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As explained above, Beta distributions can be used to describe uncertainty about θ.

Depending on the choices of both the parameters, beta distributions can be unimodal, U-shaped, strictly increasing/decreasing or uniform.

This is likely to occur since the Beta distribution can be positively or negatively skewed in such cases.

While considering the event history in the Bayesian framework, the expected value of beta distribution can be written as.

Using the method of moments (NIST-SEMATECH 2012), the parameters of a standard beta distribution can be estimated from the estimated mean and standard deviation as shown in Eqs. 2 and 3: p={mu}_xBig[left(frac{mu_xleft 1-{mu}_xright)}{sigma_x^2}right)-1 (2) q=lefrac{mu_xleft 1-{mu}_xrightfrac{mu_xleft(1-{mu}_xright)}{sigma_x^2}right -1right] (3).

Like the skew-normal distribution, the beta distribution can be positively or negatively skewed or symmetric.

The parameters in this new beta distribution can be obtained by matching the first two moments: ∫ 0 1 f (p t | p 0, N e ) f (p 0 | x 0 ) d p 0 ≈ beta (α ′ = δ (x 0 + 1 ) 2 n + 2 + δ, β ′ = δ (2 n − x 0 + 1 ) 2 n + 2 + δ ).

The corresponding Beta prior distributions can be found in Additional file 2: Table S2.

In principle, a uniform prior distribution or Beta prior distribution can be used over the set of unknown parameters.

For example, the generalized beta prime distribution can be written as (13) G ˙ (x ) ∝ x m α 1 + x m k - β.

The results proved that the beta-binomial distribution can be very useful for analyzing vegetation landscapes.