I need to find the likelihood that a set of molecules was instantaneously released at time $t_0$, say $t_0=0$.
Toy System Example:
Let $N$ be the set of molecules released from a specific point in a 3D environment. The released molecules diffuse in the environment according to the following:
$$ r[t] = r[t-1] + (\Delta r_1, \Delta r_2, \Delta r_3)$$ $$ \Delta r_i \sim \mathcal{N}(0,\, 2D\Delta t)$$ where $r[t]$, $r_i$, $D$, and $\Delta t$ are the location vector at time $t$, $i$-th component of the location vector, diffusion coefficient, and the time step, respectively.
If there is an absorbing spherical body at a distance $d$, the hitting rate of the molecules until time $t>0$ is:
$$ f_{hit}(t)= \frac{r_{bod}}{d+r_{bod}} \, \text{erfc} \left( \frac{d}{\sqrt{4Dt}} \right) $$ where $r_{bod}$ is the radius of the absorbing spherical body.
I need to estimate $t_0$ using the information of $f_{hit}$.
So, I thought of using the optimal maximum likelihood technique. I am not very familiar with the maximum likelihood technique. As per this article, I have come up with this likelihood function.
$$L(\hat{t_0}|t) = \arg\max\limits_{t_0} f_{hit}(t|t_0)\\ =\prod f_{hit}(t|t_0)$$
But knowing that $t_0$ is not a parameter, this does not seem correct, right? I am really confused. How do I work around this to find $\hat{t_0}$?
Any help would be appreciated.
