Assume that a signal $ y $ is a noisy superposition of time-shifted copies of a given waveform $ f(t) $ on a finite time interval $ [0, T] $: \begin{equation} y(t) = \sum_{i=1}^{n} f(t - \tau_j) + \eta(t) \end{equation} where $ \eta(t) $ is drawn from a Gaussian distribution of variance $ \sigma^2 $ and $ \{ \tau_j \}_{j \in \{1, \cdots, n\}} $ are a Poisson process with rate $ \mu $.
What would be the Maximum a Posteriori (MAP) estimator for $ \theta = \{ \tau_j \}_{j \in \{1, \cdots, n\}} $?
More precisely, by definition of the MAP estimator: \begin{equation} \theta_{MAP} = \underset{\theta}{\arg\max} \; f(y \mid \theta) f(\theta) \end{equation} where: \begin{equation} f(y \mid \theta) = \frac{1}{\sigma \sqrt{2 \pi}} \exp \left( -\frac{1}{2} \frac{\left\| y(t) - \hat{y}(t) \right\|_2^2}{\sigma^2} \right) \end{equation} with: \begin{equation} \hat{y}(t) = \sum_{i=1}^n f(t - \tau_i) \end{equation}
What would be the expresion of the prior $ f(\theta) $ ?
The prior $ f(\theta) $ where $ \theta = \{ \tau_j \}_{j \in \{1, \cdots, n\}} $ are a Poisson process with rate $ \mu $ is given by: \begin{align} f(\theta) &= f_{\tau_{(1)}, \cdots, \tau_{(n)}}(t_1, \cdots, t_n) \\ & = f_{\tau_{(1)}, \cdots, \tau_{(n)} \mid N(T) = n}(t_1, \cdots, t_n) \times \mathbb{P}(N(T) = n) \\ & = \frac{n!}{t^n} \mathbb{1}_{\left[ 0 < t_1 < \cdots < t_n \leq T \right]} \times \frac{(\mu t)^n}{n!} e^{-\mu t} \\ & = \mu^n e^{-\mu t} \mathbb{1}_{\left[ 0 < t_1 < \cdots < t_n \leq T \right]} \end{align} Then if we consider only the "ordered" parameters, the MAP estimator is: \begin{equation} \theta_{MAP} = \frac{1}{\sigma \sqrt{2 \pi}} \exp \left( - \frac{1}{2} \frac{\left\| y(t) - \hat{y}(t) \right\|_2^2}{\sigma^2} \right) \mu^n e^{-\mu t} \end{equation}