the auxiliary distribution q(y) (since q is a function, the free energy is called a functional, which is a mapping that depends on a function)

Michael Gutmann Variational Inference and Learning 15 / 36

Decomposition of the log marginal

I We can re-write the free energy as

F(x, q) = Eq(y) [ log p(x, y)q(y)

] = Eq(y)

[ log p(y|x)p(x)q(y)

] = Eq(y)

[ log p(y|x)q(y) + log p(x)

] = Eq(y)

[ log p(y|x)q(y)

] + log p(x)

= −KL(q(y)||p(y|x)) + log p(x)

I Hence: log p(x) = KL(q(y)||p(y|x)) + F(x, q) I KL ≥ 0 implies the bound log p(x) ≥ F(x, q). I KL(q||p) = 0 iff q = p implies that for q(y) = p(y|x), the free

energy is maximised and equals log p(x) .

Michael Gutmann Variational Inference and Learning 16 / 36

Variational principle

I By maximising the free energy

F(x, q) = Eq(y) [ log p(x, y)q(y)

] we can split the joint p(x, y) into p(x) and p(y|x)

log p(x) = max q(y) F(x, q)

p(y|x) = argmax q(y)

F(x, q)

I You can think of free energy maximisation as a “function” that takes as input a joint p(x, y) and returns as output the (log) marginal and the conditional.

Michael Gutmann Variational Inference and Learning 17 / 36

Variational principle

I Given p(x, y), consider inference tasks 1. compute p(x) =

∫ p(x, y)dy

2. compute p(y|x) I Variational principle: we can formulate the marginal inference

problems as an optimisation problem. I Maximising the free energy

F(x, q) = Eq(y) [ log p(x, y)q(y)

] gives 1. log p(x) = maxq(y) F(x, q) 2. p(y|x) = argmaxq(y) F(x, q)

I Inference becomes optimisation. I Note: while we use q(y) to denote the variational distribution,

it depends on (fixed) x. Better (and rarer) notation is q(y|x). Michael Gutmann Variational Inference and Learning 18 / 36

Solving the optimisation problem

F(x, q) = Eq(y) [ log p(x,y)q(y)

] I Difficulties when maximising the free energy:

I optimisation with respect to pdf/pmf q(y) I computation of the expectation

I Restrict search space to family of variational distributions q(y) for which F(x, q) is computable.

I Family Q specified by I independence assumptions, e.g. q(y) =

∏ i q(yi), which

corresponds to “mean-field” variational inference I parametric assumptions, e.g. q(yi) = N (yi ;µi , σ2i )

I Optimisation is generally challenging: lots of research on how to do it (keywords: stochastic variational inference, black-box variational inference)

Michael Gutmann Variational Inference and Learning 19 / 36

Program

1. Preparations

2. The variational principle Variational lower bound Free energy and the decomposition of the log marginal Free energy maximisation to compute the marginal and

conditional from the joint

3. Application to inference and learning

Michael Gutmann Variational Inference and Learning 20 / 36

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