# Information bounds and MCMC parameters estimation for the pile-up model with application to fluorescence measurements

Tabea Rebafka, François Roueff, Antoine Souloumiac

► To cite this version:

Tabea Rebafka, François Roueff, Antoine Souloumiac. Information bounds and MCMC parameters estimation for the pile-up model with application to fluorescence measurements. 7th World Congress in Probability and Statistics, Bernoulli Society (BS); Institute of Mathematical Statistics (IMS), Jul 2008, Singapour, Singapore. cea-05515044

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Information bounds and MCMC parameter
estimation in the pile-up model

Tabea Rebafka

Joint work with F. Roueff & A. Souloumiac

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>
## PILE-UP MODEL

* Application in time-resolved fluorescence

* Statistical model

## INFORMATION BOUNDS

* Cramér-Rao bound

* Optimal pile-up model

## ESTIMATION

* Gibbs sampler

* Simulation results

<figure><img src="image_5.png" /></figure>
<figure><img src="image_6.png" /></figure>

* Fluorescence Lifetime: Duration from exciting molecules up to the emission of fluorescence photons [Lakowicz, 1999; Valeur, 2002]

* Characteristic for molecules

* Exponential distribution

<figure><img src="image_7.png" /></figure>
● Pile-up affected observation Z

$$
Z = \begin{cases} \min\{Y_1, \dots, Y_N\} & \text{if } N > 0 \\ \infty & \text{if } N = 0. \end{cases}
$$

where {$Y_k, k \ge 1$} are i.i.d. r.v. with density $f$ and
$N$ is a r.v. taking its values in $\mathbb{N}$ independent of {$Y_k, k \ge 1$}

* Parametric setting: $f = f_0$ belongs to a parametric family of densities {$f_0, \theta \in \Theta$} with parameter set $\Theta \subset \mathbb{R}^d$. The distribution of $N$ belongs to a parametric family {$p_\lambda, \lambda \in \Lambda$} where $\Lambda \subset \mathbb{R}$ and $p_\lambda(k) = P_\lambda(N = k)$.

* The collection {$g_{\lambda,\theta}, (\lambda, \theta) \in \Lambda \times \Theta$} is called the pile-up model associated with {$f_0, \theta \in \Theta$} and {$p_\lambda, \lambda \in \Lambda$} where

$$
g_{\lambda,\theta}(z) = \begin{cases} f_0(z) \dot{M}(1 - F_0(z)) & \text{if } 0 < z < \infty \\ M(0) & \text{if } z = \infty \end{cases}
$$

with $M(u) = \mathbb{E}[u^N]$ and $\dot{M}(u) = \mathbb{E}[Nu^N]$ for $u \in [0, 1]$.

<figure><img src="image_8.png" /></figure>
● Pile-up affected observation Z

$$
Z = \begin{cases} \min\{Y_1, \dots, Y_N\} & \text{if } N > 0 \\ \infty & \text{if } N = 0. \end{cases}
$$

where {$Y_k, k \ge 1$} are i.i.d. r.v. with density $f$ and
$N$ is a r.v. taking its values in $\mathbb{N}$ independent of {$Y_k, k \ge 1$}

* Parametric setting: $f = f_{\theta_0}$ belongs to a parametric family of densities $\{f_{\theta}, \theta \in \Theta\}$ with parameter set $\Theta \subset \mathbb{R}^d$. The distribution of $N$ belongs to a parametric family $\{p_{\lambda}, \lambda \in \Lambda\}$ where $\Lambda \subset \mathbb{R}$ and $p_{\lambda}(k) = P_{\lambda}(N = k)$.

* The collection {$g_{\lambda,\theta}$, ($\lambda, \theta$) $\in \Lambda \times \Theta$} is called the **pile-up** model associated with {$f_{\theta}$, $\theta \in \Theta$} and {$p_{\lambda}$, $\lambda \in \Lambda$} where

$$
g_{\lambda,\theta}(z) = \begin{cases} f_{\theta}(z) \dot{M}(1 - F_{\theta}(z)) & \text{if } 0 < z < \infty \\ M(0) & \text{if } z = \infty \end{cases}
$$

with $M(u) = \mathbb{E}[u^N]$ and $\dot{M}(u) = \mathbb{E}[Nu^N]$ for $u \in [0, 1]$.
INFORMATION BOUNDS

* Cramér-Rao bound

* Optimal pile-up model

<figure><img src="image_9.png" /></figure>
* Hellinger differentiability of {$f_{\theta}, \theta \in \Theta$} is differentiability in $L^2$-norm of the square root of the density $\theta \mapsto \xi_{\theta} := \sqrt{f_{\theta}}$. Denote the Hellinger derivative by $\dot{\xi}_{\theta}$.

* Fisher information matrix $I_{\theta} = 4 \int \dot{\xi}_{\theta} \dot{\xi}_{\theta}^{\top} d\nu$.

* Theorem [Pollard, 2001]:
  Let {$f_{\theta}, \theta \in \Theta$} be Hellinger differentiable and let $I_{\theta_0}$ be nonsingular.
  Then for every estimator $T(X)$ satisfying $\sup_{\theta \in U_0} \mathbb{E}_{\theta}[T(X)^2] < \infty$ in a neighborhood $U_0$ of $\theta_0$,

$$ \mathrm{var}(T(X)) \geq \gamma_{\theta_0} I_{\theta_0}^{-1} \gamma_{\theta_0}, $$

where $\gamma_{\theta_0} = 2 \int \xi_{\theta_0} \dot{\xi}_{\theta_0} T d\nu$ is the derivative of $\theta \mapsto \mathbb{E}_{\theta}[T(X)]$ at $\theta_0$.

<figure><img src="image_10.png" /></figure>
Theroem

Let {$f_{\theta}, \theta \in \Theta$} be Hellinger differentiable w.r.t. $\mathcal{L}_+$ at $\theta_0$.
Then the pile-up model {$g_{\lambda,\theta}, (\lambda, \theta) \in \Lambda \times \Theta$} is Hellinger
differentiable w.r.t. $\overline{\mathcal{L}}_+$ at $(\lambda_0, \theta_0)$ if the following conditions on
{$p_{\lambda}, \lambda \in \Lambda$} hold

(i) $\mathbb{E}_{\lambda_0}[N^3] < \infty$,

(ii) $p_{\lambda}(1) > 0$,

(iii) the function $\lambda \mapsto \{p_{\lambda}(k)\}_{k \ge 1}$ is differentiable at $\lambda_0$ as a function defined on $\Lambda$ with values in the space of sequences

$$
\ell_1 = \{(u_k)_{k \in \mathbb{N}} : \sum_{k \ge 1} k |u_k| < \infty\},
$$

(iv) either (a) $p_{\lambda_0}(0) > 0$ and $\lambda \mapsto p_{\lambda}(0)$ is differentiable at $\lambda_0$ or (b) $p_{\lambda}(0) = 0$ for all $\lambda$ in a neighborhood of $\lambda_0$.

<figure><img src="image_11.png" /></figure>
* Assume that conditions (i)–(iv) hold and that {$f_\theta, \theta \in \Theta$} is Hellinger differentiable and that the Fisher information $I(\lambda_0, \theta_0)$ of the pile-up model is nonsingular.

* Denote the Cramér-Rao bound of the pile-up model by

$$ \textcolor{magenta}{CR}(\lambda_0, \theta_0) = I^{-1}(\lambda_0, \theta_0) = \begin{bmatrix} CR_{\lambda}(\lambda_0, \theta_0) & CR_{\lambda,\theta}(\lambda_0, \theta_0) \\ CR_{\lambda,\theta}^{T}(\lambda_0, \theta_0) & CR_{\theta}(\lambda_0, \theta_0) \end{bmatrix} $$

Theorem

Let $p_\lambda$ be the Poisson distribution with parameter $\lambda$.

Then $CR_\theta(\lambda_0, \theta_0) \sim \frac{M_{\theta_0}}{\lambda_0}$ as $\lambda_0 \to 0$.

<figure><img src="image_12.png" /></figure>
* Let {$f_\theta, \theta > 0$} be a scale family, i.e. $f_\theta(x) = \theta f_1(\theta x)$, for all $\theta > 0, x > 0$

* Then the Hellinger derivative $\dot{\zeta}_{\theta_0}$ of the pile-up model satisfies $\dot{\zeta}_{\theta_0}(x) = \frac{1}{\sqrt{\theta_0}} \zeta_1(\theta_0 x)$

* $CR_\theta(\lambda_0; \theta_0) = \theta_0^2 CR_\theta(\lambda_0, 1)$

## Consequence

$\arg\min_{\lambda_0} CR_\theta(\lambda_0, \theta_0) = \arg\min_{\lambda_0} CR_\theta(\lambda_0, 1)$.

The point of minimum $\lambda_0$ of $CR_\theta(\lambda_0, \theta_0)$ does not depend on $\theta_0$.

<figure><img src="image_13.png" /></figure>
* Let {$f_\theta, \theta > 0$} be a scale family, i.e. $f_\theta(x) = \theta f_1(\theta x)$, for all $\theta > 0, x > 0$

* Then the Hellinger derivative $\dot{\zeta}_{\theta_0}$ of the pile-up model satisfies $\dot{\zeta}_{\theta_0}(x) = \frac{1}{\sqrt{\theta_0}} \dot{\zeta}_1(\theta_0 x)$

* $CR_\theta(\lambda_0; \theta_0) = \theta_0^2 CR_\theta(\lambda_0, 1)$

Consequence

$\arg\min_{\lambda_0} CR_\theta(\lambda_0, \theta_0) = \arg\min_{\lambda_0} CR_\theta(\lambda_0, 1)$.

The point of minimum $\lambda_0$ of $CR_\theta(\lambda_0, \theta_0)$ does not depend on $\theta_0$.

<figure><img src="image_14.png" /></figure>
* Let {$f_\theta, \theta > 0$} be a scale family, i.e. $f_\theta(x) = \theta f_1(\theta x)$, for all $\theta > 0, x > 0$

* Then the Hellinger derivative $\dot{\zeta}_{\theta_0}$ of the pile-up model satisfies $\dot{\zeta}_{\theta_0}(x) = \frac{1}{\sqrt{\theta_0}} \dot{\zeta}_1(\theta_0 x)$

* $CR_\theta(\lambda_0, \theta_0) = \theta_0^2 CR_\theta(\lambda_0, 1)$

Consequence

$\arg\min_{\lambda_0} CR_\theta(\lambda_0, \theta_0) = \arg\min_{\lambda_0} CR_\theta(\lambda_0, 1)$.

The point of minimum $\lambda_0$ of $CR_\theta(\lambda_0, \theta_0)$ does not depend on $\theta_0$.

<figure><img src="image_15.png" /></figure>
<figure><img src="image_16.png" /></figure>

Cramér-Rao bound $\lambda \mapsto CR_{\theta}(\lambda, 1)$ of $\theta$ for the pile-up model.
The minimum is achieved at $\lambda_{opt} \approx 1.3275$

<figure><img src="image_17.png" /></figure>
<figure><img src="image_18.png" /></figure>

Cramér-Rao bound $\lambda \mapsto CR(\lambda, \theta_0)$ for a two-component mixture.

$\lambda_0 \in [1, 2]$ remains a good choice of the parameter.

<figure><img src="image_19.png" /></figure>
ESTIMATION

* Gibbs sampler

* Simulation results

<figure><img src="image_20.png" /></figure>
* Exponential mixture density: $f_{\theta}(y) = \sum_{k=1}^{K} \alpha_k \nu_k e^{-\nu_k y}$ where $\sum_k \alpha_k = 1$ and $\nu_1 > \dots > \nu_K > 0$ (identifiability) and the order of the mixture K fixed and known

* $N \sim \text{Poisson}(\lambda_0)$

* Reparametrization [Gruet, Philippe and Robert, 1999]
  Exponential parameters: $\varphi = \nu_1, \tau_K = \frac{\nu_K}{\nu_{K-1}}, k = 2, \dots, K$
  Mixing weights: $q_k = \frac{\alpha_k}{1 - \sum_{j=1}^{k-1} \alpha_j}, k = 1, \dots, K-1$

* Noninformative prior

$$ \frac{1}{\lambda\varphi} \prod_{k=1}^{K-1} \mathbb{1}\{q_k \in [0, 1]\} \prod_{k=2}^{K} \mathbb{1}\{\tau_k \in [0, 1]\} $$

<figure><img src="image_21.png" /></figure>
DATA AUGMENTATION STEP

* Missing variables **B** = (B<sub>1</sub>, ..., B<sub>K</sub>)<sup>T</sup> where B<sub>k</sub> is the number of photons generated by the k-th mixture component

* Generation of the missing variables in two steps

* Generate the total number of photons *N* according to the conditional distribution of (*N*|*Z*, *θ*)

$$ (N|Z, \theta) \stackrel{d}{=} \begin{cases} \text{Poisson}(\lambda(1 - F_{\theta}(Z))) + 1 & \text{if } Z \in (0, T) \\ \text{Poisson}(\lambda(1 - F_{\theta}(T))) & \text{if } Z = T \end{cases} $$

* Generate **B** = (B<sub>1</sub>, ..., B<sub>K</sub>)<sup>T</sup> according to the conditional distribution (B|N, Z, θ).

$$ P(\mathbf{B} = (b_1, \dots, b_K) | N = n, Z) = \begin{cases} \frac{(n-1)! \prod_k \alpha_k^{b_k} \sum_k b_k \nu_k}{l_\theta(Z)(1-F_\theta(Z))^{n-1} \prod_k b_k!} e^{-Z \sum_k b_k \nu_k} & \text{for } n \ge 1, \text{ if } Z \in (0, T) \\ \frac{n! \prod_k \alpha_k^{b_k} \frac{b_k}{(1-F_\theta(Z))^n} \prod_k b_k!}{(1-F_\theta(Z))^n} e^{-T \sum_k b_k \nu_k} & \text{for } n \ge 0, \text{ if } Z = T \end{cases} $$

<figure><img src="image_22.png" /></figure>
GENERATION OF PARAMETERS

* Update of the posterior distributions: Gamma and Beta distributions and a mixture of Gammas restricted on [0, 1].

$$
\begin{align*}
\lambda |(\mathbf{Z}, \mathbf{B}, \theta^{(-\lambda)}) &\sim \Gamma \left( \sum_{i=1}^{M} \sum_{k=1}^{K} B_{ik}, M \right) \\
\phi |(\mathbf{Z}, \mathbf{B}, \theta^{(-\phi)}) &\sim \Gamma \left( \sum_{i=1}^{M} 1\{\sum_{k=1}^{K} B_{ik} > 0\}, \sum_{k=1}^{K} \tau_1 \dots \tau_k \sum_{i=1}^{M} B_{ik} Z_i \} \right) \\
q_i |(\mathbf{Z}, \mathbf{B}, \theta^{(-q_i)}) &\sim \text{Beta} \left( \sum_{i=1}^{M} B_{ii} + 1, \sum_{i=1}^{M} \sum_{k=i+1}^{K} B_{ik} + 1 \right) \\
\rho(\tau_i | \mathbf{Z}, \mathbf{B}, \theta^{(-\tau_i)}) &\propto \exp \left\{ -\phi \sum_{k=1}^{K} \tau_1 \dots \tau_k \sum_{i=1}^{M} B_{ik} Z_i \right\} \prod_{i:Z_i < T} \left( \sum_{k=1}^{K} \tau_1 \dots \tau_k B_{ik} \right) 1\{\tau_i \in [0, 1]\}
\end{align*}
$$

* Generate parameters $\lambda$, $\phi$, $q_i$ and $\tau_i$ from the updated posterior distributions

<figure><img src="image_23.png" /></figure>
● Exponential distribution, sample size 5,000

<table><thead><tr><th>true parameters</th><th>λ = 1</th><th>ν<sub>1</sub> = 1</th><th>λ = 1.32</th><th>ν<sub>1</sub> = 1</th><th>λ = 2</th><th>ν<sub>1</sub> = 1</th></tr></thead><tbody><tr><td>empirical mean</td><td>1.0015</td><td>0.9997</td><td>1.3218</td><td>0.9997</td><td>2.0004</td><td>0.9996</td></tr><tr><td>empirical covariance 10<sup>-3</sup></td><td>0.3358</td><td>-0.1267<br>0.5016</td><td>0.5728</td><td>-0.1700<br>0.4442</td><td>1.0894</td><td>-0.4012<br>0.4822</td></tr><tr><td>Cramér-Rao bound 10<sup>-3</sup></td><td>0.3394</td><td>-0.0988<br>0.4240</td><td>0.5392</td><td>-0.1613<br>0.4088</td><td>1.2480</td><td>-0.3812<br>0.4501</td></tr></tbody></table>

● Two-component mixture, sample size 5,000

<table><thead><tr><th>true parameters</th><th>λ = 1.5</th><th>α<sub>1</sub> = 0.3</th><th>ν<sub>1</sub> = 2</th><th>ν<sub>2</sub> = 0.5</th></tr></thead><tbody><tr><td>empirical mean</td><td>1.5006</td><td>0.3051</td><td>2.0306</td><td>0.5086</td></tr><tr><td>empirical covariance 10<sup>-3</sup></td><td>0.6158</td><td>-0.0371<br>3.1370</td><td>-0.9596<br>-14.7681<br>85.6508</td><td>-0.0345<br>-1.2015<br>5.5287<br>0.7102</td></tr><tr><td>Cramér-Rao bound 10<sup>-3</sup></td><td>0.6960</td><td>-0.1371<br>2.3448</td><td>-0.0180<br>-10.6867<br>60.3615</td><td>-0.0456<br>-1.0793<br>4.5698<br>0.7326</td></tr></tbody></table>

<figure><img src="image_24.png" /><figcaption>Tabea Rebafka Parameter estimation in the pile-up model</figcaption></figure>
## Exponential distribution, sample size 5,000

<table><thead><tr><th>true parameters</th><th>λ = 1</th><th>ν<sub>1</sub> = 1</th><th>λ = 1.32</th><th>ν<sub>1</sub> = 1</th><th>λ = 2</th><th>ν<sub>1</sub> = 1</th></tr></thead><tbody><tr><td>empirical mean</td><td>1.0015</td><td>0.9997</td><td>1.3218</td><td>0.9997</td><td>2.0004</td><td>0.9996</td></tr><tr><td>empirical covariance 10<sup>-3</sup></td><td>0.3358</td><td>-0.1267</td><td>0.5728</td><td>-0.1700</td><td>1.0894</td><td>-0.4012</td></tr><tr><td></td><td></td><td>0.5016</td><td></td><td>0.4442</td><td></td><td>0.4822</td></tr><tr><td>Cramér-Rao bound 10<sup>-3</sup></td><td>0.3394</td><td>-0.0988</td><td>0.5392</td><td>-0.1613</td><td>1.2480</td><td>-0.3812</td></tr><tr><td></td><td></td><td>0.4240</td><td></td><td>0.4088</td><td></td><td>0.4501</td></tr></tbody></table>

## Two-component mixture, sample size 5,000

<table><thead><tr><th>true parameters</th><th>λ = 1.5</th><th>α<sub>1</sub> = 0.3</th><th>ν<sub>1</sub> = 2</th><th>ν<sub>2</sub> = 0.5</th></tr></thead><tbody><tr><td>empirical mean</td><td>1.5006</td><td>0.3051</td><td>2.0306</td><td>0.5086</td></tr><tr><td>empirical covariance 10<sup>-3</sup></td><td>0.6158</td><td>-0.0371</td><td>-0.9596</td><td>-0.0345</td></tr><tr><td></td><td></td><td>3.1370</td><td>-14.7681</td><td>-1.2015</td></tr><tr><td></td><td></td><td></td><td>85.6508</td><td>5.5287</td></tr><tr><td></td><td></td><td></td><td></td><td>0.7102</td></tr><tr><td>Cramér-Rao bound 10<sup>-3</sup></td><td>0.6960</td><td>-0.1371</td><td>-0.0180</td><td>-0.0456</td></tr><tr><td></td><td></td><td>2.3448</td><td>-10.6867</td><td>-1.0793</td></tr><tr><td></td><td></td><td></td><td>60.3615</td><td>4.5698</td></tr><tr><td></td><td></td><td></td><td></td><td>0.7326</td></tr></tbody></table>

<figure><img src="image_25.png" /></figure>
* Two-component mixture

* Exponential parameters: $\nu_1 = 0.5$, $\nu_2 = 6$

* Mixing weights: $\alpha_1 = 1/3$, $\alpha_2 = 2/3$

<table><thead><tr><th colspan="6">Standard method (λ = 0.05)</th></tr><tr><th>nb of obs</th><th>1000</th><th>3000</th><th>5000</th><th>10000</th></tr></thead><tbody><tr><th>α<sub>1</sub></th><td>bias</td><td>0.0051</td><td>0.0040</td><td>0.0036</td><td>0.0028</td></tr><tr><th></th><td>var</td><td>0.0112</td><td>0.0035</td><td>0.0021</td><td>0.0010</td></tr><tr><th>ν<sub>1</sub></th><td>bias</td><td>0.0488</td><td>0.0143</td><td>0.0098</td><td>0.0056</td></tr><tr><th></th><td>var</td><td>0.0447</td><td>0.0092</td><td>0.0054</td><td>0.0025</td></tr><tr><th>ν<sub>2</sub></th><td>bias</td><td>0.6523</td><td>0.2198</td><td>0.1626</td><td>0.1080</td></tr><tr><th></th><td>var</td><td>6.6068</td><td>0.9788</td><td>0.5520</td><td>0.2606</td></tr></tbody></table>

<table><thead><tr><th colspan="6">Gibbs sampler (λ = 1.32)</th></tr><tr><th>nb of obs</th><th>1000</th><th>3000</th><th>5000</th><th>10000</th></tr></thead><tbody><tr><th>α<sub>1</sub></th><td>bias</td><td>0.0006</td><td>0.0001</td><td>0.0001</td><td>.492e-4</td></tr><tr><th></th><td>var</td><td>0.0008</td><td>0.0003</td><td>0.0002</td><td>0.0001</td></tr><tr><th>ν<sub>1</sub></th><td>bias</td><td>0.0049</td><td>0.0018</td><td>0.0010</td><td>0.0006</td></tr><tr><th></th><td>var</td><td>0.0023</td><td>0.0008</td><td>0.0005</td><td>0.0002</td></tr><tr><th>ν<sub>2</sub></th><td>bias</td><td>0.0278</td><td>0.0069</td><td>0.0057</td><td>0.0023</td></tr><tr><th></th><td>var</td><td>0.1499</td><td>0.0491</td><td>0.0294</td><td>0.0147</td></tr></tbody></table>
* Two-component mixture

* Exponential parameters: $\nu_1 = 0.5$, $\nu_2 = 6$

* Mixing weights: $\alpha_1 = 1/3$, $\alpha_2 = 2/3$

<table><thead><tr><th colspan="5">Standard method (λ = 0.05)</th></tr><tr><th>nb of obs</th><th>1000</th><th>3000</th><th>5000</th><th>10000</th></tr></thead><tbody><tr><th>α<sub>1</sub></th><td>bias</td><td>0.0051</td><td>0.0040</td><td>0.0036</td><td>0.0028</td></tr><tr><th>α<sub>1</sub></th><td>var</td><td>0.0112</td><td>0.0035</td><td>0.0021</td><td>0.0010</td></tr><tr><th>ν<sub>1</sub></th><td>bias</td><td>0.0488</td><td>0.0143</td><td>0.0098</td><td>0.0056</td></tr><tr><th>ν<sub>1</sub></th><td>var</td><td>0.0447</td><td>0.0092</td><td>0.0054</td><td>0.0025</td></tr><tr><th>ν<sub>2</sub></th><td>bias</td><td>0.6523</td><td>0.2198</td><td>0.1626</td><td>0.1080</td></tr><tr><th>ν<sub>2</sub></th><td>var</td><td>6.6068</td><td>0.9788</td><td>0.5520</td><td>0.2606</td></tr></tbody></table>

<table><thead><tr><th colspan="5">Gibbs sampler (λ = 1.32)</th></tr><tr><th>nb of obs</th><th>1000</th><th>3000</th><th>5000</th><th>10000</th></tr></thead><tbody><tr><th>α<sub>1</sub></th><td>bias</td><td>0.0006</td><td>0.0001</td><td>0.0001</td><td>.492e-4</td></tr><tr><th>α<sub>1</sub></th><td>var</td><td>0.0008</td><td>0.0003</td><td>0.0002</td><td>0.0001</td></tr><tr><th>ν<sub>1</sub></th><td>bias</td><td>0.0049</td><td>0.0018</td><td>0.0010</td><td>0.0006</td></tr><tr><th>ν<sub>1</sub></th><td>var</td><td>0.0023</td><td>0.0008</td><td>0.0005</td><td>0.0002</td></tr><tr><th>ν<sub>2</sub></th><td>bias</td><td>0.0278</td><td>0.0069</td><td>0.0057</td><td>0.0023</td></tr><tr><th>ν<sub>2</sub></th><td>var</td><td>0.1499</td><td>0.0491</td><td>0.0294</td><td>0.0147</td></tr></tbody></table>
## Conclusion

* Optimization of the data acquisition by optimal choice of the tuning parameter

* Adequate estimator for the pile-up effect

* Acquisition time reduced by 90%

## Future work

* Include additive noise in the model for real data application

* Continuous mixture model (model with indefinitely many exponential components)

<figure><img src="image_26.png" /></figure>
* R., Roueff, Souloumiac - Information bounds and MCMC parameter estimation for the pile-up model, submitted

* R. - An MCMC approach for estimating a fluorescence lifetime with pile-up distortion, 2007

Thank you for your attention !

rebafka@telecom-paristech.fr

<figure><img src="image_27.png" /></figure>