# Marginal Log Likelihood of Count Data to Simplify MLE

My question concerns the calculation of a marginal likelihood given some priors for my underlying exponential mixture distribution.

### Background

My data is the (orderd) set of integers $$\{N_\ell\}$$. My underlying model leads to an expected $$\mu_\ell$$ value for each bin

$$\mu_\ell(a_1,a_2,\tau_1,\tau_2) = \frac{a_1}{\tau_1}\exp(-t_\ell/\tau_1) + \frac{a_2}{\tau_2}\exp(-t_\ell/\tau_2) .$$ $$a_{1,2}$$ are the amplitudes of the decay components and $$t_{1,2}$$ their respective decay times. I want to find these given the data with MLE.

My resulting likelihood of observing the set $$\{N_\ell\}$$ by Gaussian approximation is:

$$\begin{eqnarray} prob(\{N_\ell\}|a_1,a_2,\tau_1,\tau_2) &\propto& \prod_{\ell} \exp\left(-\frac{(\mu_\ell(a_1,a_2,\tau_1,\tau_2)-N_\ell)^2}{2N_\ell}\right) \\ &=&\exp\left(-\sum_\ell \frac{(\mu_\ell(a_1,a_2,\tau_1,\tau_2)-N_\ell)^2}{2N_\ell}\right) \end{eqnarray}$$ Now maximizing the log likelihood would lead me to Least Squares Regression.

### What I hoped to achieve

I have a fairly good idea of the ranges of $$\tau_1$$ and $$\tau_2$$. Those parameters are what make the Least Squares Regression nonlinear and somewhat unstable. Thus hoping to get rid of them I want to calculate the marginal likelihood, i.e. $$prob(\{N_\ell\}|a_1,a_2) \propto \int d\tau_1 \int d\tau_2 \,prob(\{N_\ell\}|a_1,a_2,\tau_1,\tau_2) prob(\tau_1) prob(\tau_2) .$$

Assume I can give an analytical expression for the priors. They could be uniform (in a range), Gaussian or Jeffrey's (in a range).

### My Questions

1. Hardest Question: What is an efficient way of calculating the coefficients $$a_1$$,$$a_2$$ from the marginal likelihood? By efficient I mean: computationally faster and more stable than performing the whole nonlinear regression of $$a_1$$,$$a_2$$,$$\tau_1$$,$$\tau_2$$ on the original likelihood?
2. Can the marginal log likelihood be approximated from the integral?
3. Is my idea useful at all? Maybe I am naive in assuming that marginalization over the decay times can just magically lead me to a simpler solution.

### What I have tried

• Analytical integration: Even after rearranging I cannot se a way of dealing with the integrals of $$e^{e^{1/\tau}}$$.
• Evaluating the marginal likelihood (not log likelihood) with numerical integration (using OCTAVE): Quickly gave up because everything evaluated to zero, of course.
• Using Laplace's Approximation: I considered it, but did not give it a try, since it effectively requires me to solve the minimization problem for $$\tau_1$$,$$\tau_2$$ if I understood correctly. If I do this, I might just as well go with the whole nonlinear least squares from the start. Maybe I am wrong?

As always any help is greatly appreciated.