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Expectation Propagation for approximate Bayesian inference

Thomas P. Minka

article2001UAI2,004 citations

Introduces Expectation Propagation, a deterministic approximate inference framework that unifies assumed-density filtering and belief propagation to deliver higher accuracy than variational Bayes or Laplace approximations in hybrid networks at comparable computational cost.

bayesian inferenceclassifiersdeterministicPropagationExpectation PropagationApproximate
arXiv: 1301.2294
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Abstract

This paper presents a new deterministic approximation technique in Bayesian networks. This method, "Expectation Propagation", unifies two previous techniques: assumed-density filtering, an extension of the Kalman filter, and loopy belief propagation, an extension of belief propagation in Bayesian networks. All three algorithms try to recover an approximate distribution which is close in KL divergence to the true distribution. Loopy belief propagation, because it propagates exact belief states, is useful for a limited class of belief networks, such as those which are purely discrete. Expectation Propagation approximates the belief states by only retaining certain expectations, such as mean and variance, and iterates until these expectations are consistent throughout the network. This makes it applicable to hybrid networks with discrete and continuous nodes. Expectation Propagation also extends belief propagation in the opposite direction - it can propagate richer belief states that incorporate correlations between nodes. Experiments with Gaussian mixture models show Expectation Propagation to be convincingly better than methods with similar computational cost: Laplace's method, variational Bayes, and Monte Carlo. Expectation Propagation also provides an efficient algorithm for training Bayes point machine classifiers.

Topics

bayesian inferenceclassifiersdeterministicefficient algorithmexpectation propagationgaussian mixture modelgaussian mixturesgaussiansinferencekalman filters

Keywords

PropagationExpectation PropagationApproximateBayesianInferenceBeliefPointBayesian InferenceApproximate Bayesian InferenceBayes Point

Table of Contents

  • 1 INTRODUCTION
  • 2 ASSUMED-DENSITY FILTERING
  • 3 EXPECTATION PROPAGATION
  • 3.1 THE CLUTTER PROBLEM
  • 3.2 RESULTS
  • 3.3 CONVERGENCE
  • 4 LOOPY BELIEF PROPAGATION
  • 5 BAYES POINT MACHINE
  • 5.1 RESULTS
  • 6 SUMMARY
  • Acknowledgment
  • References

Knowls

  1. Knowl 1 — The Expectation Propagation Algorithm

    algorithm

    Expectation Propagation (EP) is a deterministic iterative approximation framework for Bayesian inference. Let p(D,x)=ildet0(x)imesildet1(x)imesildet2(x)imesildet3(x)imesildet4(x)imesildet5(x)p(D, x) = ilde{t}_0(x) imes ilde{t}_1(x) imes ilde{t}_2(x) imes ilde{t}_3(x) imes ilde{t}_4(x) imes ilde{t}_5(x) be a target joint distribution where t0(x)=p(x)t_0(x) = p(x) is a prior distribution, each ti(x)=p(yi∣x)t_i(x) = p(y_i|x) is an observation likelihood term, D={y1,…,yn}D = \{y_1, \dots, y_n\} is the observed data, and xx is the hidden latent variable vector. EP approximates each factor ti(x)t_i(x) by an unnormalized exponential family term t~i(x)\tilde{t}_i(x) such that the global approximate posterior q(x)q(x) belongs to the same exponential family.

    Input: Prior term t0(x)t_0(x), likelihood terms ti(x)t_i(x) for i=1,…,ni = 1, \dots, n, approximating exponential family F\mathcal{F}
    Output: Approximate posterior distribution q(x)q(x), approximate model evidence p(D)p(D)
    Initialize t~0(x)=t0(x)\tilde{t}_0(x) = t_0(x) (if t0(x)∈Ft_0(x) \in \mathcal{F}, otherwise initialize t~0\tilde{t}_0)
    Initialize t~i(x)=1\tilde{t}_i(x) = 1 for i=1,…,ni = 1, \dots, n
    Compute initial posterior: q(x)=∏i=0nt~i(x)∫∏i=0nt~i(x)dxq(x) = \frac{\prod_{i=0}^n \tilde{t}_i(x)}{\int \prod_{i=0}^n \tilde{t}_i(x) dx}
    repeat
        for i=1,…,ni = 1, \dots, n do
            Compute the cavity (leave-one-out) distribution:
                q\i(x)∝q(x)t~i(x)q^{\backslash i}(x) \propto \frac{q(x)}{\tilde{t}_i(x)}
            Compute the normalizer ZiZ_i and the tilted distribution p^(x)\hat{p}(x):
                Zi=∫ti(x)q\i(x)dxZ_i = \int t_i(x) q^{\backslash i}(x) dx
                p^(x)=1Ziti(x)q\i(x)\hat{p}(x) = \frac{1}{Z_i} t_i(x) q^{\backslash i}(x)
            Update q(x)q(x) by projecting p^(x)\hat{p}(x) onto F\mathcal{F} via KL minimization (moment matching):
                q(x)=arg⁡min⁡q∈FD(p^(x)∥q(x))q(x) = \arg\min_{q \in \mathcal{F}} D(\hat{p}(x) \parallel q(x))
            Update the term approximation t~i(x)\tilde{t}_i(x):
                t~i(x)=Ziq(x)q\i(x)\tilde{t}_i(x) = Z_i \frac{q(x)}{q^{\backslash i}(x)}
        end for
    until all t~i(x)\tilde{t}_i(x) converge
    Compute the model evidence approximation: p(D)≈∫∏i=0nt~i(x)dxp(D) \approx \int \prod_{i=0}^n \tilde{t}_i(x) dx
    return q(x)q(x), p(D)p(D)

    Minimizing D(p^(x)∥q(x))D(\hat{p}(x) \parallel q(x)) when qq is in an exponential family is equivalent to setting the expected sufficient statistics under q(x)q(x) equal to those under p^(x)\hat{p}(x): Eq[fj(x)]=Ep^[fj(x)]\mathbb{E}_q[f_j(x)] = \mathbb{E}_{\hat{p}}[f_j(x)]. Each step yields an update to both the parameter estimates and the marginal evidence.

  2. Knowl 2 — Energy Objective and Fixed Points of Expectation Propagation

    theoretical result

    When the posterior approximations belong to an exponential family with sufficient statistics f1(x),…,fJ(x)f_1(x), \dots, f_J(x) such that q(x)∝p(x)exp⁡(∑j=1Jfj(x)νj)q(x) \propto p(x) \exp\left(\sum_{j=1}^J f_j(x) \nu_j\right) and the leave-one-out cavity distributions satisfy q\i(x)∝p(x)exp⁡(∑j=1Jfj(x)λij)q^{\backslash i}(x) \propto p(x) \exp\left(\sum_{j=1}^J f_j(x) \lambda_{ij}\right), the fixed points of Expectation Propagation correspond to the stationary points of the min-max objective function:

    min⁡νmax⁡λ[(n−1)log⁡∫p(x)exp⁡(∑j=1Jfj(x)νj)dx−∑i=1nlog⁡∫ti(x)p(x)exp⁡(∑j=1Jfj(x)λij)dx]\min_{\nu} \max_{\lambda} \left[ (n-1) \log \int p(x) \exp\left( \sum_{j=1}^J f_j(x) \nu_j \right) dx - \sum_{i=1}^n \log \int t_i(x) p(x) \exp\left( \sum_{j=1}^J f_j(x) \lambda_{ij} \right) dx \right]

    subject to the linear consistency constraint:

    (n−1)νj=∑i=1nλijfor each j∈{1,…,J}.(n - 1) \nu_j = \sum_{i=1}^n \lambda_{ij} \quad \text{for each } j \in \{1, \dots, J\}.

    Differentiating with respect to the parameters yields the moment matching stationarity conditions ∫fj(x)q(x)dx=∫fj(x)p^(x)dx\int f_j(x) q(x) dx = \int f_j(x) \hat{p}(x) dx.

    If every likelihood term ti(x)t_i(x) is bounded above by a constant cc (ti(x)≤ct_i(x) \le c), then by the concavity of y(n−1)/ny^{(n-1)/n}, setting λij=n−1nνj\lambda_{ij} = \frac{n-1}{n} \nu_j bounds the objective function from below by −nlog⁡c-n \log c. Consequently, stationary points are guaranteed to exist, ensuring that the Expectation Propagation iterations always have at least one fixed point.

  3. Knowl 3 — Equivalence of Expectation Propagation to Loopy Belief Propagation on Factorized Networks

    theoretical result

    When the approximating distribution for hidden variables x=(x1,…,xK)x = (x_1, \dots, x_K) is constrained to be completely disconnected (fully factorized):

    q(x)=∏k=1Kqk(xk)q(x) = \prod_{k=1}^K q_k(x_k)

    minimizing the Kullback-Leibler divergence D(p^(x)∥q(x))D(\hat{p}(x) \parallel q(x)) preserves the individual marginals: qk(xk)=∑x∖xkp^(x)q_k(x_k) = \sum_{x \setminus x_k} \hat{p}(x).

    Under this approximation structure:

    1. The term approximations factorize as t~i(x)=∏kt~ik(xk)\tilde{t}_i(x) = \prod_k \tilde{t}_{ik}(x_k).
    2. The cavity distribution qk\i(xk)∝∏j≠it~jk(xk)q_k^{\backslash i}(x_k) \propto \prod_{j \neq i} \tilde{t}_{jk}(x_k) represents the partial belief state at node xkx_k excluding messages from term ii.
    3. The updated term component t~ik(xk)=Ziqk(xk)qk\i(xk)\tilde{t}_{ik}(x_k) = Z_i \frac{q_k(x_k)}{q_k^{\backslash i}(x_k)} corresponds directly to the message sent from factor/node ii to node xkx_k in belief propagation.
    4. When likelihood terms ti(x)t_i(x) correspond to conditional probability tables or clique potentials, the EP updates reproduce the message-passing recursions of loopy belief propagation (and factor graphs).
    5. For undirected networks with pairwise potentials, the EP energy objective corresponds to the dual representation of the Bethe free energy.

  4. Knowl 4 — Expectation Propagation for the Bayes Point Machine

    algorithm

    The Bayes Point Machine (BPM) solves linear binary classification yi=sign(wTxi)y_i = \text{sign}(w^T x_i) for training data D={(x1,y1),…,(xn,yn)}D = \{(x_1, y_1), \dots, (x_n, y_n)\} using the likelihood p(yi∣xi,w)=Φ(yiwTxiϵ)p(y_i|x_i, w) = \Phi\left(\frac{y_i w^T x_i}{\epsilon}\right) with standard normal cumulative distribution Φ(z)=∫−∞zN(t;0,1)dt\Phi(z) = \int_{-\infty}^z \mathcal{N}(t; 0, 1) dt in the zero-slack limit ϵ→0\epsilon \to 0. The prior is w∼N(0,Id)w \sim \mathcal{N}(0, I_d). EP approximates the posterior by a multivariate Gaussian q(w)=N(mw,Vw)q(w) = \mathcal{N}(m_w, V_w) using Gaussian term approximations t~i(w)=siexp⁡(−12vi(wTxi−mi)2)\tilde{t}_i(w) = s_i \exp\left( -\frac{1}{2 v_i} (w^T x_i - m_i)^2 \right), where xix_i is implicitly scaled by yi/ϵy_i / \epsilon.

    Input: Data points and labels combined as xi←yixi/ϵx_i \leftarrow y_i x_i / \epsilon with ϵ→0\epsilon \to 0, dimension dd
    Output: Posterior mean mwm_w (Bayes point), covariance VwV_w, model evidence p(D)p(D)
    Initialize prior: mw=0m_w = 0, Vw=IdV_w = I_d
    Initialize term approximations: vi=∞v_i = \infty, mi=0m_i = 0, si=1s_i = 1 for i=1,…,ni = 1, \dots, n
    repeat
        for i=1,…,ni = 1, \dots, n do
            Compute cavity parameters:
                Vw\i=Vw+(Vwxi)(Vwxi)Tvi−xiTVwxiV_w^{\backslash i} = V_w + \frac{(V_w x_i)(V_w x_i)^T}{v_i - x_i^T V_w x_i}
                mw\i=mw+(Vw\ixi)vi−1(xiTmw−mi)m_w^{\backslash i} = m_w + (V_w^{\backslash i} x_i) v_i^{-1} (x_i^T m_w - m_i)
            Compute intermediate scalars:
                zi=(mw\i)TxixiTVw\ixi+1z_i = \frac{(m_w^{\backslash i})^T x_i}{\sqrt{x_i^T V_w^{\backslash i} x_i + 1}}
                αi=1xiTVw\ixi+1N(zi;0,1)Φ(zi)\alpha_i = \frac{1}{\sqrt{x_i^T V_w^{\backslash i} x_i + 1}} \frac{\mathcal{N}(z_i; 0, 1)}{\Phi(z_i)}
            Update posterior parameters:
                mw=mw\i+Vw\iαixim_w = m_w^{\backslash i} + V_w^{\backslash i} \alpha_i x_i
                Vw=Vw\i−(Vw\ixi)(αixiTmwxiTVw\ixi)(Vw\ixi)TV_w = V_w^{\backslash i} - (V_w^{\backslash i} x_i) \left( \frac{\alpha_i x_i^T m_w}{x_i^T V_w^{\backslash i} x_i} \right) (V_w^{\backslash i} x_i)^T
            Update term parameters t~i\tilde{t}_i:
                vi=xiTVw\ixi(1αixiTmw−1)v_i = x_i^T V_w^{\backslash i} x_i \left( \frac{1}{\alpha_i x_i^T m_w} - 1 \right)
                mi=xiTmw\i+(vi+xiTVw\ixi)αim_i = x_i^T m_w^{\backslash i} + (v_i + x_i^T V_w^{\backslash i} x_i) \alpha_i
                si=Φ(zi)1+vi−1xiTVw\ixiexp⁡(−12xiTVw\ixixiTmwαi)s_i = \frac{\Phi(z_i) \sqrt{1 + v_i^{-1} x_i^T V_w^{\backslash i} x_i}}{\exp\left( -\frac{1}{2} \frac{x_i^T V_w^{\backslash i} x_i}{x_i^T m_w} \alpha_i \right)}
        end for
    until parameter changes are less than 10−410^{-4}
    B=mwTVw−1mw−∑i=1nmi2viB = m_w^T V_w^{-1} m_w - \sum_{i=1}^n \frac{m_i^2}{v_i}
    p(D)≈∣Vw∣1/2exp⁡(B/2)∏i=1nsip(D) \approx |V_w|^{1/2} \exp(B / 2) \prod_{i=1}^n s_i
    return mw,Vw,p(D)m_w, V_w, p(D)

    Each data point update requires O(d2)O(d^2) operations, resulting in an overall complexity of O(nd2)O(n d^2) per sweep. Classification of a new point x∗x_* is performed via sign(mwTx∗)\text{sign}(m_w^T x_*).

  5. Knowl 5 — Expectation Propagation for Gaussian Mean Estimation with Clutter

    algorithm

    Consider estimating a dd-dimensional mean vector xx from observations D={y1,…,yn}D = \{y_1, \dots, y_n\} drawn from a two-component Gaussian mixture model with known clutter ratio w∈[0,1]w \in [0, 1]:

    p(yi∣x)=(1−w)N(yi;x,Id)+wN(yi;0,10Id),p(x)=N(0,100Id).p(y_i|x) = (1-w)\mathcal{N}(y_i; x, I_d) + w\mathcal{N}(y_i; 0, 10I_d), \quad p(x) = \mathcal{N}(0, 100 I_d).

    EP models the posterior with a spherical Gaussian q(x)=N(mx,vxId)q(x) = \mathcal{N}(m_x, v_x I_d) and term approximations t~i(x)=siexp⁡(−12vi(x−mi)T(x−mi))\tilde{t}_i(x) = s_i \exp\left(-\frac{1}{2 v_i} (x - m_i)^T (x - m_i)\right).

    Input: Data points y1,…,yn∈Rdy_1, \dots, y_n \in \mathbb{R}^d, clutter ratio ww
    Output: Posterior mean mxm_x, variance vxv_x, model evidence p(D)p(D)
    Initialize prior term: v0=100v_0 = 100, m0=0m_0 = 0, s0=(2πv0)−d/2s_0 = (2\pi v_0)^{-d/2}
    Initialize data terms: vi=∞v_i = \infty, mi=0m_i = 0, si=1s_i = 1 for i=1,…,ni = 1, \dots, n
    Initialize posterior: mx=m0m_x = m_0, vx=v0v_x = v_0
    repeat
        for i=1,…,ni = 1, \dots, n do
            Compute cavity parameters:
                (vx\i)−1=vx−1−vi−1(v_x^{\backslash i})^{-1} = v_x^{-1} - v_i^{-1}
                mx\i=mx+vx\ivi−1(mx−mi)m_x^{\backslash i} = m_x + v_x^{\backslash i} v_i^{-1} (m_x - m_i)
            Compute tilted distribution moments:
                Zi=(1−w)N(yi;mx\i,(vx\i+1)Id)+wN(yi;0,10Id)Z_i = (1 - w)\mathcal{N}(y_i; m_x^{\backslash i}, (v_x^{\backslash i} + 1)I_d) + w\mathcal{N}(y_i; 0, 10I_d)
                ri=1−wZiN(yi;0,10Id)r_i = 1 - \frac{w}{Z_i} \mathcal{N}(y_i; 0, 10I_d)
                mx=mx\i+vx\iriyi−mx\ivx\i+1m_x = m_x^{\backslash i} + v_x^{\backslash i} r_i \frac{y_i - m_x^{\backslash i}}{v_x^{\backslash i} + 1}
                vx=vx\i−ri(vx\i)2vx\i+1+ri(1−ri)(vx\i)2∥yi−mx\i∥2d(vx\i+1)2v_x = v_x^{\backslash i} - r_i \frac{(v_x^{\backslash i})^2}{v_x^{\backslash i} + 1} + r_i (1 - r_i) \frac{(v_x^{\backslash i})^2 \|y_i - m_x^{\backslash i}\|^2}{d (v_x^{\backslash i} + 1)^2}
            Update term parameters t~i\tilde{t}_i:
                vi−1=vx−1−(vx\i)−1v_i^{-1} = v_x^{-1} - (v_x^{\backslash i})^{-1}
                mi=mx\i+(vi+vx\i)(vx\i)−1(mx−mx\i)m_i = m_x^{\backslash i} + (v_i + v_x^{\backslash i}) (v_x^{\backslash i})^{-1} (m_x - m_x^{\backslash i})
                si=Zi(2πvi)d/2N(mi;mx\i,(vi+vx\i)Id)s_i = \frac{Z_i}{(2\pi v_i)^{d/2} \mathcal{N}(m_i; m_x^{\backslash i}, (v_i + v_x^{\backslash i})I_d)}
        end for
    until changes in (mi,vi,si)(m_i, v_i, s_i) are less than 10−410^{-4}
    B=mxTmxvx−∑i=0nmiTmiviB = \frac{m_x^T m_x}{v_x} - \sum_{i=0}^n \frac{m_i^T m_i}{v_i}
    p(D)≈(2πvx)d/2exp⁡(B/2)∏i=0nsip(D) \approx (2\pi v_x)^{d/2} \exp(B / 2) \prod_{i=0}^n s_i
    return mx,vx,p(D)m_x, v_x, p(D)

  6. Knowl 6 — Performance Comparison on UCI Classification Benchmarks for Bayes Point Machines

    data/table

    The Bayes Point Machine trained via Expectation Propagation (EP) was evaluated against the Billiard algorithm (500 iterations, initialized at the SVM solution) and Support Vector Machines (SVM) across four UCI repository benchmark datasets. Datasets were randomly split into 60% training and 40% testing sets across 40 trials. All classifiers used zero slack and a Gaussian kernel with σ=3\sigma = 3; features were normalized to zero mean and unit variance on the training set.

    Dataset EP Billiard SVM
    Heart .203±.069.203 \pm .069 .207±.069.207 \pm .069 .232±.069.232 \pm .069
    Thyroid .037±.037.037 \pm .037 .037±.038.037 \pm .038 .053±.035.053 \pm .035
    Ionosphere .099±.057.099 \pm .057 .113±.064.113 \pm .064 .115±.066.115 \pm .066
    Sonar .140±.077.140 \pm .077 .147±.072.147 \pm .072 .129±.075.129 \pm .075

    Values represent test error rate mean ±\pm two standard deviations over 40 splits. EP achieves a lower average test error rate than SVM on Heart, Thyroid, and Ionosphere (statistically significant with 99% probability), while performing as well as or better than the Billiard algorithm across all datasets at substantially lower computational cost.

  7. Knowl 7 — Accuracy and Efficiency of EP in Gaussian Clutter Estimation

    empirical result

    In synthetic experiments on the Gaussian clutter problem (w=0.5w = 0.5, true mean x=2x = 2) comparing EP against Laplace's method, Variational Bayes (VB), importance sampling (using the prior as importance distribution), and Gibbs sampling:

    1. Posterior Mean and Evidence Accuracy: EP achieves orders-of-magnitude lower absolute error in estimating the posterior mean E[x∣D]\mathbb{E}[x|D] and model evidence p(D)p(D) compared to Laplace's method and VB at equivalent or lower floating point operations (FLOPS).
    2. Iterative Refinement over Assumed-Density Filtering: The first pass of EP corresponds exactly to ADF. Subsequent iterations (typically converging within 4 to 5 passes) reduce estimation errors by several orders of magnitude.
    3. Scaling with Sample Size: As sample size increases from n=20n = 20 to n=200n = 200, the true posterior becomes more unimodal and Gaussian, leading to substantial accuracy gains for deterministic methods (EP, Laplace, VB), whereas sampling methods fail to exploit the Gaussianity of the posterior efficiently.

  8. Knowl 8 — Divergence and Inadequacy of EP under Multi-modal Posteriors

    limitation

    Expectation Propagation relies on projecting the posterior onto a single unimodal exponential family distribution. When the true posterior exhibits significant multimodality (e.g., in small sample regimes n=20n=20 in the clutter problem where data supports multiple distinct mode hypotheses):

    1. Canonical EP iterations can fail to converge or diverge entirely.
    2. When restricted or damped versions of EP are forced to converge, the resulting unimodal approximation captures only a single mode, severely underestimating the global uncertainty and misrepresenting the multifold nature of the true posterior.
    3. The non-convergence of canonical EP serves as a diagnostic indicating that the chosen approximating distribution family is fundamentally mismatched with the structure of the true posterior.

Coverage note — None was omitted; all key algorithmic contributions (general EP, clutter model, BPM), theoretical properties (energy/fixed point duality, Loopy BP connection), empirical evaluations (clutter experiments, UCI classification benchmarks), and documented limitations are fully covered.

References

  1. 1.Blake, C. L., & Merz, C. J. (1998). UCI repository of machine learning databases. www.ics.uci.edu/~mlearn/MLRepository.html
  2. 2.Boyen, X., & Koller, D. (1998). Tractable inference for complex stochastic processes. Uncertainty in AI.
  3. 3.Frey, B. J., & MacKay, D. J. (1997). A revolution: Belief propagation in graphs with cycles. NIPS 10.
  4. 4.Frey, B. J., Patrascu, R., Jaakkola, T., & Moran, J. (2000). Sequentially fitting inclusive trees for inference in noisy-OR networks. NIPS 13.
  5. 5.Herbrich, R., Graepel, T., & Campbell, C. (1999). Bayes point machines: Estimating the Bayes point in kernel space. IJCAI Workshop Support Vector Machines.
  6. 6.Koller, D., Lerner, U., & Angelov, D. (1999). A general algorithm for approximate inference and its application to hybrid Bayes nets. Uncertainty in AI (pp. 324-333).
  7. 7.Kschischang, F. R., Frey, B. J., & Loeliger, H.-A. (2000). Factor graphs and the sum-product algorithm. IEEE Trans Info Theory, to appear. www.comm.toronto.edu/frank/factor/.
  8. 8.Lauritzen, S. L. (1992). Propagation of probabilities, means and variances in mixed graphical association models. J American Statistical Association, 87, 1098-1108.
  9. 9.Maybeck, P. S. (1982). Stochastic models, estimation and control, chapter 12.7. Academic Press.
  10. 10.Minka, T. P. (2001). A family of algorithms for approximate Bayesian inference. Doctoral dissertation, Massachusetts Institute of Technology. vismod.www.media.mit.edu/~tpminka/.
  11. 11.Murphy, K., Weiss, Y., & Jordan, M. (1999). Loopy-belief propagation for approximate inference: An empirical study. Uncertainty in AI.
  12. 12.Opper, M., & Winther, O. (1999). A Bayesian approach to on-line learning. On-Line Learning in Neural Networks. Cambridge University Press.
  13. 13.Opper, M., & Winther, O. (2000). Gaussian processes for classification: Mean field algorithms. Neural Computation, 12, 2655-2684.
  14. 14.Shachter, R. (1990). A linear approximation method for probabilistic inference. Uncertainty in AI.
  15. 15.Yedidia, J. S., Freeman, W. T., & Weiss, Y. (2000). Generalized belief propagation (Technical Report TR2000-26). MERL. www.merl.com/reports/TR2000-26/index.html

Citation

MLA
Minka, T. P. “Expectation Propagation for Approximate Bayesian Inference”. arXiv, 2013, http://arxiv.org/abs/1301.2294v1.
APA
Minka, T. P. (2013). Expectation Propagation for approximate Bayesian inference. arXiv. http://arxiv.org/abs/1301.2294v1
Chicago
Minka, T. P. 2013. “Expectation Propagation for Approximate Bayesian Inference”. arXiv. http://arxiv.org/abs/1301.2294v1.
Harvard
Minka, T.P. (2013) “Expectation Propagation for approximate Bayesian inference”, arXiv [Preprint]. Available at: http://arxiv.org/abs/1301.2294v1.
Vancouver
1. Minka TP (2013) Expectation Propagation for approximate Bayesian inference. arXiv

BibTeX

@article{minka2013expectation,
  title = {Expectation Propagation for approximate Bayesian inference},
  author = {Minka, Thomas P.},
  year = {2013},
  journal = {arXiv},
  url = {http://arxiv.org/abs/1301.2294v1},
  eprint = {1301.2294}
}
Metadata:arXiv

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