Ranking sources at variable(random) frequencies

Question

Hi, I have this math modeling problem that I need help with. If I have 3 data sources, each being updated at different frequencies, what would be the best way to rank them so the less frequent sources don't get pushed down too quickly? In other words, I want the less likely source to be "weighed down" in its spot. Also, the frequencies of the data source are not consistent.

I was trying to think of the problem in some sort of a bayesian update/inference/learning way but I couldn't properly describe it. Something like "what is the probability that data source A will be in rank position 1" but I'm not sure if that's the right track. I have limited probability experience. Anyone have any thoughts on this?

I don't need the exact solution or anything, just want to be pointed in the direction. I don't have a breadth of mathematical knowledge, but I should be proficient enough to figure it out if anyone can provide articles, papers, examples, etc.

Thanks for the help.

Answer 1

You can model the arrivals as a poisson distribution with different arrival rates λ_i for each one of your sources.

$f(n_i;\lambda_i) = \dfrac{\lambda_i^{n_i} e^{-\lambda_i}} { n_i!}$

You can then assume that the arrival rates are random draws from a gamma distribution which would let you pool information across all your data sources.

$\lambda_i$ ~ $\Gamma(k,\theta)$.

Assume suitable distributions for $k$ and $\theta$. You can then construct the posterior distribution for $\lambda_i$, $k$ and $\theta$. The gamma distribution will help you 'shrink' your estimates for $\lambda_i$. In other words, if the frequencies of arrival is too low then it will be nudged a bit higher and if it is too high it will be nudged a bit lower depending on the data you have. Basically, the pooling of information will enable you to arrive at more robust estimates of the arrival rates.

Estimate these parameters conditional on the data you observe using standard Bayesian methods. Once you have the estimates you can find out the probability that source $i$ is ranked the highest etc from the estimates. Do note that if the observed arrivals are of order of magnitude different from one another then you would be better of simply ordering on the actual arrival numbers as the bayesian computation above would not give any different results.

Google the following terms to get some traction: hierarchical bayesian models, gibbs sampler, mcmc sampler, conjugate priors etc.