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Gaussian Process Structural Equation Models with Latent Variables

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Gaussian Process Structural Equation Models with Latent Variables

Ricardo SilvaDepartment of Statistical Science

University College London

Robert B. Gramacy

Statistical laboratory

University of Cambridge

- A Bayesian approach for graphical models with measurement error
- Model: nonparametric DAG + linear measurement model
- Related literature: structural equation models (SEM), error-in-variables regression

- Applications: dimensionality reduction, density estimation, causal inference
- Evaluation: social sciences/marketing data, biological domain

- Approach: Gaussian process prior + MCMC
- Bayesian pseudo-inputs model + space-filling priors

Calorie intake

Weight

Calorie intake

Reported calorie intake

Weight

Notation corner:

Observed

Latent

- Task: estimate error and f()
- Error estimation can be treated separately

- Caveat emptor: outrageously hard in theory
- If errors are Gaussian, best (!) rate of convergence is O((1/log N)2), N sample size
- Don’t panic

Calorie intake

Reported calorie intake = Calorie intake + error

Weight = f(Calorie intake) + error

Reported calorie intake

Weight

(Fan and Truong, 1993)

Calorie intake

Weight

Reported calorie intake

Reported weight

Calorie intake

Weight

Self-reported calorie intake

Weight recorded in the morning

Assisted report ofcalorie intake

Weight recorded in the evening

Widely studied as Structural Equations Models (SEMs) with latent variables

Calorie intake

Weight

Well-being

Reported calorie intake

Reported time to fall asleep

Reported weight

(Bollen, 1989)

Industrialization Level 1960

DemocratizationLevel 1965

DemocratizationLevel 1960

GNP etc.

GNP etc.

Fairness of elections etc.

GNP etc.

Fairness of elections etc.

GNP etc.

(Palomo et al., 2007)

- Some assumptions
- assume DAG structure
- assume (for simplicity only) no observed variable has children in the

- Linear functional relationships:
- Parentless vertices ~ Gaussian

Xi = i0 + XTP(i)Bi + i

Yj = j0 + XTP(j)j + j

Notation corner:

Y

X

Functional relationships:where each fi() belongs to some functional space.

Parentless latent variables follow a mixture of Gaussians, error terms are Gaussian

Xi = fi(XP(i)) + i

Yj = j0 + XTP(j)j + j

j ~ N(0, vj)

i ~ N(0, vi)

- GP Networks (Friedman and Nachman, 2000):
- Reduces to our likelihood for Yi = “Xi”

- Gaussian process latent variable model (Lawrence, 2005):
- Module networks (Segal et al., 2005):
- Shared non-linearitiese.g., Y4 = 40+41f(IL) + error, Y5 = 50+51f(IL) + error

- Dynamic models (e.g., Ko and Fox, 2009)
- Functions between different data points, symmetry

- Given observed marginal M(Y) and DAG, are M(X), {}, {v} unique?
- Relevance for causal inference and embedding
- Embedding: problematic MCMC for latent variable interpretation if unidentifiable
- Causal effect estimation: not resolved from data
- Note: barring possible MCMC problems, not essential for prediction

- Illustration:
- Yj = X1 + error, for j = 1, 2, 3; Yj = 2X2 + error, j = 4, 5, 6
- X2 = 4X12 + error

Assumed

structure

(In this model, regression coefficients are fixed for Y1 and Y4.)

Assumed

structure

(Nothing fixed, and all Y freely depend on both X1 and X2.)

- Many roads to identifiability via different sets of assumptions
- We will ignore estimation issues in this discussion!

- One generic approach boils down to a reduction to multivariate deconvolutionso that the density of X can be uniquely obtained from the (observable) density of Y and (given) density of error
- But we have to nail the measurement error identification problem first.

Y = X + error

Hazelton and Turlach (2009)

- The assumption of three or more “pure” indicators:
- Scale, location and sign of Xi is arbitrary, so fix Y1i = Xi + i1
- It follows that remaining linear coefficients inYji = 0ji + 1jiXi +ji are identifiable, and so is the variance of each error term

Xi

Y1i

Y2i

Y3i

(Bollen, 1989)

Select one pure indicator per latent variable to form set Y1 (Y11, Y12, ..., Y1L) and E1 ( 11, 12, ..., 1L)

Fromobtain the density of X, since Gaussian assumption for error terms results in density of E1 being known

Notice: since density of X is identifiable, identifiability of directionality Xi Xj vs. Xj Xi is achievable in theory

Y1 = X + E1

(Hoyer et al., 2008)

- Three “pure indicators” per variable might not be reasonable
- Alternatives:
- Two pure indicators, non-zero correlation between latent variables
- Repeated measurements (e.g., Schennach 2004)
- X* = X + error
- X** = X + error
- Y = f(X) + error

- Also related: results on detecting presence of measurement error (Janzing et al., 2009)
- For more: Econometrica, etc.

- Measurement model: standard linear regression priors
- e.g., Gaussian prior for coefficients, inverse gamma for conditional variance
- Could use the standard normal-gamma priors so that measurement model parameters are marginalized
- In the experiments, we won’t use such normal-gamma priors, though, because we want to evaluate mixing in general

- Samples using P(Y | X, f(X))p(X, f(X)) instead of P(Y | X, f(X), )p(X, f(X))p()

- Function f(XPa(i)): Gaussian process prior
- f(XPa(i)(1)), f(XPa(i) (2)), ..., f(XPa(i) (N)) ~ jointly Gaussian with particular kernel function

- Computational issues:
- Scales as O(N3), N being sample size
- Standard MCMC might converge poorly due to high conditional association between latent variables

- Hierarchical approach
- Recall: standard GP from {X(1), X(2), ..., X(N)}, obtain distribution over {f(X(1)), f(X(2)), ..., f(X(N))}
- Predictions of “future” observations f(X*(1)), f(X*(2)), ..., etc. are jointly conditionally Gaussian too
- Idea:
- imagine you see a pseudo training set X
- your “actual” training set {f(X(1)), f(X(2)), ..., f(X(N))} is conditionally Gaussian given X
- however, drop all off-diagonal elements of the conditional covariance matrix

(Snelson and Ghahramani, 2006; Banerjee et al., 2008)

Standard model

Pseudo-inputs model

- Snelson and Ghaharamani (2006): empirical Bayes estimator for pseudo-inputs
- Pseudo-inputs rapidly amounts to many more free parameters sometimes prone to overfitting

- Here: “space-filling” prior
- Let pseudo-inputs X have bounded support
- Set p(Xi) det(D), where D is some kernel matrix
- A priori, “spreads” points in some hyper-cube

- No fitting: pseudo-inputs are sampled too
- Essentially no (asymptotic) extra cost since we have to sample latent variables anyway
- Possible mixing problems?

REFERENCES HERE

- Squared exponential kernel, hyperparameterl
- exp(–|xi – xj|2 / l)

- 1-dimensional pseudo-input space, 2 pseudo-data points
- X(1), X(2)

- Fix X(1) to zero, sample X(2)
- NOT independent. It should differ from the uniform distribution at different degrees according to l

- Having a prior
- treats overfitting
- “blurs” pseudo-inputs, which theoretically leads to a bigger coverage
- if number of pseudo-inputs is “insufficient,” might provide some edge over models with fixed pseudo-inputs, but care should be exercised

- Example
- Synthetic data with quadratic relationship

Sampling 150 latent points from the predictive distribution, 2 fixed pseudo-inputs

(Average predictive log-likelihood: -4.28)

Sampling 150 latent points from the predictive distribution, 2 fixed pseudo-inputs

(Average predictive log-likelihood: -4.47)

Sampling 150 latent points from the predictive distribution, 2 free pseudo-inputs with priors

(Average predictive log-likelihood: -3.89)

With 3 free pseudo-inputs

(Average predictive log-likelihood: -3.61)

- Metropolis-Hastings, low parent dimensionality ( 3 parents in our examples)
- Mostly standard. Main points:
- It is possible to integrate away pseudo-functions.
- Sampling function values {f(Xj(1)), ... f(Xj(N))} is done in two-stages:
- Sample pseudo-functions for Xj conditioned on all but function values
- Conditional covariance of pseudo-functions (“true” functions marginalized)
- Then sample {f(Xj(1)), ... f(Xj(N))} (all conditionally independent)

- Sampling function values {f(Xj(1)), ... f(Xj(N))} is done in two-stages:

- It is possible to integrate away pseudo-functions.

(N = number of training points, M = number of pseudo-points)

- When sampling pseudo-input variable XPa(i)(d)
- Factors: pseudo-functions and “regression weights”

- Metropolis-Hastings step:
- Warning: for large number of pseudo-points, p(fi(d) | f\i(d), X) can be highly peaked
- Alternative: propose and sample fi(d)() jointly

In order to calculate the ratio iterativelyfast submatrix updates are necessary for to obtain O(NM) cost per pseudo-point, i.e., total of O(NM2)

- Evaluation of Markov chain behaviour
- “Objective” model evaluation via predictive log-likelihood
- Quick details
- Squared exponential kernel
- Prior for a (and b): mixture of Gamma (1, 20) + Gamma(20, 20)
- M = 50

- Our old friend
- Yj = X1 + error, for j = 1, 2, 3; Yj = 2X2 + error, j = 4, 5, 6
- X2 = 4X12 + error

- Visualization: comparison against GPLVM
- Nonparametric factor-analysis, independent Gaussian marginals for latent variables

GPLVM: (Lawrence, 2005)

- Example: consumer data
- Identify the factors that affect willingness to pay more to consume environmentally friendly products
- 16 indicators of environmental beliefs and attitudes, measuring 4 hidden variables
- X1: Pollution beliefs
- X2: Buying habits
- X3: Consumption habits
- X4: Willingness to spend more

- 333 datapoints.

- Latent structure
- X1 X2, X1 X3, X2 X3, X3 X4

(Bartholomew et al., 2008)

Unidentifiable

model

SparseModel 1.1

- Goal: compare predictive loglikelihood of
- Pseudo-input GPSEM, linear and quadratic polynomial models, GPLVM and subsampled full GPSEM

- Dataset 1: Consumer data
- Dataset 2: Abalone (also found in UCI)
- Postulate two latent variables, “Size” and “Weight.” Size has as indicators the length, diameter and height of each abalone specimen, while Weight has as indicators the four weight variables. 3000+ points.

- Dataset 3: Housing (also found in UCI)
- Includes indicators about features of suburbs in Boston that are relevant for the housing market. 3 latent variables, ~400 points

Pseudo-input GPSEM at least an order of magnitude faster than “full” GPSEM model (undoable in Housing). Even when subsampled to 300 points, full GPSEM still slower.

- Even Metropolis-Hastings does a somewhat decent job (for sparse models)
- Potential problems with ordinal/discrete data.

- Evaluation of high-dimensional models
- Structure learning
- Hierarchical models
- Comparisons against
- random projection approximations
- mixture of Gaussian processes with limited mixture size

- Full MATLAB code available

Thanks to Patrik Hoyer, Ed Snelson and IriniMoustaki.

S. Banerjee, A. Gelfand, A. Finley and H. Sang (2008). “Gaussian predictive process models for large spatial data sets”. JRSS B.

D. Janzing, J. Peters, J. M. Mooij and B. Schölkopf. (2009). Identifying confounders using additive noise models. UAI.

M. Hazelton and B. Turlach (2009). “Nonparametric density deconvolution by weighted kernel estimators”. Statistics and Computing.

S. Schennack (2004). “Estimation of nonlinear models with measurement error”. Econometric 72.