Computing relative likelihoods of sequences relative to Poisson process I have a problem where I need to compute the likelihood of a given sequence of events (in time) given a Poisson process with a particular lambda (events per second). I only need it to within a constant, for comparison to other possible sequences.  A likelihood ratio, for example:
p(sequence_1 | lambda) / p(sequence_2 | lambda)
These probabilities could be computed by computing the value of the Poisson distribution over the full time span.  For example, if the whole interval is 10 seconds, I could compute the value of the Poisson distribution with 10*lambda and the total number of events.
However, I could also break the full interval down into subsets, such as 1 second intervals.  I think the total probability for a sequence ought to be related to the product individual Poisson probabilities for each sub-interval, but I think I must be missing a normalizing factor. Furthermore, at an intuitive level, if all the events were bunched up in time, analyzing over the full time may appear likely while looking at the sub intervals is unlikely.
Is there a way to do this where the result is not a function of the intervals used for processing?
EDIT:  For clarification, my goal is to check a sequence against the Poisson process and see how well it matches. I need the score to be in units of probability such that if the score is twice as high, it is twice as likely to be from the specified Poisson.

This is a basic probability result: if $Y_1, Y_2, \ldots, Y_n \sim \text{Poisson}(\lambda)$ then $\sum_{i=1}^n Y_i \sim \text{Poisson} (n\lambda)$. So differing from the premise of your question: the "result" is never a function of the intervals used for processing.
The combined and the separate analyses produce the same "result" whether a prediction, inference, estimation, etc. Dividing up a Poisson sample into the independent contributions makes no impact on a final evaluation of probability or likelihood like we would intuitively expect.
