Table 1.

NEBLS Estimation Formulas for a Variety of Observation Processes, as Listed in the Left Column.

Observation Process	Observation Density: PY|X(y|x)	Numerator of Estimator: L{PY}(y)
General discrete	A (matrix)	(A ∘ X ∘ A−1)PY(y)
Additive:
General (section 6.1)	PW(y − x)	$\mathit{y}{P}_Y\left(\mathit{y}\right)-{\mathcal{F}}^{-1}\left\{i{\nabla }_w\ln \left(\widehat{P_W}\left(\omega \right)\right)\widehat{P_Y}\right(\omega \left)\right\}\left(\mathit{y}\right)$
Gaussian (Miyasawa, 1961; Stein, 1981*)	$\frac{\exp -\frac{1}{2}{(\mathbf{y}-\mathbf{x}-\mu )}^T{\wedge }^{-1}(\mathbf{y}-\mathbf{x}-\mu )}{\sqrt{\mid 2\pi \wedge \mid }}$	(y − μ)PY(y) + Λ ∇y(y)
Laplacian	$\frac{1}{2\alpha }{e}^{-\mid (y-x)∕\alpha \mid }$	$y{P}_Y\left(y\right)+2{\alpha }^2\{P_W^{\prime }\star P_Y\}\left(y\right)$
Poisson	$\sum \frac{{\lambda }^k{e}^{-\lambda }}{k!}\delta (y-x-ks)$	yPY(y) − λs PY(y − s)
Cauchy	$\frac{1}{\pi }\left(\frac{\alpha }{{\left(\alpha \right(y-x\left)\right)}^2+1}\right)$	$y{P}_Y\left(y\right)-\{\frac{1}{2\pi \alpha y}\star P_Y\}\left(y\right)$
Uniform	$\{\begin{array}{cc}\frac{1}{2a},\hfill & \mid y-x\mid \le a\hfill \\ 0,\hfill & \mid y-x\mid >a\hfill \end{array}\phantom{\}}$	$y{P}_Y\left(y\right)+a{\sum }_k\mathrm{sgn}\left(k\right)P_Y(y-ak)-\frac{1}{2}\int P_Y\left(\tilde{y}\right)\mathrm{sgn}(y-\tilde{y})d\tilde{y}$
Random number of components	PW(y − x), where:  $W\sim {\sum }_{k=0}^K{W}_k$,  Wk i.i.d. (Pc), K ~ Poiss(λ)	yPY(y) − λ{(yPc) ⋆ PY}(y)
Gaussian scale mixture	$\mathit{Y}=\mathit{x}+\sqrt{Z}\mathit{U}$, U ~ N(0, Λ), Z ~ pZ	$\mathit{y}{P}_Y\left(\mathit{y}\right)+\left({\mathcal{F}}^{-1}\left\{\frac{{\int }_0^{\infty }z{p}_z\left(z\right)e^{-z\frac{1}{2}{\omega }^T\wedge \omega }dz}{{\int }_0^{\infty }{p}_z\left(z\right)e^{-z\frac{1}{2}{\omega }^T\wedge \omega }dz}\right\}\star \wedge {\nabla }_{\mathit{y}}{P}_Y\right)\left(y\right)$
Discrete exponential:
General (section B.1)   (Maritz & Lwin, 1989; Hwang, 1982*)	h(x)g(n)xn	$\frac{g\left(n\right)}{g(n+1)}{P}_Y(n+1)$
Inverse (Hwang, 1982)	h(x)g(n)x−n	$\frac{g\left(n\right)}{g(n-1)}{P}_Y(n-1)$
Poisson (section 6.1)   (Robbins, 1956; Hwang, 1982*)	$\frac{x^n{e}^{-x}}{n!}$	(n + 1)PY(n + 1)
Continuous exponential:
General (section B.3)  (Maritz & Lwin, 1989; Berger, 1980*)	h(x)g(y)eT(y)x	$\frac{g\left(y\right)}{T^{\prime }\left(y\right)}\frac{d}{dy}\left(\frac{P_Y\left(y\right)}{g\left(y\right)}\right)$
Inverse (Berger, 1980*)	h(x)g(y)eT(y)/x	$-g\left(y\right){\int }_y^{\infty }\frac{T^{\prime }\left(\tilde{y}\right)}{g\left(\tilde{y}\right)}{P}_Y\left(\tilde{y}\right)d\tilde{y}$
Laplacian scale mixture	$\frac{1}{x}{e}^{-\frac{y}{x}}$, x, y > 0	Pr{Y > y}
Power of fixed:
General (section B.3)	$\widehat{P_{Y\mid X}}(\omega \mid x)={\left[\widehat{P_W}\right(\omega \left)\right]}^x$	$\left({\mathcal{F}}^{-1}\left\{\frac{1}{i\frac{d}{dw}\ln \left(\widehat{P_W}\right(\omega \left)\right)}\right\}\star \left(y{P}_Y\right)\right)\left(y\right)$
Gaussian scale mixture	$\frac{1}{\sqrt{2\pi x}}{e}^{-\frac{y^2}{2x}}$	EY{Y; Y > y}
Signal-dependent AWGN	$Y=ax+\sqrt{x}W$, $W\sim \mathcal{N}(0,1)$	$\mathrm{sgn}\left(a\right)\left(\left(e^{ax}{I}_{\{ax<0\}}\right)\star \left(y{P}_Y\right)\right)\left(y\right)$
Multiplicative α-stable	$Y=x^{\frac{1}{\alpha }}W$,  W α-stable	$\left({\mathcal{F}}^{-1}\left\{\frac{i}{\mathrm{sgn}\left(\omega \right){\mid \omega \mid }^{\alpha -1}}\right\}\star \left(y{P}_Y\right)\right)\left(y\right)$
Multiplicative lognormal (section B.4)	Y = xeW, W Gaussian	$e^{\frac{3}{2}{\sigma }^2}{P}_Y\left(e^{{\sigma }^2}y\right)y$
Uniform mixture (section B.5)	$\begin{array}{cc}\hfill \frac{1}{2x},\hfill & \hfill \mid y\mid \le x\hfill \\ \hfill 0,\hfill & \hfill \mid y\mid >x\hfill \end{array}$	∣y∣PY(y) + Pr{Y > ∣y∣}

Notes: Expressions in parentheses indicate the section containing the derivation and brackets contain the bibliographical references for operators L, with the asterisk denoting references for the parametric dual operator, L*. Middle column gives the measurement density (note that variable n replaces y for discrete measurements). Right column gives the numerator of the NEBLS estimator, L{P_Y}(y). The symbol * indicates convolution, a hat (e.g., $\widehat{P_W}$) indicates a Fourier transform, and ${\mathcal{F}}^{-1}$ is the inverse Fourier transform.