The recursive variational Gaussian approximation (R-VGA)

Abstract

We consider the problem of computing a Gaussian approximation to the posterior distribution of a parameter given N observations and a Gaussian prior. Owing to the need of processing large sample sizes N, a variety of approximate tractable methods revolving around online learning have flourished over the past decades. In the present work, we propose to use variational inference to compute a Gaussian approximation to the posterior through a single pass over the data. Our algorithm is a recursive version of variational Gaussian approximation we have called recursive variational Gaussian approximation. We start from the prior, and for each observation, we compute the nearest Gaussian approximation in the sense of Kullback–Leibler divergence to the posterior given this observation. In turn, this approximation is considered as the new prior when incorporating the next observation. This recursive version based on a sequence of optimal Gaussian approximations leads to a novel implicit update scheme which resembles the online Newton algorithm and which is shown to boil down to the Kalman filter for Bayesian linear regression. In the context of Bayesian logistic regression, the implicit scheme may be solved, and the algorithm is shown to perform better than the extended Kalman filter, while being less computationally demanding than its sampling counterparts.

Appendix

1.1 Generalization of Theorem 1

Theorem 1 can be generalized to any target distribution belonging to an exponential family of natural parameter $\eta$ as follows:

Theorem 4

Considering an exponential family $q_\eta$ of natural parameter $\eta$, mean parameter m, and a strictly convex log partition function F such that $q_\eta (\theta )=h(\theta )\exp (<\eta ,\theta >-F( {\eta }))$, the solution to the recursive variational approximation problem between a target distribution $q_\eta$ and the one-sample posterior $p(\theta |y_t) \propto p(y_t|\theta )q_{\eta _{t-1}}(\theta )$:

$$\begin{aligned} \underset{\eta _t}{\arg \min } \quad KL(q_{\eta _{t}}(\theta )|p(\theta |y_t)), \end{aligned}$$

(95)

must satisfy the following implicit fixed point equation on the natural parameter:

$$\begin{aligned} \eta _{t}&=\eta _{t-1} +(\nabla ^2 F(\eta _t))^{-1} \nabla _{\eta _t} {\mathbf {E}}_{q_{\eta _t}}[\log p(y_t|\theta )] \end{aligned}$$

(96)

$$\begin{aligned}&=\eta _{t-1} + \nabla _{m_t} {\mathbf {E}}_{q_{m_t}}[\log p(y_t|\theta )]. \end{aligned}$$

(97)

If $q_\eta$ is a Gaussian distribution, this fixed point equation is equivalent to the R-VGA updates given in Theorem 1.

This theorem extends the results found in Khan and Lin (2017) and Khan and Nielsen (2018) to the recursive variational inference framework. Here, the implicit updates come directly from the critical point of our recursive form (95) and not from the application of an explicit natural gradient.

Proof

The recursive variational approximation for the target distribution $q_\eta$ of log-partition function F may be rewritten as:

$$\begin{aligned}&KL(q_{\eta _{t}}(\theta )|p(\theta |y_t)) \nonumber \\&\quad ={\mathbf {E}}_{q_{\eta _{t}}}[-\log p(y_t|\theta )] + KL(q_{\eta _{t}}(\theta )|q_{\eta _{t-1}}(\theta )) + c \end{aligned}$$

(98)

$$\begin{aligned}&\quad = {\mathbf {E}}_{q_{\eta _{t}}}[-\log p(y_t|\theta )] + B_F(\eta _{t-1},\eta _{t}) + c, \end{aligned}$$

(99)

where c is a normalization constant and $B_F$ is the Bregman divergence associated with the strictly convex log partition function F:

$$\begin{aligned}&B_F(\eta _1,\eta _2)=F(\eta _1)-F(\eta _2)-<\eta _1-\eta _2,\nabla F(\eta _2)>. \end{aligned}$$

(100)

The derivative of the variational divergence (99) with respect to the natural parameter $\eta _t$ gives:

$$\begin{aligned}&\nabla _{\eta _t} {\mathbf {E}}_{q_{\eta _{t}}}[-\log p(y_t|\theta )]+(\eta _{t}-\eta _{t-1})\nabla ^2 F(\eta _t) =0 \end{aligned}$$

(101)

$$\begin{aligned}&\iff \nonumber \\ \eta _{t}&=\eta _{t-1} + (\nabla ^2 F(\eta _t))^{-1} \nabla _{\eta _t} {\mathbf {E}}_{q_{\eta _{t}}}[\log p(y_t|\theta )] \end{aligned}$$

(102)

$$\begin{aligned}&=\eta _{t-1} + \nabla _{m_t} {\mathbf {E}}_{q_{m_{t}}}[\log p(y_t|\theta )], \end{aligned}$$

(103)

which yields the desired result. The last equality comes from the properties of exponential families with a strictly convex log-partition function F: The mean parameter m is the Legendre dual of the natural parameter $\eta$, i.e., $<\eta ,m>=F(\eta )+F^*(m)$ and $\nabla ^2 F(\eta )=\frac{\partial m}{\partial \eta }$.

Finally, if we apply this equation to the natural parameter for a Gaussian distribution $\eta =\begin{pmatrix} P^{-1} \mu \\ -\frac{1}{2}P^{-1} \end{pmatrix}$, we retrieve the R-VGA updates. Indeed, to derive the R-VGA from the implicit version of the natural gradient we use the chain rule to relate the mean parameters to the natural parameters as proposed by Khan et al., in the context of batch variational approximation, in Khan et al. (2018) (Appendix C). In the Gaussian case, the mean is $m=\begin{pmatrix} m_1=\mu \\ m_2=P+\mu \mu ^T \end{pmatrix}$ and the natural parameter is $\eta =\begin{pmatrix} \eta _1=P^{-1}\mu \\ \eta _2=-\frac{1}{2}P^{-1} \end{pmatrix}$. The gradient with respect to the mean parameters $m_1,m_2$ can be expressed as the gradient with respect to the sources parameters $\mu ,P$ using the chain rule:

$$\begin{aligned} \frac{\partial f}{\partial m_1}&=\frac{\partial f}{\partial \mu }\frac{\partial \mu }{\partial m_1}+\frac{\partial f}{\partial P}\frac{\partial P}{\partial m_1}=\frac{\partial f}{\partial \mu }-2\frac{\partial f}{\partial P}\mu \end{aligned}$$

(104)

$$\begin{aligned} \frac{\partial f}{\partial m_2}&=\frac{\partial f}{\partial \mu }\frac{\partial \mu }{\partial m_2}+\frac{\partial f}{\partial P}\frac{\partial P}{\partial m_2}=\frac{\partial f}{\partial P}. \end{aligned}$$

(105)

The updates

$$\begin{aligned}&\eta _{t} = \eta _{t-1} +\nabla _{m_t} {\mathbf {E}}_{q_{\eta _t}}[\log p(y_t|\theta )], \end{aligned}$$

(106)

become:

$$\begin{aligned}&\begin{pmatrix}P_t^{-1}\mu _t \\ -\frac{1}{2}P_t^{-1} \end{pmatrix} =\begin{pmatrix} P_{t-1}^{-1}\mu _{t-1} \\ -\frac{1}{2}P_{t-1}^{-1} \end{pmatrix} \\&\quad + \begin{pmatrix} \frac{\partial }{\partial \mu _t} {\mathbf {E}}_{q_{\eta _t}}[\log p(y_t|\theta )] -2\frac{\partial }{\partial P_t}{\mathbf {E}}_{q_{\eta _t}}[\log p(y_t|\theta )] \mu _t\\ \frac{\partial }{\partial P_t} {\mathbf {E}}_{q_{\eta _t}}[\log p(y_t|\theta )] \end{pmatrix} \end{aligned}$$

$$\begin{aligned}&\begin{pmatrix}(P_t^{-1}+2\frac{\partial }{\partial P_t}{\mathbf {E}}_{q_{\eta _t}}[\log p(y_t|\theta )] )\mu _t \\ P_t^{-1} \end{pmatrix} \\&\quad =\begin{pmatrix} P_{t-1}^{-1}\mu _{t-1} \\ P_{t-1}^{-1} \end{pmatrix} + \begin{pmatrix} \frac{\partial }{\partial \mu _t} {\mathbf {E}}_{q_{\eta _t}}[\log p(y_t|\theta )] \\ -2 \frac{\partial }{\partial P_t} {\mathbf {E}}_{q_{\eta _t}}[\log p(y_t|\theta )] \end{pmatrix} \end{aligned}$$

$$\begin{aligned} \begin{pmatrix}\mu _t \\ P_t^{-1} \end{pmatrix}=\begin{pmatrix} \mu _{t-1} \\ P_{t-1}^{-1} \end{pmatrix} + \begin{pmatrix} P_{t-1} \frac{\partial }{\partial \mu _t} {\mathbf {E}}_{q_{\eta _t}}[\log p(y_t|\theta )] \\ -2 \frac{\partial }{\partial P_t} {\mathbf {E}}_{q_{\eta _t}}[\log p(y_t|\theta )] \end{pmatrix}, \end{aligned}$$

which are the R-VGA equations indeed. $\square$

1.2 Proof of Theorem 2

Proof

In the case where p is a multivariate Gaussian distribution $p(y_t|\theta ) \sim {{\mathcal {N}}}(y_t | H_t\theta , R_t)$, where $H_t$ is the observation matrix, we have the relation:

$$\begin{aligned} \log p(y_t|\theta )&=-\frac{1}{2}(y_t-H_t\theta )^TR_t^{-1}(y_t-H_t\theta ) \\&\quad +C=-\ell _t(\theta )\\ \nabla _{\mu _t} {\mathbf {E}}_{q_t}[\log p(y_t|\theta )]&= H_t^TR_t^{-1}(y_t-H_t\mu _t)=-\nabla \ell _t(\theta ) \\ -2\nabla _{P_t} {\mathbf {E}}_{q_t}[\log p(y_t|\theta )]&= H_t^TR_t^{-1}H_t=\nabla ^2 \ell _t(\theta ), \end{aligned}$$

and this last relation gives directly the second R-VGA update equation, rewriting (2) as:

$$\begin{aligned}&P_t^{-1}=P_{t-1}^{-1} + H_t^TR_t^{-1}H_t, \end{aligned}$$

(107)

which is the information update equation. We then rewrite the first R-VGA update Eq. (1) as:

$$\begin{aligned}&\mu _t=\mu _{t-1}+P_{t-1}H_t^TR_t^{-1}(y_t-H_t\mu _t) \end{aligned}$$

(108)

$$\begin{aligned}&\iff \mu _t({\mathbf {I}}_d+P_{t-1}H_t^TR_t^{-1}H_t)=\mu _{t-1}+P_{t-1}H_t^TR_t^{-1}y_t \nonumber \\&\iff \mu _t=(P_{t-1}^{-1}+H_t^TR_t^{-1}H_t)^{-1}P_{t-1}^{-1}(\mu _{t-1}+P_{t-1}H_t^TR_t^{-1}y_t) \nonumber \\&\iff \mu _t=\mu _{t-1}+(P_{t-1}^{-1}+H_t^TR_t^{-1}H_t)^{-1}(-H_t^TR_t^{-1}H_t\mu _{t-1} \nonumber \\&\qquad +H_t^TR_t^{-1}y_t) \nonumber \\&\iff \mu _t=\mu _{t-1}+P_tH_t^TR_t^{-1}(y_t-H_t\mu _{t-1}). \end{aligned}$$

(109)

If we note the Kalman gain $K_t=P_tH_t^TR_t^{-1}$, we find the Kalman update equations for state (42), indeed:

$$\begin{aligned}&K_t=P_{t}H_{t}^TR_{t}^{-1}=(P_{t-1}^{-1}+H_{t}^TR_{t}^{-1}H_{t})^{-1}H_{t}^TR_{t}^{-1} \nonumber \\&\quad =P_{t-1}H_{t}^T(R_{t}+H_{t}P_{t-1}H_{t}^T)^{-1}, \end{aligned}$$

(110)

where we have used the matrix formula: $(A^{-1}+B^TC^{-1}B)^{-1}B^TC^{-1}=AB^T(BAB^T+C)^{-1}$. The Kalman update equation for the covariance matrix (42) is then deduced from (107) using the Woodbury formula:

$$\begin{aligned} P_t^{-1}&=P_{t-1}^{-1} + H_t^TR_t^{-1}H_t \nonumber \\ \iff P_{t}&=P_{t-1}-P_{t-1}H_{t}^T(R_{t}+H_{t}P_{t-1}H_{t}^T)^{-1}H_{t} P_{t-1} \nonumber \\&=({\mathbb {I}}-K_{t}H_{t})P_{t-1}. \end{aligned}$$

(111)

Equivalence to linear Kalman filter has thus been proved.

Equation (109) can be rewritten to find, combined with (107), the online Newton descent. Indeed, if we pose $Q_t=P_t^{-1}$ as the estimation of the Hessian matrix up to the iteration t, we find directly:

$$\begin{aligned} \mu _t&=\mu _{t-1}-Q_t^{-1} \nabla \ell _t(\theta ) \end{aligned}$$

(112)

$$\begin{aligned} Q_t&=Q_{t-1} + H_t^TR_t^{-1}H_t. \end{aligned}$$

(113)

This proves the equivalence to the online Newton descent. Now, it is well known the Kalman filter is optimal for the least mean squares problem; let us reformulate the proof to better show the connection to stochastic optimization.

Let us recall the form of the least mean squares cost function for t observations:

$$\begin{aligned} \sum _{i=1}^t \ell _i(\theta ) = \frac{1}{2} \theta ^TQ_t\theta - v_t^T\theta + C. \end{aligned}$$

(114)

We express now the optimal $\theta ^*_t=Q_t^{-1}v_t$ at time t in function of the optimal at time $t-1$:

$$\begin{aligned} \theta ^*_{t}&=Q^{-1}_{t}v_{t} \nonumber \\&=Q^{-1}_{t}\left( v_{t-1}+H_{t}^T R_{t}^{-1} y_{t}\right) \nonumber \\&=Q^{-1}_{t}\left( Q_{t-1}\theta ^*_{t-1} +H_{t}^T R_{t}^{-1} y_{t} \right) \nonumber \\&=Q^{-1}_{t}\left( \left( Q_{t} - H_{t}^TR_{t}^{-1}H_{t}\right) \theta ^*_{t-1} +H_{t}^T R_{t}^{-1} y_{t} \right) \nonumber \\&=\theta ^*_{t-1}-Q^{-1}_{t}H_{t}^TR_{t}^{-1}H_{t} \theta ^*_t + Q^{-1}_{t} H_{t}^TR_{t}^{-1}y_{t} \nonumber \\&=\theta ^*_{t-1}+Q^{-1}_{t}H_{t}^TR_{t}^{-1}(y_{t}-H_{t}\theta ^*_{t-1})\nonumber \\&=\theta ^*_{t-1}-Q^{-1}_{t}\nabla \ell _t(\theta ) \end{aligned}$$

(115)

$$\begin{aligned}&=\theta ^*_{t-1}+K_t(y_{t}-H_{t}\theta ^*_{t-1}). \end{aligned}$$

(116)

The two last equations show that the recursive least mean squares estimate is found by both online Newton and the linear Kalman filter. $\square$

1.3 Proof of Theorem 3

Proof

The proof is quite similar to the proof in the linear case. The update Eq. (3) can be rewritten using the same manipulations as in the linear case:

$$\begin{aligned} \mu _t&=\mu _{t-1}-P_{t-1}{\mathbf {E}}_{q_t}[\nabla _\theta \ell _t(\theta )] \end{aligned}$$

(117)

$$\begin{aligned}&=\mu _{t-1}+P_{t-1}{\mathbf {E}}_{q_t}[H_t^T R_t^{-1}(y_t-h(\mu _{t-1}) - H_t (\theta -\mu _{t-1})] \nonumber \\&=\mu _{t-1}+P_{t-1}H_t^T R_t^{-1}(y_t-h(\mu _{t-1})+H_t \mu _{t-1}) \nonumber \\&\qquad - P_{t-1}H_t^TR_t^{-1}H_t\mu _t \nonumber \\&\iff \nonumber \\ \mu _t&=({\mathbf {I}} + P_{t-1}H_t^TR_t^{-1}H_t)^{-1}(\mu _{t-1} \nonumber \\&\qquad +P_{t-1}H_t^T R_t^{-1}(y_t-h(\mu _{t-1}) +H_t \mu _{t-1}))\nonumber \\&=(P_{t-1}^{-1}+ H_t^TR_t^{-1}H_t)^{-1}P_{t-1}^{-1}(\mu _{t-1} \nonumber \\&\qquad +P_{t-1}H_t^T R_t^{-1}(y_t-h(\mu _{t-1})+H_t \mu _{t-1}))\nonumber \\&=\mu _{t-1}+(P_{t-1}^{-1}+ H_t^TR_t^{-1}H_t)^{-1}(-H_t^TR_t^{-1}H_t\mu _{t-1}\nonumber \\&\qquad +H_t^T R_t^{-1}(y_t-h(\mu _{t-1})+H_t \mu _{t-1}))\nonumber \\&=\mu _{t-1}+P_tH_t^TR_t^{-1}(y_t-h(\mu _{t-1}))\nonumber \\&=\mu _{t-1}-P_t\nabla _\theta \ell _t(\mu _{t-1}) \nonumber \\&=\mu _{t-1}-P_t{\mathbf {E}}_{q_{t-1}}[\nabla _\theta \tilde{\ell _t}(\theta )] \end{aligned}$$

(118)

$$\begin{aligned}&=\mu _{t-1}+K_t(y_t-h(\mu _{t-1})). \end{aligned}$$

(119)

From (50) and (118), we deduce that R-VGA is equivalent to the explicit scheme.

From (50) and (119), we deduce that R-VGA is equivalent to the extended Kalman filter using the same formula (110) and (111) as in the linear case. The equivalence between the extended Kalman filter and the natural gradient is already known (Ollivier 2018), we recall the main argument. In the update Eq. (50), the information matrix $P_t^{-1}$ is of growing size as long as we observe new data. If we pose $J_t=\frac{1}{t+1}P_t^{-1}$, we can reformulate the update (50) as a moving average:

$$\begin{aligned} J_t&=\frac{t}{t+1}J_{t-1} + \frac{1}{t+1} H_t^TR_t^{-1}H_t \end{aligned}$$

(120)

$$\begin{aligned}&=\frac{t}{t+1}J_{t-1} + \frac{1}{t+1} {\mathbf {E}}_y\left[ \frac{\partial \log p(y|\theta )}{\partial \theta } \frac{\partial \log p(y|\theta )}{\partial \theta }^T \right] \Bigr |_{\begin{array}{c} \mu _{t-1} \end{array}} \\&=\frac{t}{t+1}J_{t-1} - \frac{1}{t+1} {\mathbf {E}}_y\left[ \frac{\partial ^2 \log p(y|\theta )}{\partial \theta ^2}\right] \Bigr |_{\begin{array}{c} \mu _{t-1} \end{array}} \nonumber \\&=\frac{t}{t+1}J_{t-1} + \frac{1}{t+1} {\mathbf {E}}_y\left[ \nabla _\theta ^2 \ell _t(\theta )\right] \Bigr |_{\begin{array}{c} \mu _{t-1} \end{array}} \nonumber , \end{aligned}$$

(121)

which is the Fisher matrix update. The derivation from (120) to (121) is not obvious. Martens (2020) introduces it in the context of the generalized Gauss–Newton. To better understand where it comes from, we rather use the proof proposed in Ollivier (2018). Using the relation $\frac{\partial \log p(y|\theta )}{\partial \eta }=y-{\bar{y}}$ which holds for exponential families, we can write:

$$\begin{aligned} \frac{\partial {\bar{y}}}{\partial \theta }R_t^{-1}\frac{\partial {\bar{y}}}{\partial \theta }^T&=\frac{\partial {\bar{y}}}{\partial \theta }R_t^{-1}R_tR_t^{-1}\frac{\partial {\bar{y}}}{\partial \theta }^T\\&=\frac{\partial {\bar{y}}}{\partial \theta }\frac{\partial \eta }{\partial {\bar{y}}}R_t\frac{\partial \eta }{\partial {\bar{y}}}^T\frac{\partial {\bar{y}}}{\partial \theta }^T=\frac{\partial \eta }{\partial \theta }R_t\frac{\partial \eta }{\partial \theta }^T\\&=\frac{\partial \eta }{\partial \theta }{\mathbf {E}}_y \left[ (y-{\bar{y}})(y-{\bar{y}})^T\right] \frac{\partial \eta }{\partial \theta }^T\\&=\frac{\partial \eta }{\partial \theta }{\mathbf {E}}_y\left[ \frac{\partial \log p(y|\theta )}{\partial \eta } \frac{\partial \log p(y|\theta )}{\partial \eta }^T\right] \frac{\partial \eta }{\partial \theta }^T\\&={\mathbf {E}}_y\left[ \frac{\partial \log p(y|\theta )}{\partial \theta } \frac{\partial \log p(y|\theta )}{\partial \theta }^T\right] . \end{aligned}$$

$\square$

1.4 Output uncertainty assessment

Fig. 11

Iso-probabilities of the outputs in function of the inputs for the R-VGA implicit and explicit schemes (left), the EKF, QKF (middle), and the Laplace (right). We have considered $\sigma _0=10$ and $\mu _0=0$ with $N=300$ and $s=5$

Fig. 12

Values of the function $F_{\alpha _0,\nu _0,y}$ for $\alpha _0=100$, $\nu _0=300$ and $y=0$. In the upper row, we show the initial value $(\alpha _0, \nu _0)$ which is located on the bound of the domain (blue dot) and the optimal value (red dot)

Fig. 13

Bounds for the roots $\alpha$, $\nu$ of F through the proposed iterative scheme (in green) for $\nu _0=1, 5$, and 10. Iteration on F for different initial conditions is shown in red and blue

Fig. 14

$s=2$. Upper row: 2D synthetic dataset (left) and KL divergence. Lower row: confidence ellipsoids for the R-VGA implicit and explicit versions (left) and the EKF and QKF (right) at different times (final $t=N$ in black). The batch Laplace covariance is shown in red and is considered accurate

Fig. 15

Same as Fig. 14 in the case where $s=5$

Fig. 16

Same as Fig. 14 in the case where $s=10$

Given an unseen input, a prediction based on the estimated distribution $q_t$ may either be obtained through the maximum a posteriori (MAP) estimate of the parameter, that is, $P[y|x]=\sigma (x^T\theta ^*)$, or it may be obtained using the entire (Bayesian) distribution, that is, $P[y|x]={\mathbf {E}}[y|x]={\mathbf {E}}_{q \sim {{\mathcal {N}}}(\mu ^*,P^*)}[\sigma (x^T\theta )] \approx \sigma (k x^T\mu ^*)$ with $k=\frac{\beta }{\sqrt{ x^TP^*x +\beta ^2}}$. Because of this, the latter is less confident than the former MAP based-approach. Indeed, we have the following relation for the sigmoid:

$$\begin{aligned} |{\mathbf {E}}_q[\sigma (x)]-0.5|< |\sigma ({\mathbf {E}}_q[x])-0.5| \text { for } k<1. \end{aligned}$$

(122)

This relation comes from the fact the sigmoid is convex for $x<0$ and concave for $x>0$. The prediction based on the Bayesian approach is shown in Fig. 11 where we have drawn the iso-probabilities of the outputs in function of the inputs. On this separable dataset, the Laplace gives low probabilities prediction for the unseen inputs, whereas the QKF tends to predict with high probabilities. The R-VGA gives prediction probabilities between both of them.

1.5 Details on the fixed point method

The roots of F are the fixed point of the function ${\tilde{F}}$ defined by:

$$\begin{aligned} {\tilde{F}}_{\alpha _0,\nu _0,y}:{\mathbb {R}}^2&\rightarrow {\mathbb {R}}^2&\end{aligned}$$

(123)

$$\begin{aligned} (\alpha ,\nu )&\rightarrow {\tilde{F}}(\alpha ,\nu )=({\tilde{f}},{\tilde{g}})&\end{aligned}$$

(124)

where:

$$\begin{aligned} {\tilde{f}}(\alpha ,\nu )&=-\nu _0\sigma (\alpha k(\nu ) ) + \alpha _0 +\nu _0 y \end{aligned}$$

(125)

$$\begin{aligned} {\tilde{g}}(\alpha ,\nu )&=\frac{\nu _0}{1+\nu _0 k(\nu ) \sigma ^\prime (\alpha k(\nu ) )}. \end{aligned}$$

(126)

Function F is displayed in Fig. 12.

We found that ${\tilde{F}}$ is not contractive, so that fixed point iterations shall oscillate and do not converge, as shown in Fig. 13. However, we can further restrict the admissible domain $[\alpha _\mathrm{min}, \alpha _\mathrm{max}] \times [\nu _\mathrm{min}, \nu _\mathrm{max}]$ in which the searched point lies as follows.

Coarse bounds are given by:

$$\begin{aligned} \alpha _\mathrm{min}&=\alpha _0 + \nu _0(y-1) \le \alpha \le \alpha _0 + \nu _0y=\alpha _\mathrm{max} \end{aligned}$$

(127)

$$\begin{aligned} \nu _\mathrm{min}&=\nu _0(1-\frac{\nu _0}{4+\nu _0}) \le \nu \le \nu _0= \nu _\mathrm{max}. \end{aligned}$$

(128)

Using the fact that $0 \le k(\nu _\mathrm{max}) \le k(\nu ) \le k(\nu _\mathrm{min})$, the first inequality (127) gives:

$$\begin{aligned}&a_1 \le \alpha k(\nu ) \le a_2 \end{aligned}$$

(129)

$$\begin{aligned}&\text {where:} \nonumber \\&\alpha _\mathrm{min} \ge 0\Rightarrow a_1 =\alpha _\mathrm{min} k(\nu _\mathrm{max}),\nonumber \\ {}&\alpha _\mathrm{min}<0 \Rightarrow a_1=\alpha _\mathrm{min} k(\nu _\mathrm{min}) \end{aligned}$$

(130)

$$\begin{aligned}&\alpha _\mathrm{max} \ge 0\Rightarrow a_2 =\alpha _\mathrm{max} k(\nu _\mathrm{min}),\nonumber \\&\alpha _\mathrm{max}<0 \Rightarrow a_2=\alpha _\mathrm{max} k(\nu _\mathrm{max}). \end{aligned}$$

(131)

And we find the following new bound for $\alpha$:

$$\begin{aligned} \alpha _0 + \nu _0y-\nu _0 \sigma (a_2) \le \alpha \le \alpha _0 + \nu _0y-\nu _0 \sigma (a_1). \end{aligned}$$

(132)

For the second inequality (128), we use (129) to bound:

$$\begin{aligned} b_1\le \sigma (\alpha k(\nu ))(1-\sigma (\alpha k(\nu )))=\sigma ^{\prime }(\alpha k(\nu ))\le b_2, \end{aligned}$$

(133)

where $b_1$ and $b_2$ depend on the sign of $a_1a_2$:

$$\begin{aligned} a_1a_2&>0 \Rightarrow b_1=\min (\sigma ^{\prime }(a_1),\sigma ^{\prime }(a_2)) \text { and } \nonumber \\&\quad b_2=\max (\sigma ^{\prime }(a_1),\sigma ^{\prime }(a_2)) \end{aligned}$$

(134)

$$\begin{aligned} a_1a_2&\le 0 \Rightarrow b_1=\min (\sigma ^{\prime }(a_1),\sigma ^{\prime }(a_2)) \text { and } b_2=1/4. \end{aligned}$$

(135)

And we find the following new bound for $\nu$:

$$\begin{aligned}&\frac{\nu _0}{1+\nu _0k(\nu _{min})b_2} \le \nu \le \frac{\nu _0}{1+\nu _0k(\nu _{max})b_1}. \end{aligned}$$

(136)

This scheme can be iterated to restrict the search domain.

For moderate values of $\nu _0$, that is, moderately uncertain prior, the domain shrinks fast. But for highly uncertain priors it does not, as shown in Fig. 13, hence the need to resort to a 2D optimization algorithm.

1.6 Influence of separability of the dataset

We plot here results that reflect the sensitivity to the separability factor s with $s=2$ (Fig. 14), $s=5$ (Fig. 15), and $s=10$ (Fig. 16). The evolution of the ellipsoids over the iterations is also displayed. We see RVG-A consistently yields good performance.