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Using the example of an inhomogeneous Poisson process in 1 dimension for simplicity, with a varying rate parameter $\lambda (t)$.

Let's say I am trying to find the form of $\lambda (t)$, using data which is a bunch of measured event values $\{t_i\}_{i=1}^N$. If I split the space of $t$ into disjoint regions $\{B_j\}_{j=1}^M$ (i.e bins), where I now have an "average" rate $\lambda_j$ in each bin and a number of events in each bin $D_j$, I can write the likelihood as something like:

\begin{align} L(\{\lambda_j\}) & = \prod_{j=1}^{M} \lambda_j^{D_j} \exp( - \lambda_j B_j) \end{align}

and try to find the rates in each bin, $\{\lambda_j\}$; or in a Bayesian formulation - with a prior - to find the posteriors on $\{\lambda_j\}$.

Now, I'm trying to figure out what I would do if my event values $t_i$ were all uncertain. In this situation, I wouldn't know $D_j$ precisely: it wouldn't be a fixed integer but a random variable.

As far as I understand, this can be treated with a hierarchical model, but would there be a simple approximate way? Something like bootstrapping the event values to find what proportion fall into each bin, causing the number of events in each bin, $D_j$, to not be an integer. In other words, say I have just two bins and one observed event, with an uncertain $t_1$. For example, our measurements might say that there is a 30% chance of the event having been in $B_1$ and a 70% chance of having been in $B_2$. Would it not be a valid approximation to just set $D_1 = 0.3$ and $D_2 = 0.7$ and use the same equation for likelihood above?

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  • $\begingroup$ How can you have a non-integer count of the number of events occurring in a given time interval? $\endgroup$
    – jbowman
    Jan 9, 2020 at 14:49
  • $\begingroup$ Well that's what I mean. There is an integer count of total events; however, as their exact time is uncertain, the total number of events within a region/bin is a random variable. Its expected value will not be an integer for example. So I guess the question is whether it is valid to replace the number of events in a bin $D_j$ with the "expected number of events in the bin," or $E[D_j]$. Where by expected, I mean expected w.r.t to the data and its uncertainty. $\endgroup$
    – Marses
    Jan 9, 2020 at 15:14
  • $\begingroup$ Maybe you have in mind using Expectation-Maximisation (EM), considering the true numbers of events as latent variables. As I understand you do not have the complete observations $t_i$ but only the observed counts $O_j$ for the bins. What can be the distribution of the r.vs $O_j$ given the values of $D_j$? $\endgroup$
    – Yves
    Jan 9, 2020 at 18:15
  • $\begingroup$ My observations would be in the form of uncertainty distributions on the $t_i$'s. So for example, Gaussian distributions, with a mean value and $1\sigma$. Expectation-Minimisation would work for the max-likelihood of the rates, though this would mean I would actually have to fully write this out as a hierarchical/multi-level model. However in my case, I'm treating this in a Bayesian way, so I want the posterior distrbutions on the rates, with an MCMC to sample them for example. I'm just wondering if it makes sense to simplify this the way I described and avoid going with a hierarchical model. $\endgroup$
    – Marses
    Jan 9, 2020 at 22:17
  • $\begingroup$ Treating this properly as a hierarchical Bayesian model, I would have the data (estimators for the uncertainty distribution of the $t_i$'s: the mean and variance of each measurement), conditioned on latent parameters representing the "true" and unknown $t_i$'s as random variables, conditioned on the rates. For the posterior distribution on the rates, I would need to marginalise over the latent parameters. This would lead to something very similar in form to just replacing the event counts with non-integers that represent "expected event counts". I'll come back to this tomorrow. $\endgroup$
    – Marses
    Jan 9, 2020 at 22:22

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