Beta distribution parameter estimation: method of moments

In a paper: Topics over time, method of moments was applied to estimate $\alpha$ and $\beta$ for a Beta distribution.

My question is that how $\alpha$ or $\beta$ should be calculated if there are no samples or just one sample? If there are just one sample, the variance will be zero so that the formula can not be used due to zero division (variance will be zero in this case). Or, is it possible to set the parameters such the density value on that sample (say $x$) will be $p(x) = \infty$

• Well, with only one observation it will be hard to estimate two parameters! Either go for more data, or consult expert opinion, maybe construct a bayesian prior. – kjetil b halvorsen Jul 9 '18 at 9:19

As suggested by kjetil b halvorsen there is always a Bayesian approach to the problem. This however requires the selection of a prior distribution on the pair $(\alpha,\beta)$ that must reflect prior or expert knowledge on the parameter $(\alpha,\beta)$ as the posterior will reflect as much the prior as it does the information contained in the single sample. Or only the prior if there is no data. Given such a prior $p(\alpha,\beta)$ the posterior is (a) the prior for no data and (b) the distribution proportional to $$p(\alpha,\beta)\, \alpha^x\,\beta^{1-x}\, \Gamma(\alpha)\,\Gamma(\beta)\big/ \Gamma(\alpha+\beta)$$ which is not a standard distribution unless $p(\alpha,\beta)$ cancels the terms $\Gamma(\alpha)\,\Gamma(\beta)\big/ \Gamma(\alpha+\beta)$ as for instance $$p(\alpha,\beta)\propto e^{-\lambda\alpha-\mu\beta}\, \Gamma(\alpha+\beta)\big/\Gamma(\alpha)\,\Gamma(\beta)$$which is particularly delicate to caliber (and justify).