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Though widely used in applications, reinforced random walk on graphs havenever been the subject of a valid statistical inference. We develop in thispaper a statistical framework for a general two-colored urn model. Theprobability to draw a ball at each step depends on the number of balls of eachcolor and on a multidimensional parameter $\theta$ through a function $f$,called a choice function. We introduce two estimators of $\theta$: the maximumlikelihood estimator and a weighted least squares estimator which is lessefficient, but is closer to the calibration techniques used in the appliedliterature. In general, the model is an inhomogeneous Markov chain and thisproperty makes the estimation of the parameter impossible on a single path,even if it were infinite. Therefore we assume that we observe i.i.d.experiments, each of a predetermined finite length. This is coherent with theusual experimental set-ups. We apply the statistical framework to a real lifeexperiment: the selection of a path among pre-existing channels by an antcolony. We performed experiments, which consisted of letting ants pass throughthe branches of a fork. We consider the particular urn model proposed by J.-L.Deneubourg in 1990 to describe this phenomenon. We simulate this model forseveral parameter values in order to assess the accuracy of the MLE and theWLSE. Then we estimate the parameter from the experimental data and evaluateconfident regions with Bootstrap algorithms. The findings of this paper do notcontradict the biological literature, but give statistical significance to thevalues of the parameter found therein.

Parameter estimation of a two-colored urn model class