FREMnotes.16.lyx

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\begin_body

\begin_layout Section
Fisher Information for a Poisson variable
\end_layout

\begin_layout Standard
This is derivation of the fisher information for Poisson distributed variable
 
\begin_inset Formula $X$
\end_inset

 with mean 
\begin_inset Formula $\lambda$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
X\sim\textrm{Po}(n,\lambda)=p(n|\theta)=\frac{\lambda^{n}e^{-\lambda}}{n!}
\]

\end_inset


\end_layout

\begin_layout Subsection
Likelihood
\end_layout

\begin_layout Standard
Likelihood of the Poisson distributed variable with detection 
\begin_inset Formula $n_{k}$
\end_inset

 in K pixels: 
\begin_inset Formula 
\begin{equation}
l(\theta)=\prod_{k=1}^{K}l_{k}=\prod_{k=1}^{K}\frac{\lambda_{k}^{n_{k}}e^{-\lambda_{k}}}{n_{k}!}\label{eq:Likelihood of Poisson}
\end{equation}

\end_inset

where 
\begin_inset Formula $l_{k}(\theta)=p(n_{k}|\theta)$
\end_inset

 to emphasise the dependency on the parameter 
\begin_inset Formula $\theta$
\end_inset

.
\end_layout

\begin_layout Standard
Log-Likelihood:
\begin_inset Formula 
\[
\mathcal{L}=\sum_{k}\left(n_{k}\log\lambda_{k}-\lambda_{k}-\log n_{k}!\right)
\]

\end_inset


\end_layout

\begin_layout Subsection
Fisher Information
\end_layout

\begin_layout Standard
Fisher information:
\begin_inset Formula 
\begin{equation}
I(\theta)=-\mathbb{E}\left[\frac{\partial^{2}\mathcal{L}}{\partial\theta^{2}}\right]=\mathbb{E}\left[\left(\frac{\partial\mathcal{L}}{\partial\theta}\right)^{2}\right]=\mathbb{E}\left[\left(\sum_{k}\frac{\partial\log(l_{k})}{\partial\theta}\right)^{2}\right]=\mathbb{E}\left[\left(\sum_{k}\frac{1}{l_{k}}\frac{\partial l_{k}}{\partial\theta}\right)^{2}\right]\label{eq:Fisher Info Definition}
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{alignat*}{1}
I(\theta) & =\mathbb{E}\left[\left(\sum_{k}\frac{1}{l_{k}}\frac{\partial l_{k}}{\partial\theta}\right)\left(\sum_{m}\frac{1}{l_{m}}\frac{\partial l_{m}}{\partial\theta}\right)\right]\\
 & =\mathbb{E}\left[\sum_{k}\frac{1}{l_{k}^{2}}\left(\frac{\partial l_{k}}{\partial\theta}\right)^{2}\right]+\mathbb{E}\left[\sum_{k}\sum_{m\neq k}\frac{1}{l_{k}}\frac{\partial l_{k}}{\partial\theta}\frac{1}{l_{m}}\frac{\partial l_{m}}{\partial\theta}\right]
\end{alignat*}

\end_inset

as 
\begin_inset Formula $n_{k}$
\end_inset

 are iid then the second term can be expressed as 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align*}
\mathbb{E}\left[\sum_{k}\sum_{m\neq k}\frac{1}{l_{k}}\frac{\partial l_{k}}{\partial\theta}\frac{1}{l_{m}}\frac{\partial l_{m}}{\partial\theta}\right] & =\sum_{k}\sum_{m\neq k}\mathbb{E}_{k}\left[\frac{1}{l_{k}}\frac{\partial l_{k}}{\partial\theta}\right]\mathbb{E}_{m}\left[\frac{1}{l_{m}}\frac{\partial l_{m}}{\partial\theta}\right]
\end{align*}

\end_inset

where
\begin_inset Formula 
\[
\mathbb{E}_{k}\left[f(n_{k})\right]=\sum_{n_{k}\geq0}p(n_{k}|\theta)f(n_{k})
\]

\end_inset

But 
\begin_inset Formula 
\[
\mathbb{E}_{k}\left[\frac{1}{l_{k}}\frac{\partial l_{k}}{\partial\theta}\right]=\sum_{n_{k}}l_{k}\frac{1}{l_{k}}\frac{\partial l_{k}}{\partial\theta}=\sum_{n_{k}}\frac{\partial l_{k}}{\partial\theta}=\frac{\partial\sum_{n_{k}}l_{k}}{\partial\theta}=0
\]

\end_inset

as 
\begin_inset Formula $\sum_{n_{k}}l_{k}=\sum_{n_{k}}p(n_{k}|\theta)=1$
\end_inset

.
 The Fisher Information can then be expressed 
\begin_inset Formula 
\begin{align*}
I(\theta) & =\mathbb{E}\left[\sum_{k}\frac{1}{l_{k}^{2}}\left(\frac{\partial l_{k}}{\partial\theta}\right)^{2}\right]\\
 & =\sum_{k=1}^{k}\sum_{n_{k}\geq0}l_{k}\frac{1}{l_{k}^{2}}\left(\frac{\partial l_{k}}{\partial\theta}\right)^{2}\\
 & =\sum_{k=1}^{k}\sum_{n_{k}\geq0}\frac{1}{l_{k}}\left(\frac{\partial l_{k}}{\partial\theta}\right)^{2}
\end{align*}

\end_inset

Derivatives of likelihood Eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Likelihood of Poisson"

\end_inset

: 
\begin_inset Formula 
\[
\frac{\partial l_{k}}{\partial\theta}=\frac{l_{k}(n_{k}-\lambda_{k})}{\lambda_{k}}\frac{\partial\lambda_{k}}{\partial\theta}
\]

\end_inset

And we get:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{align*}
I(\theta) & =\sum_{k=1}^{k}\sum_{n_{k}\geq0}\frac{l_{k}(n_{k}-\lambda_{k})^{2}}{\lambda_{k}^{2}}\left(\frac{\partial\lambda_{k}}{\partial\theta}\right)^{2}\\
 & =\sum_{k=1}^{k}\frac{1}{\lambda_{k}^{2}}\left(\frac{\partial\lambda_{k}}{\partial\theta}\right)^{2}\mathbb{E}_{k}\left[(n_{k}-\lambda_{k})^{2}\right]
\end{align*}

\end_inset

for Poisson 
\begin_inset Formula $\mathrm{var}(n)=\mathrm{mean}(n)=\lambda$
\end_inset

 gives 
\begin_inset Formula 
\[
\mathbb{E}_{k}\left[(n_{k}-\lambda_{k})^{2}\right]=\textrm{var}(n_{k})=\lambda_{k}
\]

\end_inset

and 
\begin_inset Formula 
\begin{equation}
I(\theta)=\sum_{k=1}^{K}\frac{1}{\lambda_{k}}\left(\frac{\partial\lambda_{k}}{\partial\theta}\right)^{2}\label{eq:Fisher Info for Poisson}
\end{equation}

\end_inset

This is the pixelised version (detection of the photons in K detectors -
 CCD camera and 
\begin_inset Formula $\lambda_{k}=\int_{C_{k}}\lambda(x)dx$
\end_inset

 where 
\begin_inset Formula $C_{k}$
\end_inset

 is an area of the pixels of the detector).
 
\end_layout

\begin_layout Standard
Non pixelised version 
\begin_inset CommandInset citation
LatexCommand cite
key "Ram2006"

\end_inset


\begin_inset Formula 
\[
I(\theta)=\int\frac{1}{\lambda(x)}\left(\frac{\partial\lambda(x)}{\partial\theta}\right)^{2}dx
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Newpage newpage
\end_inset


\end_layout

\begin_layout Subsection
Two sources separated by a distance 
\begin_inset Formula $d$
\end_inset


\begin_inset CommandInset label
LatexCommand label
name "sub:Two-sources-separated"

\end_inset


\end_layout

\begin_layout Standard
These are comment on Fisher Information estimation as described in 
\begin_inset CommandInset citation
LatexCommand cite
key "Ram2006"

\end_inset

.
\end_layout

\begin_layout Standard
For two sources separated by a distance 
\begin_inset Formula $d$
\end_inset

 we have a mean value of the intensity:
\begin_inset Formula 
\[
\lambda=\Lambda_{1}f_{1}+\Lambda_{2}f_{2}
\]

\end_inset

 where 
\begin_inset Formula $f_{i}$
\end_inset

 and 
\begin_inset Formula $\Lambda_{i}$
\end_inset

 is the response function and intensity, respectively, of the source 
\begin_inset Formula $i$
\end_inset

.
 For translationally invariant PSF and in-focus sources: 
\begin_inset Formula $f_{1}=q(x-\frac{d}{2})$
\end_inset

 and 
\begin_inset Formula $f_{2}=q(x+\frac{d}{2})$
\end_inset

 
\begin_inset Formula 
\[
\lambda(d)=\Lambda_{1}q(x-\frac{d}{2})+\Lambda_{2}q(x+\frac{d}{2})
\]

\end_inset

where 
\begin_inset Formula $q$
\end_inset

 is the PSF of the sources.
 For pixelised version (integral over pixel area 
\begin_inset Formula $C_{k}$
\end_inset

)
\begin_inset Formula 
\[
\lambda_{k}(d)=\Lambda_{1}\int_{C_{k}}q(x-\frac{d}{2})dx+\Lambda_{2}\int_{C_{k}}q(x+\frac{d}{2})dx
\]

\end_inset

so we get (as described in
\begin_inset CommandInset citation
LatexCommand citet
key "Ram2006"

\end_inset

)
\begin_inset Formula 
\begin{equation}
I(d)=\frac{1}{4}\sum_{k=1}^{K}\frac{\left(\Lambda_{1}\int_{C_{k}}\partial_{x}q(x-\frac{d}{2})dx-\Lambda_{2}\int_{C_{k}}\partial_{x}q(x+\frac{d}{2})dx\right)^{2}}{\Lambda_{1}\int_{C_{k}}q(x-\frac{d}{2})dx+\Lambda_{2}\int_{C_{k}}q(x+\frac{d}{2})dx}\label{eq:Fisher Info Pixelised - Ram}
\end{equation}

\end_inset


\end_layout

\begin_layout Subparagraph*
Limit 
\begin_inset Formula $d=0$
\end_inset


\end_layout

\begin_layout Standard
If 
\begin_inset Formula $\Lambda_{1}=\Lambda_{2}$
\end_inset

 then 
\begin_inset Formula $I(d=0)=0$
\end_inset

 which means 
\begin_inset Formula $\textrm{var}(d=0)\rightarrow\infty$
\end_inset

.
 (This does not hold for 
\begin_inset Formula $\Lambda_{1}\neq\Lambda_{2}$
\end_inset

).
 
\end_layout

\begin_layout Subparagraph*
Limit 
\begin_inset Formula $d\rightarrow\infty$
\end_inset


\end_layout

\begin_layout Standard
When sources are far apart then the mixing term in nominator in 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Fisher Info Pixelised - Ram"

\end_inset

 
\begin_inset Formula $\Lambda_{1}\Lambda_{2}\partial_{x}q(x-\frac{d}{2})\partial q(d+\frac{d}{2})=0$
\end_inset

 as the 
\begin_inset Formula $\partial_{x}q(x-\frac{d}{2})$
\end_inset

 (
\begin_inset Formula $q(x-\frac{d}{2})$
\end_inset

) and 
\begin_inset Formula $\partial_{x}q(x+\frac{d}{2})$
\end_inset

 (
\begin_inset Formula $q(x+\frac{d}{2})$
\end_inset

 ) do not have any overlap.
 The 
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Fisher Info Pixelised - Ram"

\end_inset

 then decomposes into two individual terms (sum of Fisher Information for
 localisation of individual sources.) 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{alignat*}{1}
I(d) & =\frac{1}{4}\sum_{k=1}^{K}\left[\frac{\left(\Lambda_{1}\int_{C_{k}}\partial_{x}q(x-\frac{d}{2})dx\right)^{2}}{\Lambda_{1}\int_{C_{k}}q(x-\frac{d}{2})dx}+\frac{\left(\Lambda_{2}\int_{C_{k}}\partial_{x}q(x+\frac{d}{2})dx\right)^{2}}{\Lambda_{2}\int_{C_{k}}q(x+\frac{d}{2})dx}\right]\\
 & =\frac{1}{4}\sum_{k=1}^{K}\frac{\left(\int_{C_{k}}\partial_{x}q(x)dx\right)^{2}}{\int_{C_{k}}q(x)dx}\left[\Lambda_{1}+\Lambda_{2}\right]
\end{alignat*}

\end_inset


\end_layout

\begin_layout Subparagraph
Limit 
\begin_inset Formula $\Lambda_{i}=0$
\end_inset


\end_layout

\begin_layout Standard
If 
\begin_inset Formula $\Lambda_{1}=0$
\end_inset

 or 
\begin_inset Formula $\Lambda_{2}=0$
\end_inset

 
\begin_inset Formula $I(d)\neq0$
\end_inset

.
 So the variance is finite even if one of the sources is not present.
\end_layout

\begin_layout Subsection
An alternative way to derive Fisher information for two sources separated
 by 
\begin_inset Formula $d$
\end_inset

:
\end_layout

\begin_layout Standard
This is a suggestion how to fix the problems with limits for Fisher Information
 derived above.
 This gives infinite variance when one of the sources is no present.
 Also fix weird behaviour of the 
\begin_inset Formula $I(d)$
\end_inset

 for 
\begin_inset Formula $d=0$
\end_inset

.
 
\end_layout

\begin_layout Standard
For two sources 
\begin_inset Formula $f_{1}=q(x-c_{1})$
\end_inset

 and 
\begin_inset Formula $f_{2}=q(x-c_{2})$
\end_inset

 we have 
\begin_inset Formula $\lambda=\Lambda_{1}f_{1}+\Lambda_{2}f_{2}$
\end_inset

.
 The distance between the two sources is 
\begin_inset Formula $d=c_{1}-c_{2}$
\end_inset

.
 This is a linear combination 
\begin_inset Formula $\bm{a}^{T}\cdot\bm{c}$
\end_inset

 of the variable 
\begin_inset Formula $\bm{c}=(c_{1},c_{2})^{T}$
\end_inset

 where 
\begin_inset Formula $\bm{a}=(1,-1)^{T}$
\end_inset

.
 The variance of 
\begin_inset Formula $d$
\end_inset

 is given by 
\begin_inset Formula 
\[
\textrm{var}(d)=\textrm{var}(\bm{a}^{T}\cdot\bm{c})=\bm{a}^{T}\cdot\bm{Q}\cdot\bm{a}=Q_{11}+Q_{22}-2Q_{12}
\]

\end_inset

 where 
\begin_inset Formula $\bm{Q}$
\end_inset

 is a covariance matrix 
\begin_inset Formula $\bm{Q}=\bm{I}^{-1}(\theta)$
\end_inset

 and 
\begin_inset Formula $\bm{I}(\theta)$
\end_inset

 is the Fisher information matrix (symmetric 
\begin_inset Formula $I_{12}=I_{21}$
\end_inset

) 
\begin_inset Formula 
\[
\bm{I}(\theta)=\left(\begin{array}{cc}
I_{11} & I_{12}\\
I_{12} & I_{22}
\end{array}\right)
\]

\end_inset

given by generalisation of Eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Fisher Info for Poisson"

\end_inset

 
\begin_inset Formula 
\begin{equation}
I_{ij}(\theta)=\sum_{k=1}^{K}\frac{1}{\lambda_{k}}\frac{\partial\lambda_{k}}{\partial\theta_{i}}\frac{\partial\lambda_{k}}{\partial\theta_{j}}\label{eq:Fisher Info gerneral lambda}
\end{equation}

\end_inset

The covariance matrix 
\begin_inset Formula $\bm{Q}$
\end_inset

 is then 
\begin_inset Formula 
\[
\bm{Q}=\bm{I}^{-1}(\theta)=\frac{1}{I_{11}I_{12}-I_{12}^{2}}\left(\begin{array}{cc}
I_{22} & -I_{12}\\
-I_{12} & I_{11}
\end{array}\right)
\]

\end_inset

and the variance of 
\begin_inset Formula $d=c_{1}-c_{2}$
\end_inset

 
\begin_inset Formula 
\begin{equation}
\textrm{var}(d)=(1,-1)^{T}\cdot\bm{Q}\cdot(1,-1)=\frac{I_{11}+I_{22}+2I_{12}}{I_{11}I_{12}-I_{12}^{2}}=\frac{p}{q}\label{eq:variance alternative}
\end{equation}

\end_inset

The individual terms of the Fisher Information matrix 
\begin_inset Formula 
\[
I_{11}=\sum_{k=1}^{K}\frac{1}{\lambda_{k}}\left(\frac{\partial\lambda_{k}}{\partial c_{1}}\right)^{2}=\sum_{k=1}^{K}\frac{\left(\Lambda_{1}q'_{k}(c_{1})\right)^{2}}{\Lambda_{1}q_{k}(c_{1})+\Lambda_{2}q_{k}(c_{2})}
\]

\end_inset

where 
\begin_inset Formula $q_{k}(c_{i})$
\end_inset

 is the pixelised version (pixel area 
\begin_inset Formula $\Gamma_{k}$
\end_inset

) of the PSF
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{alignat*}{1}
q_{k}(c_{i}) & =\int_{\Gamma_{k}}q(x-c_{i})dx\\
q'_{k}(c_{i}) & =\int_{\Gamma_{k}}\frac{\partial q(x-c_{i})}{\partial x}dx
\end{alignat*}

\end_inset

If we use 
\begin_inset Formula 
\[
f_{k}(c_{1},c_{2})=\Lambda_{1}q_{k}(c_{1})+\Lambda_{2}q_{k}(c_{2}),
\]

\end_inset

we get expressions for the terms of the Fisher information matrix: 
\begin_inset Formula 
\[
\]

\end_inset


\begin_inset Formula 
\begin{equation}
\begin{aligned}I_{ii} & =\Lambda_{i}^{2}\sum_{k=1}^{K}\frac{\left(q'_{k}(c_{i})\right)^{2}}{f_{k}(c_{1},c_{2})}\\
I_{ij} & =\Lambda_{i}\Lambda_{j}\sum_{k=1}^{K}\frac{q'_{k}(c_{i})q'_{k}(c_{j})}{f_{k}(c_{1},c_{2})}
\end{aligned}
\label{eq:Fisher Information alternative - Individual}
\end{equation}

\end_inset

 
\end_layout

\begin_layout Standard
Numerator 
\begin_inset Formula $p=I_{11}+I_{22}+2I_{12}$
\end_inset

 in Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:variance alternative"

\end_inset


\begin_inset Formula 
\[
p=\sum_{k=1}^{K}\frac{1}{f_{k}(c_{1},c_{2})}\left[\Lambda_{1}^{2}q'^{2}{}_{k}(c_{1})+\Lambda_{2}^{2}q'{}_{k}^{2}(c_{2})+2\Lambda_{1}\Lambda_{2}q'_{k}(c_{1})q'_{k}(c_{2})\right]
\]

\end_inset


\end_layout

\begin_layout Standard
The terms in the denominator 
\begin_inset Formula $r=I_{11}I_{22}-I_{12}^{2}$
\end_inset

 in Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:variance alternative"

\end_inset

 
\begin_inset Formula 
\[
\begin{alignedat}{1}I_{11}I_{22} & =\Lambda_{1}^{2}\Lambda_{2}^{2}\sum_{k,l}^{K}\frac{\left(q'_{k}(c_{1})q'_{l}(c_{2})\right)^{2}}{f_{k}(c_{1},c_{2})f_{l}(c_{1},c_{2})}\\
I_{12}^{2} & =\Lambda_{1}^{2}\Lambda_{2}^{2}\sum_{k,l}^{K}\frac{q'_{k}(c_{1})q'_{k}(c_{2})q'_{l}(c_{1})q'_{l}(c_{2})}{f_{k}(c_{1},c_{2})f_{l}(c_{1},c_{2})}
\end{alignedat}
\]

\end_inset


\end_layout

\begin_layout Subparagraph
Limit 
\begin_inset Formula $c_{1}\rightarrow c_{2},\,(d\rightarrow0)\Rightarrow q_{k}(c_{1})\rightarrow q_{k}(c_{2})$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
p=(\Lambda_{1}^{2}+\Lambda_{2}^{2}+2\Lambda_{1}\Lambda_{2})\sum_{k=1}^{K}\frac{q'{}_{k}^{2}(c)}{f_{k}(c,c)}
\]

\end_inset

which can be further simplified by explicitly writing 
\begin_inset Formula $f_{k}(c_{1},c_{2})$
\end_inset


\begin_inset Formula 
\[
p=(\Lambda_{1}+\Lambda_{2})\sum_{k=1}^{K}\frac{q'{}_{k}^{2}(c)}{q_{k}(c)}
\]

\end_inset


\begin_inset Formula $q_{k}$
\end_inset

 as a PSF is positive function, therefore the sum is not zero and 
\begin_inset Formula $p$
\end_inset

 is non-zero for any 
\begin_inset Formula $\Lambda_{1},\,\Lambda_{2}$
\end_inset

.
 
\end_layout

\begin_layout Standard
The two terms in the denominator in Eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:variance alternative"

\end_inset

 are identical for 
\begin_inset Formula $c_{1}=c_{2}$
\end_inset


\begin_inset Formula 
\[
I_{11}I_{22}=I_{12}^{2}=\frac{\Lambda_{1}^{2}\Lambda_{2}^{2}}{\left(\Lambda_{1}+\Lambda_{2}\right)^{2}}\sum_{k,l=1}^{K}\frac{q'_{k}(c)}{q_{k}(c)}\frac{q'_{l}(c)}{q_{l}(c)}
\]

\end_inset

and therefore 
\begin_inset Formula 
\begin{eqnarray*}
r=I_{11}I_{22}-I_{12}^{2}=\det\left[\bm{I}(\theta)\right] & \equiv & 0
\end{eqnarray*}

\end_inset

for any 
\begin_inset Formula $\Lambda_{i}$
\end_inset

.
 
\begin_inset Formula $\bm{I}(\theta)$
\end_inset

 is therefore a singular matrix for 
\begin_inset Formula $d=0$
\end_inset

 and inversion 
\begin_inset Formula $\bm{I}^{-1}(\theta)$
\end_inset

 does not exist for 
\begin_inset Formula $c_{1}=c_{2}$
\end_inset

, but the limit 
\begin_inset Formula $c_{1}\rightarrow c_{2},\,(d\rightarrow0)$
\end_inset

 gives 
\begin_inset Formula $p\neq0,\, r\rightarrow0$
\end_inset

 and 
\begin_inset Formula $\textrm{var}(d\rightarrow0)=\frac{p}{r}\rightarrow\infty$
\end_inset

.
 
\end_layout

\begin_layout Subparagraph*
Limit 
\begin_inset Formula $d\rightarrow\infty$
\end_inset


\end_layout

\begin_layout Standard
The cross term 
\begin_inset Formula $I_{ij}$
\end_inset

 in Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Fisher Information alternative - Individual"

\end_inset

 vanishes (
\begin_inset Formula $I_{ij}=0,\: i\neq j$
\end_inset

) because of the multiplication 
\begin_inset Formula $q'_{k}(c_{1})q'_{k}(c_{2})$
\end_inset

 which is zero for largely separated PSF with finite support (if the support
 of 
\begin_inset Formula $q_{k}(c_{1})$
\end_inset

 and 
\begin_inset Formula $q_{k}(c_{2})$
\end_inset

 do not have mutual overlap).
 Then from Eq.
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:variance alternative"

\end_inset

 
\begin_inset Formula 
\[
\textrm{var}(d)=\frac{1}{I_{11}}+\frac{1}{I_{22}}
\]

\end_inset

which is the sum of variances for estimation two separated sources.
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
I_{ii}=\Lambda_{i}^{2}\sum_{k=1}^{K}\frac{q_{k}'^{2}(c_{i})}{\Lambda_{1}q_{k}(c_{1})+\Lambda_{2}q_{k}(c_{2})}=\Lambda_{i}\sum_{k=1}^{K}\frac{q_{k}'^{2}(c_{i})}{q_{k}(c_{i})}
\]

\end_inset

if 
\begin_inset Formula $q_{k}(c_{i})$
\end_inset

 (and 
\begin_inset Formula $q_{k}'(c_{i})$
\end_inset

) have a finite support.
\end_layout

\begin_layout Standard
For non-pixelised version and for Gaussian approximation of the PSF (
\begin_inset Formula $q(x-a)\propto\exp\left(-\frac{(x-a)^{2}}{2\sigma^{2}}\right)$
\end_inset

 (with 
\begin_inset Formula $\sigma=\frac{\sqrt{2}}{2\pi}\frac{\lambda}{NA}$
\end_inset

 
\begin_inset CommandInset citation
LatexCommand cite
key "Zhang2007"

\end_inset

) we have 
\begin_inset Formula $q'(x)=\frac{1}{\sigma^{2}}xq(x)$
\end_inset

 and 
\begin_inset Formula $q'^{2}/q=\frac{1}{\sigma^{4}}x^{2}q$
\end_inset

 which gives 
\begin_inset Formula $\int q'^{2}/qdx=\frac{1}{\sigma^{4}}\int qx^{2}=\frac{1}{\sigma^{4}}\sigma^{2}=\frac{1}{\sigma^{2}}$
\end_inset

 and therefore 
\begin_inset Formula $I_{ii}=\frac{\Lambda_{i}}{\sigma^{2}}$
\end_inset

 
\begin_inset Formula 
\begin{alignat*}{1}
\textrm{var}(d\rightarrow\infty) & =\sigma^{2}\left(\frac{1}{\Lambda_{1}}+\frac{1}{\Lambda_{2}}\right)
\end{alignat*}

\end_inset


\end_layout

\begin_layout Subparagraph
Limit 
\begin_inset Formula $\Lambda_{i}=0,\ \Lambda_{j}\neq0$
\end_inset


\end_layout

\begin_layout Standard
then 
\begin_inset Formula $I_{ii}\equiv0$
\end_inset

 and 
\begin_inset Formula $I_{ij}\equiv0$
\end_inset

 and so 
\begin_inset Formula $\det(\bm{I}(\theta))\equiv0$
\end_inset

, and matrix is singular.
 In the limit 
\begin_inset Formula $\Lambda_{i}\rightarrow0$
\end_inset

 the variance Eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:variance alternative"

\end_inset

 
\begin_inset Formula $\textrm{var}(d)\rightarrow\infty$
\end_inset

.
 
\end_layout

\begin_layout Subsection
Time distribution of the intensities (blinking)
\end_layout

\begin_layout Standard
For likelihood dependent on parameter 
\begin_inset Formula $\Lambda_{t}$
\end_inset

 (T different time slices)
\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
l_{T}(d,\Lambda)=\prod_{k=1}^{K}\prod_{t=1}^{T}p(n_{k}|d,\Lambda_{t})p(\Lambda_{t})
\]

\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
\mathcal{L}_{T}(d,\Lambda)=\sum_{k=1}^{K}\sum_{t=1}^{T}\left[\log\left(l_{k}(d,\Lambda_{t})\right)+\log\left(p(\Lambda_{t})\right)\right]
\]

\end_inset

as 
\begin_inset Formula $p(\Lambda)$
\end_inset

 is not dependent on 
\begin_inset Formula $d$
\end_inset

 then
\begin_inset Formula 
\[
\frac{\partial^{2}\mathcal{L}_{T}(d,\Lambda)}{\partial d^{2}}=\sum_{t=1}^{T}\frac{\partial^{2}\mathcal{L}(d,\Lambda_{t})}{\partial d^{2}}
\]

\end_inset

but in the expectation equation Eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Fisher Info Definition"

\end_inset

 the time dependence appears as
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{alignat*}{1}
I_{T}(\theta) & =-\mathbb{E}_{T}\left[\sum_{t=1}^{T}\frac{\partial^{2}\mathcal{L}(d,\Lambda_{t})}{\partial d^{2}}\right]=\sum_{t=1}^{T}-\mathbb{E}_{T}\left[\frac{\partial^{2}\mathcal{L}(d,\Lambda_{t})}{\partial d^{2}}\right]=\sum_{t=1}^{T}\mathbb{E}_{T}\left[\left(\frac{\partial\mathcal{L}(d,\Lambda_{t})}{\partial d}\right)^{2}\right]\\
 & =\sum_{t=1}^{T}\int_{\Lambda_{t}}p(\Lambda_{t})I(\theta)d\Lambda_{t}=\sum_{t,k}\int_{\Lambda_{t}}p(\Lambda_{t})\frac{1}{\lambda_{k}(\Lambda_{t})}\left(\frac{\partial\lambda_{k}(\Lambda_{t})}{\partial d}\right)^{2}d\Lambda_{t}
\end{alignat*}

\end_inset


\end_layout

\begin_layout Subsection
Time distribution of the intensities - integrating out 
\begin_inset Formula $\Lambda$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula 
\[
l_{k}(d)=\int_{\Lambda}l_{k}(d,\Lambda)d\Lambda=\int_{\Lambda}p(n_{k}|d,\Lambda)p(\Lambda)d\Lambda
\]

\end_inset

for four state model of two sources: 
\begin_inset Formula $\left\{ (\Lambda_{1},0),(0,\Lambda_{2}),(\Lambda_{1},\Lambda_{2}),(0,0)\right\} $
\end_inset

 with uniform background intensity 
\begin_inset Formula $b$
\end_inset

: 
\begin_inset Formula $\lambda^{1}=\Lambda_{1}q(x-c_{1})+b$
\end_inset

, 
\begin_inset Formula $\lambda^{2}=\Lambda_{2}q(x-c_{2})+b$
\end_inset

, 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\lang english

\begin_inset Formula $\lambda^{3}=+\Lambda_{1}q(x-c_{1})+\Lambda_{2}q(x-c_{2})+b$
\end_inset

, 
\begin_inset Formula $\lambda^{4}=0+b$
\end_inset


\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit
\lang british
 with uniform distribution over these states
\begin_inset Formula 
\[
l_{k}(\theta)=\frac{1}{4}\sum_{i=1}^{4}\mathrm{Po}(\lambda_{k}^{i})
\]

\end_inset

derivatives 
\begin_inset Formula 
\[
\frac{\partial l_{k}}{\partial c_{p}}=\frac{1}{4}\sum_{i}\frac{\partial\mathrm{Po}(\lambda_{k}^{i})}{\partial c_{p}}=\frac{1}{4}\sum_{i}\left(\mathrm{Po}(\lambda_{k}^{i})\frac{(n_{k}-\lambda_{k}^{i})}{\lambda_{k}^{i}}\frac{\partial\lambda_{k}^{i}}{\partial c_{p}}\right)
\]

\end_inset

The Fisher information matrix diagonal entries:
\begin_inset Formula 
\begin{alignat}{1}
I_{pp}(\theta) & =\mathbb{E}\left[\left(\sum_{k=1}^{N}\frac{1}{l_{k}}\frac{\partial l_{k}}{\partial c_{p}}\right)^{2}\right]\nonumber \\
 & =\mathbb{E}\left[\left\{ \sum_{k=1}^{N}\left(\frac{1}{\sum_{j=1}^{4}\textrm{Po}(\lambda_{k}^{j})}\frac{\partial\sum_{i=1}^{4}\textrm{Po}(\textrm{\lambda}_{k}^{i})}{\partial c_{p}}\right)\right\} \left\{ \sum_{l=1}^{N}\left(\frac{1}{\sum_{j=1}^{4}\textrm{Po}(\lambda_{l}^{j})}\frac{\partial\sum_{i=1}^{4}\textrm{Po}(\textrm{\lambda}_{l}^{i})}{\partial c_{p}}\right)\right\} \right]\nonumber \\
 & =\sum_{k=1}^{N}\mathbb{E}_{k}\left[\frac{\left(\sum_{i=1}^{4}\frac{\partial\textrm{Po}(\textrm{\lambda}_{k}^{i})}{\partial c_{p}}\right)^{2}}{\left(\sum_{j=1}^{4}\textrm{Po}(\lambda_{k}^{j})\right)^{2}}\right]\label{eq:Fisher Info Integrated Out - diagonal entries}
\end{alignat}

\end_inset

as the cross terms (
\begin_inset Formula $k,\, l$
\end_inset

) in the sum (2nd row) are zeros: 
\begin_inset Formula 
\begin{alignat*}{1}
\mathbb{E}\left[\left(\frac{\sum_{i=1}^{4}\frac{\partial\textrm{Po}(\textrm{\lambda}_{k}^{i})}{\partial c_{p}}}{\sum_{j=1}^{4}\textrm{Po}(\lambda_{k}^{j})}\right)\left(\frac{\sum_{i=1}^{4}\frac{\partial\textrm{Po}(\textrm{\lambda}_{l}^{i})}{\partial c_{p}}}{\sum_{j=1}^{4}\textrm{Po}(\lambda_{l}^{j})}\right)\right] & =\mathbb{E}_{k}\left[\frac{\sum_{i=1}^{4}\frac{\partial\textrm{Po}(\textrm{\lambda}_{k}^{i})}{\partial c_{p}}}{\sum_{j=1}^{4}\textrm{Po}(\lambda_{k}^{j})}\right]\mathbb{E}_{l}\left[\frac{\sum_{i=1}^{4}\frac{\partial\textrm{Po}(\textrm{\lambda}_{l}^{i})}{\partial c_{p}}}{\sum_{j=1}^{4}\textrm{Po}(\lambda_{l}^{j})}\right]\\
 & =\sum_{i=1}^{4}\frac{\partial}{\partial c_{p}}\left(\sum_{n_{k}\geq0}\textrm{Po}(\textrm{\lambda}_{k}^{i})\right)\sum_{i=1}^{4}\frac{\partial}{\partial c_{p}}\left(\sum_{n_{k}\geq0}\textrm{Po}(\textrm{\lambda}_{l}^{i})\right)\\
 & =0
\end{alignat*}

\end_inset

Expressing the derivatives and the expectation from Eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Fisher Info Integrated Out - diagonal entries"

\end_inset

:
\begin_inset Formula 
\begin{alignat*}{1}
I_{pp}(\theta) & =\sum_{k=1}^{N}\mathbb{E}_{k}\left[\left\{ \frac{\sum_{i=1}^{4}\left(\mathrm{Po}(n_{k};\lambda_{k}^{i})\frac{(n_{k}-\lambda_{k}^{i})}{\lambda_{k}^{i}}\frac{\partial\lambda_{k}^{i}}{\partial c_{p}}\right)}{\sum_{j=1}^{4}\textrm{Po}(n_{k};\lambda_{k}^{j})}\right\} ^{2}\right]\\
 & =\frac{1}{4}\sum_{k=1}^{N}\sum_{n_{k}\geq0}\frac{\left\{ \sum_{i=1}^{4}\left(\mathrm{Po}(n_{k};\lambda_{k}^{i})\frac{(n_{k}-\lambda_{k}^{i})}{\lambda_{k}^{i}}\frac{\partial\lambda_{k}^{i}}{\partial c_{p}}\right)\right\} ^{2}}{\sum_{j=1}^{4}\textrm{Po}(n_{k};\lambda_{k}^{j})}
\end{alignat*}

\end_inset

For the four states model we have 
\begin_inset Formula $\lambda^{3}(c_{1},c_{2})=\lambda^{1}(c_{1})+\lambda^{2}(c_{2})-b$
\end_inset

 and so 
\begin_inset Formula $\frac{\partial\lambda^{3}}{\partial c_{p}}=\frac{\partial\lambda^{p}}{\partial c_{p}}$
\end_inset

 and 
\begin_inset Formula $\frac{\partial\lambda^{j}}{\partial c_{p}}=0,\, i\neq j$
\end_inset

 for 
\begin_inset Formula $p=\{1,2\},\: j=\{1,2,4\};$
\end_inset

 so 
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{alignat*}{1}
I_{pp}(\theta) & =\sum_{k=1}^{N}\left(\frac{\partial\lambda_{k}^{p}}{\partial c_{p}}\right)^{2}\mathbb{E}_{k}\left[\left\{ \frac{\sum_{i=\{p,3\}}\left(\mathrm{Po}(n_{k};\lambda_{k}^{i})\frac{(n_{k}-\lambda_{k}^{i})}{\lambda_{k}^{i}}\right)}{\sum_{j=1}^{4}\textrm{Po}(n_{k};\lambda_{k}^{j})}\right\} ^{2}\right]
\end{alignat*}

\end_inset


\end_layout

\begin_layout Standard
The Fisher information matrix off-diagonal entries:
\end_layout

\begin_layout Standard
\begin_inset Formula 
\begin{alignat}{1}
I_{pq}(\theta) & =\sum_{k=1}^{N}\mathbb{E}_{k}\left[\frac{\left(\sum_{i=1}^{4}\frac{\partial\textrm{Po}(\textrm{\lambda}_{k}^{i})}{\partial c_{p}}\right)\left(\sum_{l=1}^{4}\frac{\partial\textrm{Po}(\textrm{\lambda}_{k}^{l})}{\partial c_{q}}\right)}{\left(\sum_{j=1}^{4}\textrm{Po}(\lambda_{k}^{j})\right)^{2}}\right]\nonumber \\
 & =\sum_{k=1}^{N}\left(\frac{\partial\lambda_{k}^{p}}{\partial c_{p}}\right)\left(\frac{\partial\lambda_{k}^{q}}{\partial c_{q}}\right)\mathbb{E}_{k}\left[\frac{\left(\sum_{i=\{p,3\}}\mathrm{Po}(n_{k};\lambda_{k}^{i})\frac{(n_{k}-\lambda_{k}^{i})}{\lambda_{k}^{i}}\right)\left(\sum_{i=\{q,3\}}\mathrm{Po}(n_{k};\lambda_{k}^{i})\frac{(n_{k}-\lambda_{k}^{i})}{\lambda_{k}^{i}}\right)}{\left(\sum_{j=1}^{4}\textrm{Po}(n_{k};\lambda_{k}^{j})\right)^{2}}\right]\label{eq:Fisher Info Integrated Out-off diagonal entries}
\end{alignat}

\end_inset


\end_layout

\begin_layout Paragraph
Limit 
\begin_inset Formula $d\rightarrow0$
\end_inset


\end_layout

\begin_layout Standard
When 
\begin_inset Formula $c^{1}=c^{2}$
\end_inset

 then 
\begin_inset Formula $\lambda^{1}=\lambda^{2}$
\end_inset

 and 
\begin_inset Formula $\frac{\partial\text{Po}(\lambda^{1})}{\partial c^{1}}=\frac{\partial\text{Po}(\lambda^{2})}{\partial c^{2}}$
\end_inset

.
 Then all entries in 
\begin_inset Formula $I_{pq}$
\end_inset

 are equal and the matrix is singular.
 For the limit 
\begin_inset Formula $d\rightarrow0$
\end_inset

 the determianat 
\begin_inset Formula $\det(\bm{I})\rightarrow0$
\end_inset

 and the variance 
\begin_inset Formula $\text{var}(d)\rightarrow\infty$
\end_inset

.
\end_layout

\begin_layout Paragraph
Limit 
\begin_inset Formula $d\rightarrow\infty$
\end_inset


\end_layout

\begin_layout Standard
Sources are far apart and 
\begin_inset Formula $\lambda^{1}$
\end_inset

 and 
\begin_inset Formula $\lambda^{2}$
\end_inset

 do not have a common overlap.
 For 
\begin_inset Formula $k'$
\end_inset

 where 
\begin_inset Formula $\frac{\partial\lambda_{k'}^{p}}{\partial c_{p}}\neq0,\,\frac{\partial\lambda_{k'}^{q}}{\partial c_{p}}\equiv0$
\end_inset

 and 
\begin_inset Formula $\text{Po}(n_{k'},\lambda_{k'}^{3})=\mbox{Po}(n_{k'},\lambda_{k'}^{1})$
\end_inset

.
 From Eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Fisher Info Integrated Out - diagonal entries"

\end_inset

 the cross terms vanishes (
\begin_inset Formula $I_{pq}=0$
\end_inset

 because 
\begin_inset Formula $\frac{\partial\text{Po}(\lambda^{p})}{\partial c_{p}}\frac{\partial\text{Po}(\lambda^{q})}{\partial c_{q}}=0$
\end_inset

).
 For bacground free images (
\begin_inset Formula $b=0$
\end_inset

) the diagonal elements 
\begin_inset Formula 
\begin{alignat*}{1}
I_{pp} & =\sum_{k=1}^{N}\mathbb{E}_{k}\left[\frac{\left(2\frac{\partial\textrm{Po}(\textrm{\lambda}_{k}^{p})}{\partial c_{p}}\right)^{2}}{\left(2\textrm{Po}(\lambda_{k}^{p})\right)^{2}}\right]\\
 & =\sum_{k=1}^{N}\mathbb{E}_{k}\left[\frac{\left(\textrm{Po}(\textrm{\lambda}_{k}^{p})\frac{\left(n_{k}-\lambda_{k}^{p}\right)}{\lambda_{k}^{p}}\frac{\partial\lambda_{k}^{p}}{\partial c_{p}}\right)^{2}}{\left(\textrm{Po}(\lambda_{k}^{p})\right)^{2}}\right]\\
 & =\sum_{k=1}^{N}\left(\frac{1}{\lambda_{k}^{p}}\frac{\partial\textrm{\lambda}_{k}^{p}}{\partial c_{p}}\right)^{2}\mathbb{E}_{k}\left[\left(n_{k}-\lambda_{k}^{p}\right)^{2}\right]\\
 & =\sum_{k=1}^{N}\left(\frac{1}{\lambda_{k}^{p}}\frac{\partial\textrm{\lambda}_{k}^{p}}{\partial c_{p}}\right)^{2}\frac{1}{4}\sum_{n_{k}\geq0}\left(\sum_{i=1}^{4}\mbox{Po}(\lambda_{k}^{i})\left(n_{k}-\lambda_{k}^{p}\right)^{2}\right)\\
 & =\sum_{k=1}^{N}\left(\frac{1}{\lambda_{k}^{p}}\frac{\partial\textrm{\lambda}_{k}^{p}}{\partial c_{p}}\right)^{2}\frac{1}{4}\sum_{n_{k}\geq0}\left(2\mbox{Po}(\lambda_{k}^{p})\left(n_{k}-\lambda_{k}^{p}\right)^{2}\right)\\
 & =\sum_{k=1}^{N}\left(\frac{1}{\lambda_{k}^{p}}\frac{\partial\textrm{\lambda}_{k}^{p}}{\partial c_{p}}\right)^{2}\frac{1}{2}\lambda_{k}^{p}\\
 & =\frac{1}{2}\sum_{k=1}^{N}\frac{1}{\lambda_{k}^{p}}\left(\frac{\partial\textrm{\lambda}_{k}^{p}}{\partial c_{p}}\right)^{2}
\end{alignat*}

\end_inset

which is the Eq.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Fisher Info for Poisson"

\end_inset

 (up to the factor 
\begin_inset Formula $1/2$
\end_inset

).
 The factor 1/2 comes from the fact that the source appears only in 50%
 of the observations.
\end_layout

\begin_layout Standard
When the background is present 
\begin_inset Formula 
\begin{alignat*}{1}
I_{pp} & =\sum_{k=1}^{N}\mathbb{E}_{k}\left[\frac{\left(2\frac{\partial\textrm{Po}(\textrm{\lambda}_{k}^{p})}{\partial c_{p}}\right)^{2}}{\left(2\textrm{Po}(\lambda_{k}^{p})+2\mbox{Po}(b)\right)^{2}}\right]\\
 & =\sum_{k=1}^{N}\mathbb{E}_{k}\left[\frac{\left(\textrm{Po}(\textrm{\lambda}_{k}^{p})\frac{\left(n_{k}-\lambda_{k}^{p}\right)}{\lambda_{k}^{p}}\frac{\partial\lambda_{k}^{p}}{\partial c_{p}}\right)^{2}}{\left(\textrm{Po}(\lambda_{k}^{p})+\mbox{Po}(b)\right)^{2}}\right]\\
 & =\sum_{k=1}^{N}\left(\frac{1}{\lambda_{k}^{p}}\frac{\partial\textrm{\lambda}_{k}^{p}}{\partial c_{p}}\right)^{2}\mathbb{E}_{k}\left[\left(\frac{\mbox{Po}(\lambda_{k}^{p})}{\mbox{Po}(\lambda_{k}^{p})+\mbox{Po}(b)}\right)^{2}\left(n_{k}-\lambda_{k}^{p}\right)^{2}\right]\\
 & =\sum_{k=1}^{N}\left(\frac{1}{\lambda_{k}^{p}}\frac{\partial\textrm{\lambda}_{k}^{p}}{\partial c_{p}}\right)^{2}\mathbb{E}_{k}\left[\left(1-\frac{\mbox{Po}(b)}{\mbox{Po}(\lambda_{k}^{p})+\mbox{Po}(b)}\right)^{2}\left(n_{k}-\lambda_{k}^{p}\right)^{2}\right]\\
 & =\frac{1}{2}\sum_{k=1}^{N}\frac{1}{\lambda_{k}^{p}}\left(\frac{\partial\textrm{\lambda}_{k}^{p}}{\partial c_{p}}\right)^{2}-S
\end{alignat*}

\end_inset

where
\begin_inset Formula 
\[
S=\sum_{k=1}^{N}\left(\frac{1}{\lambda_{k}^{p}}\frac{\partial\textrm{\lambda}_{k}^{p}}{\partial c_{p}}\right)^{2}\mathbb{E}_{k}\left[\left\{ \frac{2\mbox{Po}(b)}{\mbox{Po}(\lambda_{k}^{p})+\mbox{Po}(b)}+\left(\frac{2\mbox{Po}(b)}{\mbox{Po}(\lambda_{k}^{p})+\mbox{Po}(b)}\right)^{2}\right\} \left(n_{k}-\lambda_{k}^{p}\right)^{2}\right].
\]

\end_inset

This term is positive and therefore reduces 
\begin_inset Formula $I_{pp}$
\end_inset

.
\end_layout

\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "Ram,Zhang"
options "apalike"

\end_inset


\end_layout

\end_body
\end_document