Aided Optimal Search: Data-Driven Target Pursuit from On-Demand Delayed Binary Observations

Abstract

We consider the following search problem: an autonomous robot (the searcher) needs to locate and reach a target moving in a field scattered with Unattended Ground Sensors (UGS). The searcher has very limited information about the target: (i) it has an initial distribution (the prior) describing the probability of the target being at a given location at the initial time, and (ii) it can interrogate nearby sensors; each sensor records a binary measurement, describing whether or not the target passed in the proximity of the sensor at some point in time. Then the goal for the searcher is to estimate the trajectory of the target, and plan a maneuver that allows reducing the uncertainty about the current target state. We refer to this problem as aided optimal search, in that the search process is aided by an external infrastructure (the ground sensors). The paper adopts a Dynamic Data-Driven Appplications Systems (DDDAS) paradigm, in which the data collected by the searcher is used to update the belief on the trajectory of the target, and the searcher actively steers the measurement process to improve its knowledge about the location of the target. In particular, we make two main contributions. The first regards the target trajectory estimation. We show how to perform optimal Bayesian inference from binary measurements using a Gaussian Mixture Model (GMM). One of the main insights is that parameterizing the GMM in the information filter (inverse covariance) form allows huge computational savings: the information matrix of each mixture component is a very sparse (block-tridiagonal) matrix, which allows us to deal with a GMM with thousands of components in a fraction of a second. The second contribution regards planning: we propose a Mixed-Integer Programming (MIP) approach to plan the optimal searcher path, which minimizes the uncertainty about the position of the target. The key idea here is the use of sampling to decouple the complexity of the MIP from the length of the trajectory of the target. We demonstrate the proposed strategy in extensive simulations, reporting statistics about success rate and computational time for different scenarios and target motion models. The proposed search strategy largely outperforms greedy strategies (e.g., visiting the most likely target position).

Prediction Equations

In this section we derive the equations for the prediction phase of our incremental smoother. We start with a lemma, which will be useful to simplify the derivation later on.

Lemma 1 (From measurement to state space)

Given a multivariate Gaussian $\mathcal {N}(A x; \eta , \Omega )$ , with $A \in { {\mathbb R}^{d \times d} }$ and full rank, then the multivariate Gaussian can be written equivalently as:

$$\displaystyle \begin{aligned} \mathcal{N}(A x; \eta, \Omega) = \mathcal{N}(x; A^{\mathsf{T}} \eta, A^{\mathsf{T}} \Omega A) \end{aligned} $$

(14.52)

Proof. We prove the claim by inspection. We write explicitly the right-hand side of (14.52) as:

$$\displaystyle \begin{aligned} \begin{array}{lll} {} \mathcal{N}(x; A^{\mathsf{T}} \eta, A^{\mathsf{T}} \Omega A) = k \; \text{exp} \bigg\{ -\frac{1}{2} [ x - (A^{\mathsf{T}} \Omega A)^{-1} A^{\mathsf{T}} \eta ]^{\mathsf{T}} \times \\ \hspace{4cm} (A^{\mathsf{T}} \Omega A) [ x - (A^{\mathsf{T}} \Omega A)^{-1} A^{\mathsf{T}} \eta ] \bigg\} = \\ k \; \text{exp} \bigg\{ -\frac{1}{2} \bigg[ x^{\mathsf{T}} (A^{\mathsf{T}} \Omega A) x - 2 \eta^{\mathsf{T}} A x + \eta^{\mathsf{T}} A (A^{\mathsf{T}} \Omega A)^{-1} A^{\mathsf{T}} \eta \bigg] \bigg\} \end{array} \end{aligned} $$

(14.53)

From the fact that A is square and full rank (hence invertible), the previous simplifies to:

$$\displaystyle \begin{aligned} \begin{array}{lll} k \; \text{exp} \bigg\{ -\frac{1}{2} \bigg[ (A x)^{\mathsf{T}} \Omega (A x) - 2 \eta^{\mathsf{T}} A x + \eta^{\mathsf{T}} \Omega^{-1} \eta \bigg] \bigg\} = \\ k \; \text{exp} \bigg\{ -\frac{1}{2} \bigg[ (A x - \Omega^{-1}\eta)^{\mathsf{T}} \Omega (A x - \Omega^{-1}\eta) \bigg] \bigg\} = \mathcal{N}(A x; \eta, \Omega) \end{array} \end{aligned} $$

(14.54)

which proves the claim. □

We can now focus on the derivation of the prediction equations. Let us start from the general prediction Eq. (14.14):

$$\displaystyle \begin{aligned} {\mathbb P}\left(y_{1:t+1} | Z_{1:t}\right) = {\mathbb P}\left(y_{t+1} | y_{t}\right) {\mathbb P}\left(y_{1:t} | Z_{1:t}\right) \end{aligned} $$

(14.55)

Substituting our choice of prior probability (14.15) and transition probability (14.7), we get:

$$\displaystyle \begin{aligned} \begin{array}{lll} {} {\mathbb P}\left(y_{1:t+1} | Z_{1:t}\right) &=& \mathcal{N}(y_{t+1} - A y_t; 0,\Omega_w) \mathcal{M}(y_{1:t} ; \{\eta_{t,j}, \Omega_{t,j}, \alpha_{t,j} \}_{j=1}^m) = \\ && \mathcal{N}(y_{t+1} - A y_t; 0,\Omega_w) \sum_{j=1}^m \alpha_{t,j} \mathcal{N}(y_{1:t} \; ; \eta_{t,j}, \Omega_{t,j}) = \\ && \sum_{j=1}^m \alpha_{t,j} \mathcal{N}(y_{t+1} - A y_t; 0,\Omega_w) \mathcal{N}(y_{1:t} ; \eta_{t,j}, \Omega_{t,j}) \end{array}\end{aligned} $$

(14.56)

Now we use the definition of S _1:t and S _t:t+1, given in (14.18), which we substitute in (14.56):

$$\displaystyle \begin{aligned} \begin{array}{lll} {} {\mathbb P}\left(y_{1:t+1} | Z_{1:t}\right) = \sum_{j=1}^m \alpha_{t,j} \mathcal{N}(S_{t:t+1} y_{1:t+1}; 0,\Omega_w) \mathcal{N}(S_{1:t} y_{1:t+1} ; \eta_{t,j}, \Omega_{t,j}) \end{array} \end{aligned} $$

(14.57)

We can develop each summand as follows:

(14.58)

Noting that the matrix $\left [\begin {array}{c} S_{1:t} \\ S_{t:t+1} \end {array}\right ]$ is square and full rank, we apply Lemma 1 and simplify the previous expression as:

$$\displaystyle \begin{aligned} \begin{array}{lll} {} \mathcal{N}(S_{t:t+1} y_{1:t+1}; 0,\Omega_w) \mathcal{N}(S_{1:t} y_{1:t+1} ; \eta_{t,j}, \Omega_{t,j}) = \\ \mathcal{N}(y_{1:t+1} ; S_{1:t}^{\mathsf{T}} \eta_{t,j}, S_{1:t}^{\mathsf{T}} \Omega_{t,j} S_{1:t} + S_{t:t+1}^{\mathsf{T}} \Omega_{t,j} S_{t:t+1} ) \end{array} \end{aligned} $$

(14.59)

Substituting (14.59) back into (14.57), we obtain:

$$\displaystyle \begin{aligned} \begin{array}{lll} {} {\mathbb P}\left(y_{1:t+1} | Z_{1:t}\right) = \sum_{j=1}^m \alpha_{t,j} \mathcal{N}(y_{1:t+1} ; S_{1:t}^{\mathsf{T}} \eta_{t,j}, S_{1:t}^{\mathsf{T}} \Omega_{t,j} S_{1:t} + S_{t:t+1}^{\mathsf{T}} \Omega_{t,j} S_{t:t+1} ) \end{array} \end{aligned} $$

(14.60)

which coincides with Eqs. (14.16) and (14.17).

14.1.1 B Update Equations

In this section we derive the equations for the update phase of our incremental smoother. We start with a lemma, which will be useful to simplify the derivation later on.

Lemma 2 (Update in Information Form)

Given two multivariate Gaussians $\mathcal {N}(x; \bar {\eta }, \bar {\Omega })$ and $\mathcal {N}(A x; \eta _a, \Omega _a)$ , with $x \in { {\mathbb R}^{d} }$ and $A \in { {\mathbb R}^{d_a \times d} }$ (full row rank, d _a ≤ d), then the following equality holds:

$$\displaystyle \begin{aligned} \mathcal{N}(A x; \eta_a, \Omega_a) \mathcal{N}(x; \bar{\eta}, \bar{\Omega}) = \kappa \mathcal{N}(x; \bar{\eta} + A^{\mathsf{T}} \eta_a, \bar{\Omega} + A^{\mathsf{T}} \Omega_a A ) \end{aligned} $$

(14.61)

where κ is a constant independent on x.

Proof. We prove the claim by inspection. We write explicitly the left-hand side of (14.61) as:

$$\displaystyle \begin{aligned} \begin{array}{lll} {} \mathcal{N}(A x; \eta_a, \Omega_a) \mathcal{N}(x; \bar{\eta}, \bar{\Omega}) = \frac{ \det(\bar{\Omega})^{\frac{1}{2}} }{ (2\pi)^{\frac{d}{2}} } \frac{ \det(\Omega_a)^{\frac{1}{2}} }{ (2\pi)^{\frac{d_a}{2}} } \times \\ \; \text{exp} \bigg\{ -\frac{1}{2} \bigg[ (Ax - \Omega_a^{-1} \eta_a)^{\mathsf{T}} \Omega_a (Ax - \Omega_a^{-1} \eta_a) + (x - \bar{\Omega}^{-1} \bar{\eta})^{\mathsf{T}} \bar{\Omega} (x - \bar{\Omega}^{-1} \bar{\eta}) \bigg] \bigg\} = \\ {\mbox{(developing the squares and introducing} {\kappa} \mbox{to denote constant factors)} } \\ \kappa \times \; \text{exp} \bigg\{ -\frac{1}{2} \bigg[ x^{\mathsf{T}} (\bar{\Omega} {+} A^{\mathsf{T}} \Omega_a A) x {-} 2 x^{\mathsf{T}} (\bar{\eta} {+} A^{\mathsf{T}} \eta_a) + \eta_a^{\mathsf{T}} \Omega_a^{-1} \eta_a + \bar{\eta}^{\mathsf{T}} \bar{\Omega}^{-1} \bar{\eta} \bigg] \bigg\} = \\ {\mbox{(including constants at the exponent in} {\kappa)} } \\ \kappa \times \; \text{exp} \bigg\{ -\frac{1}{2} \bigg[ x^{\mathsf{T}} (\bar{\Omega} + A^{\mathsf{T}} \Omega_a A) x - 2 x^{\mathsf{T}} (\bar{\eta} + A^{\mathsf{T}} \eta_a) \bigg] \bigg\} = \\ {\mbox{(reincluding more convenient constants at the exponent)} } \\ \kappa \times \text{exp} \bigg\{ -\frac{1}{2} \bigg[ x^{\mathsf{T}} (\bar{\Omega} + A^{\mathsf{T}} \Omega_a A) x - 2 x^{\mathsf{T}} (\bar{\eta} + A^{\mathsf{T}} \eta_a) + \\ (\bar{\eta} + A^{\mathsf{T}} \eta_a)^{\mathsf{T}} (\bar{\Omega} + A^{\mathsf{T}} \Omega_a A)^{-1} (\bar{\eta} + A^{\mathsf{T}} \eta_a) \bigg] \bigg\} = \\ {\text{(isolating}\ \text{the}\ \text{Gaussian}\ \text{term,}\ \text{up}\ \text{to}\ \text{constant)} } \\ \kappa \times \mathcal{N}(x; \bar{\eta} + A^{\mathsf{T}} \eta_a, \bar{\Omega} + A^{\mathsf{T}} \Omega_a A ) \end{array} \end{aligned} $$

(14.62)

A simple way to explicitly compute the constant κ is to observe that:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \int \mathcal{N}(A x; \eta_a, \Omega_a) \mathcal{N}(x; \bar{\eta}, \bar{\Omega}) \text{d}x = \end{array} \end{aligned} $$

(14.63)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \int \kappa \times \mathcal{N}(x; \bar{\eta} + A^{\mathsf{T}} \eta_a, \bar{\Omega} + A^{\mathsf{T}} \Omega_a A ) \text{d}x = \kappa \end{array} \end{aligned} $$

(14.64)

Hence κ is the result of a convolution of two Gaussian distributions, which can be computed as [68, page 209]

$$\displaystyle \begin{aligned} \begin{array}{lll} \kappa = \mathcal{N}_{\mathcal{P}}(\Omega_a^{-1} \eta_a; A \bar{\Omega}^{-1} \bar{\eta}, A \bar{\Omega}^{-1} A^{\mathsf{T}} + \Omega_a^{-1}) \end{array} \end{aligned} $$

(14.65)

and this concludes the proof. □

Detection (b _it = 1)

Let us start from the general update Eq. (14.19):

$$\displaystyle \begin{aligned} {\mathbb P}\left(y_{1:t+1} | Z_{1:t+1}\right) = \frac{ {\mathbb P}\left(z_{t+1} | y_{1:t+1} \right) {\mathbb P}\left(y_{1:t+1} | Z_{1:t}\right) }{ \int {\mathbb P}\left(z_{t+1} | y_{1:t+1} \right) {\mathbb P}\left(y_{1:t+1} | Z_{1:t}\right) dy_{1:t+1} } \end{aligned} $$

(14.66)

Let us focus on the term ${\mathbb P}\left (z_{t+1} | y_{1:t+1} \right ) {\mathbb P}\left (y_{1:t+1} | Z_{1:t}\right )$. First of all, we rewrite the measurement likelihood as:

$$\displaystyle \begin{aligned} \begin{array}{lll} {} {\mathbb P}\left( b_{it}=1 | y_{1:t}\right) = \exp{ \left\{ -\frac{ \|p^y_{\tau_{it}} - s_i \|{}^2}{r^2} \right\} } = \\ \exp{ \left\{ -\frac{ \|U_{1:t+1} y_{1:t+1} - s_i \|{}^2}{r^2} \right\} } = \frac{1}{\gamma} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \end{array} \end{aligned} $$

(14.67)

which stresses the fact that the measurement likelihood can be seen as a “scaled” multivariate Gaussian, with γ being the normalization factor (the expression of this term is irrelevant for the subsequent derivation). Let us now substitute the prior probability (14.16) and the measurement likelihood (14.67) in (14.66):

$$\displaystyle \begin{aligned} \begin{array}{lll} {} \frac{ {\mathbb P}\left(z_{t+1} | y_{1:t+1} \right) {\mathbb P}\left(y_{1:t+1} | Z_{1:t}\right) }{ \int {\mathbb P}\left(z_{t+1} | y_{1:t+1} \right) {\mathbb P}\left(y_{1:t+1} | Z_{1:t}\right) dy_{1:t+1} } = \\ {} \frac{ \frac{1}{\gamma} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) }{ \int \frac{1}{\gamma} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j} ) dy_{1:t+1} } = \\ {} \frac{ \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) }{ \int \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) dy_{1:t+1} } = \\ {} \frac{ \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) }{ \int \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) dy_{1:t+1} } \end{array} \end{aligned} $$

which, using Lemma 2, becomes:

$$\displaystyle \begin{aligned} \begin{array}{lll} {} \frac{ \sum_{j=1}^m \bar{\alpha}_{t+1,j} \; \beta_{t+1,j} \; \mathcal{N}(y_{1:t+1} \; ; \; {\eta}_{t+1,j}, {\Omega}_{t+1,j} ) }{ \int \sum_{j=1}^m \bar{\alpha}_{t+1,j} \; \beta_{t+1,j} \; \mathcal{N}(y_{1:t+1} \; ; \; {\eta}_{t+1,j}, {\Omega}_{t+1,j} ) dy_{1:t+1} } \end{array} \end{aligned} $$

(14.68)

where η _t+1,j and Ω_t+1,j are defined as in (14.25). Observing that the integral of each Gaussian at the denominator of (14.68) is one, the previous simplifies to

$$\displaystyle \begin{aligned} \begin{array}{lll} \left( \frac{\bar{\alpha}_{t+1,j} \; \beta_{t+1,j} }{ \sum_{j=1}^m \bar{\alpha}_{t+1,j} \; \beta_{t+1,j} } \right) \mathcal{N}(y_{1:t+1} \; ; \; {\eta}_{t+1,j}, {\Omega}_{t+1,j} ) \end{array} \end{aligned} $$

(14.69)

which matches the expression of (14.25).

No detection (b _it = 0)

In this case, the measurement likelihood is:

$$\displaystyle \begin{aligned} \begin{array}{lll} {} {\mathbb P}\left( b_{it}=0 | y_{1:t}\right) = 1- \exp{ \left\{ -\frac{ \|p^y_{\tau_{it}} - s_i \|{}^2}{r^2} \right\} } = 1 - \frac{1}{\gamma} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \end{array} \end{aligned} $$

(14.70)

where γ = 2πr ² (this is the inverse of the normalization factor of the Gaussian).

Let us now substitute the prior probability (14.16) and the measurement likelihood (14.70) in (14.66):

$$\displaystyle \begin{aligned} \begin{array}{lll} {} \frac{ {\mathbb P}\left(z_{t+1} | y_{1:t+1} \right) {\mathbb P}\left(y_{1:t+1} | Z_{1:t}\right) }{ \int {\mathbb P}\left(z_{t+1} | y_{1:t+1} \right) {\mathbb P}\left(y_{1:t+1} | Z_{1:t}\right) dy_{1:t+1} } = \\ {} \frac{ (1-\frac{1}{\gamma} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} )) \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) }{ \int (1-\frac{1}{\gamma} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} )) \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j} ) dy_{1:t+1} } = \\ \\ {\mbox{(integral of the GMM at the denominator is 1)} } \\ \\ \frac{ (1-\frac{1}{\gamma} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} )) \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) }{ 1 - \sum_{j=1}^m \bar{\alpha}_{t+1,j} \frac{1}{\gamma} \int \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j},\bar{\Omega}_{t+1,j}) dy_{1:t+1} } = \end{array} \end{aligned} $$

$$\displaystyle \begin{aligned} \begin{array}{lll} {\mbox{(from the definition of} {\beta_{t+1,j})} } \\ \\ \frac{ (1-\frac{1}{\gamma} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) }{ 1 - \sum_{j=1}^m \bar{\alpha}_{t+1,j} \frac{1}{\gamma} \beta_{t+1,j} } = \\ {} \frac{ \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) }{ 1 - \sum_{j=1}^m \bar{\alpha}_{t+1,j} \frac{1}{\gamma} \beta_{t+1,j} } -\\ {} \frac{ \sum_{j=1}^m \bar{\alpha}_{t+1,j} \frac{1}{\gamma} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) }{ 1 - \sum_{j=1}^m \bar{\alpha}_{t+1,j} \frac{1}{\gamma} \beta_{t+1,j} } = \\ \\ {\mbox{(note that gamma does not simplify)} } \\ \\ \frac{ \sum_{j=1}^m \bar{\alpha}_{t+1,j} \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) }{ 1 - \sum_{j=1}^m \bar{\alpha}_{t+1,j} \frac{1}{\gamma} \beta_{t+1,j} } -\\ {} \frac{ \sum_{j=1}^m \bar{\alpha}_{t+1,j} \frac{1}{\gamma} \mathcal{N}(U_{1:t+1} y_{1:t+1} \; ; \frac{s_i}{r^2} , \frac{1}{r^2} ) \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) }{ 1 - \sum_{j=1}^m \bar{\alpha}_{t+1,j} \frac{1}{\gamma} \beta_{t+1,j} } = \\ \\ {\mbox{(using Lemma \mbox{2} in each term of the second sum)} } \\ \\ \frac{ \sum_{j=1}^m \bar{\alpha}_{t+1,j} }{ 1 - \sum_{j=1}^m \bar{\alpha}_{t+1,j} \frac{1}{\gamma} \beta_{t+1,j} } \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j}, \bar{\Omega}_{t+1,j}) + \\ {} \frac{ \sum_{j=1}^m - \bar{\alpha}_{t+1,j} \frac{1}{\gamma} \beta_{t+1,j} }{ 1 - \sum_{j=1}^m \bar{\alpha}_{t+1,j} \frac{1}{\gamma} \beta_{t+1,j} } \mathcal{N}(y_{1:t+1} \; ; \; \bar{\eta}_{t+1,j} + U_{1:t+1}^{\mathsf{T}} \frac{s_i}{r^2}, \bar{\Omega}_{t+1,j} + \frac{1}{r^2} U_{1:t+1}^{\mathsf{T}} U_{1:t+1}) \end{array} \end{aligned} $$

which coincides with (14.25).