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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.

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