Novel approximations for inference and learning in nonlinear dynamical systems

We formulate the problem of inference in nonlinear dynamical systems (NLDS) in the Expectation-Propagation framework, and propose two novel inference algorithms based on Laplace approximation and the unscented transform (UT). The algorithms are compared empirically and employed as an improved E-step in a conjugate gradient learning algorithm. We illustrate its use for data mining with two high-dimensional time series from marketing research. This contribution is based on work that appeared in [1] and [2]. More information can be obtained via www.snn.kun.nl/~ypma/papers/list_of_papers.html or [email protected]. The work is supported by STW, project NNN.5321 “Graphical models for data mining”. Data was provided by BrandmarC. Model. We consider dynamical systems with nonlinearities in the stateand observation equations and additive Gaussian noise, xt = f(xt−1) + vt, vt ∼ N (0, Q); yt = g(xt) + wt, wt ∼ N (0, R) (1) where f(·) and g(·) are nonlinear functions. In the well-known Kalman filter and smoother all functions are assumed linear and posterior beliefs on the hidden states can be computed exactly. In the nonlinear model, forward and backward messages cannot be computed exactly any more, so one has to resort to approximations. Inference with unscented and Laplace approximation. One can express inference in a graphical model as a sequence of multiplications and a summation (or integral) of local factors and messages. In the NLDS model, p(x1:T , y1:T ) factorizes: p(x1:T , y1:T ) = ∏ t Ψt(xt−1, xt) = ∏ t p(xt|xt−1)p(yt|xt). Beliefs p(xt|y1:T ) are computed by p(xt|y1:T ) = α̂t(xt)β̂t(xt), where α̂t(xt) and β̂t(xt) are the forward and backward messages at xt. We express a two-slice belief as a product of a twoslice potential and ’incoming messages’, p̂t(xt−1, xt) ∝ α̂t−1(xt−1)Ψt(xt−1, xt)β̂t(xt). Belief qt(xt) is obtained as qt(xt) = collapse p̂t(xt−1, xt)dxt−1, where “collapse” involves projection to a Gaussian and marginalization (in this case over xt−1). In our first approach we collapse the nongaussian marginal onto a Gaussian by applying Laplace approximation. In our second approach, we use the unscented transform to collapse the nongaussian two-slice joint pt(xt−1, xt) to a Gaussian, in three steps: 1. prediction: approximate α̂t−1(xt−1)Ψt (xt−1, xt) p ∗ t (xt−1, xt) with UT; 2. correction: compute pt (xt) by marginalization; approximate p ∗ t (xt)Ψ b t(xt, yt) with a Gaussian pt (xt, yt) using UT; incorporate evidence into p ∗ t (yt|xt) = pt (xt, yt)/p ∗ t (xt), resulting in p ∗∗ t (yt|xt); 3. combination: compute qt(xt−1, xt) = pt (xt−1, xt)p ∗∗ t (yt|xt)β̂t(xt), and obtain qt(xt−1) and qt(xt) by marginalization. We use UT for computing moments of the joints pt (xt−1, xt) and p ∗ t (xt, yt). For example, in the prediction step we need to compute ∫ ∫ α̂t−1(xt−1)Ψt (xt−1, xt)h(xt−1, xt)dxt−1dxt ≈ ∑