Parameter estimation for a bivariate beta distribution with arbitrary beta marginals and positive correlation

Trick, Susanne; Rothkopf, Constantin A.; Jäkel, Frank Article — Published Version Parameter estimation for a bivariate beta distribution with arbitrary beta marginals and positive correlation METRON Provided in Cooperation with: Springer Nature Suggested Citation: Trick, Susanne; Rothkopf, Constantin A.; Jäkel, Frank (2023) : Parameter estimation for a bivariate beta distribution with arbitrary beta marginals and positive correlation, METRON, ISSN 2281-695X, Springer Milan, Milano, Vol. 81, Iss. 2, pp. 163-180, https://doi.org/10.1007/s40300-023-00247-2 This Version is available at: https://hdl.handle.net/10419/309571 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ METRON (2023) 81:163–180 https://doi.org/10.1007/s40300-023-00247-2 Parameter estimation for a bivariate beta distribution with arbitrary beta marginals and positive correlation Susanne Trick1,2 ·Constantin A. Rothkopf1,2,3 ·Frank Jäkel1,2 Received: 28 September 2022 / Accepted: 15 April 2023 / Published online: 18 May 2023 © The Author(s) 2023 Abstract We discuss a bivariate beta distribution that can model arbitrary beta-distributed marginals with a positive correlation. The distribution is constructed from six independent gammadistributed random variates. While previous work used an approximate and sometimes inaccurate method to compute the distribution’s covariance and estimate its parameters, here, we derive all product moments and the exact covariance, which can be computed numerically. Based on this analysis we present an algorithm for estimating the parameters of the distribution using moment matching. We evaluate this inference method in a simulation study and demonstrate its practical use on a data set consisting of predictions from two correlated forecasters. Furthermore, we generalize the bivariate beta distribution to a correlated Dirichlet distribution, for which the proposed parameter estimation method can be used analogously. Keywords Bivariate beta distribution ·Correlated beta distribution ·Covariance ·Moment matching 1 Introduction Probabilistic forecasts are important in many domains, among them finance and economics, business and marketing, politics, public health, engineering, and meteorological, ecological, Constantin A. Rothkopf and Frank Jäkel contributed equally to this work BSusanne Trick [email protected] Constantin A. Rothkopf [email protected] Frank Jäkel [email protected] 1Centre for Cognitive Science, Technical University of Darmstadt, Alexanderstr. 10, 64283 Darmstadt, Germany 2Institute of Psychology, Technical University of Darmstadt, Alexanderstr. 10, 64283 Darmstadt, Germany 3Frankfurt Institute for Advanced Studies, Goethe University, Ruth-Moufang-Str. 1, 60438 Frankfurt, Germany 123 164 S. Trick et al. and environmental science. For binary events these forecasts take the form of probability estimates that are either provided by humans or machine learning algorithms. In order to model these probability estimates one can use an arbitrary distribution on the interval [0,1], such as the beta distribution, beta-generated distributions, the Kumaraswamy distribution, or any distribution on the real numbers transformed through the logistic function. In many cases the probabilities will be correlated, e.g. if different experts forecast the outcome of an election or the probability of rain. In order to be able to model such correlated probabilities, therefore, a bivariate distribution is needed. For example, one can use bivariate generalizations of the Kumaraswamy distribution [1], bivariate beta-generated distributions [2], or a multivariate Gaussian with logistic transformations [3]. However, the most common choice for modeling such probabilities is the beta distribution, since it is the standard distribution for probabilities in Bayesian statistics and is simply more familiar to practitioners than the other distributions [4,5]. Therefore, in this paper we focus on bivariate beta distributions. While a multitude of constructions for bivariate beta distributions have been proposed, they have different constraints and properties, which limit their applicability. We will first review previous constructions of bivariate beta distributions together with their respective properties and then examine the most promising construction that can model arbitrary beta marginals with a positive correlation [5]. For this construction of modeling arbitrary beta marginals with positive correlation, so far, there has not been an exact method for parameter inference and it has thus rarely been used. Here, we therefore introduce a new estimation method for this bivariate beta distribution with arbitrary beta marginals and positive correlation. Many bivariate beta distributions have been proposed in the literature [1,2,5–21]. Among them, some approaches have been derived from general families of bivariate distributions, such as the Farlie-Gumbel-Morgenstern family of distributions [e.g. 7] and the Sarmanov family of distributions [e.g. 8]. Others derived a bivariate beta distribution from a bivariate extension of the F distribution [9,13], whereas Nadarajah and Kotz [11] proposed different bivariate beta distributions constructed from products of univariate beta-distributed random variables. In general, using different copulas one can construct different bivariate distributions with the same beta marginals [2,20]. However, as beta-distributed random variables can easily be constructed from normalized gamma-distributed random variables, it is natural to try and generalize this construction to the bivariate case. In this vein, several authors have introduced correlations through shared gamma-distributed random variables [1,5,10,14,18]. The most straightforward case of this construction has been studied by Olkin and Liu [10], building on the work of Libby and Novick [6]. They use three independent gamma-distributed random variables U1,U2,U3with respective shape parameters υ1,υ 2,υ 3and same scale parameter to construct X=U1 U1+U3 and Y=U2 U2+U3 .(1) Using the standard construction of beta variates from gamma variates, the joint distribution of the random variables Xand Yis a bivariate beta distribution with marginal distributions Beta(υ1,υ3)forXand Beta(υ2,υ3)forY. The correlation between Xand Y,whichis obtained through the shared latent variable U3and its parameter υ3, is in the range [0,1]. For high values of υ3, the correlation tends to 0 whereas for low values of υ3, it tends to 1. However, if υ3is high, the values of Xand Yalso tend to 0 and if υ3is low they tend to 1 accordingly. This behavior severely limits the usefulness of the distribution for most applications. A further limitation is the constraint that the marginal distributions share the same second parameter υ3. Thus, the bivariate beta distribution proposed by Olkin and Liu 123 Parameter estimation for a bivariate... 165 does not allow for arbitrary beta marginals, which limits its flexibility in modeling probability forecasts. Arnold and Ng [14] proposed a more flexible construction for a bivariate beta distribution. They use five independent gamma-distributed random variables U1,...,U5with shape parameters υ1,...,υ 5and scale parameter 1 to define two correlated random variables X=U1+U3 U1+U3+U4+U5 and Y=U2+U4 U2+U3+U4+U5 (2) with marginal distributions Beta(υ1+υ3,υ 4+υ5)forXand Beta(υ2+υ4,υ 3+υ5)forY. Compared to Olkin and Liu [10], this construction of a bivariate beta distribution can generate all correlations in the range [-1,1] and marginal distributions with differing second parameters. Nevertheless, because of how the two marginals share parameters, not all combinations of parameters of the marginal beta distributions are possible. For example, the marginals Beta(10,4) for Xand Beta(1,1) for Ycannot be obtained. Olkin and Trikalinos [18] base their construction of a bivariate beta distribution on the Dirichlet distribution. U=(U00,U01,U10,U11)is drawn from a 4-dimensional Dirichlet distribution with parameters υ1,...,υ 4. By just using three of its components, U00,U01,U10, new random variables X=U00 +U10 and Y=U01 +U10 (3) are constructed, with marginal distributions Beta(υ1+υ3,υ 2+υ4)forXand Beta(υ2+ υ3,υ 1+υ4)forY. As Dirichlet-distributed random variables can also be constructed from gamma random variables, we can equivalently construct Xand Yin Eq. (3) from four independent gamma-distributed random variables U1,...,U4with shape parameters υ1,...,υ 4 and equal scale parameter 1, with X=U1+U3 U1+U2+U3+U4 and Y=U2+U3 U1+U2+U3+U4 .(4) As can easily be seen from this construction, all correlations in the range [-1,1] can be generated. In particular, the correlation tends to -1 if υ3and υ4tend to 0 and U3and U4 will be negligible compared to U1and U2. In this case X≈U1 U1+U2≈1−Y. Similarly, the higher the values of υ3and υ4relative to υ1and υ2, the more negligible U1and U2will be and the correlation increases to 1 until X≈U3 U3+U4≈Y. Less obviously, a correlation of 0 is obtained in case υ1·υ2=υ3·υ4[18]. Still, this construction of a bivariate beta distribution does not allow arbitrary beta marginal distributions. Since all υiare constrained to be positive, for some combinations of marginal distributions the resulting system of linear equations for the parameters υihas no solution. For example, the two marginals Beta(2,2) for Xand Beta(1,1) for Ycannot be generated, regardless of their correlation. Magnussen [5] introduced yet another construction based on six gamma variates. While all the constructions thus far constrain the parameters of the marginal beta distributions, this construction does allow for arbitrary beta marginals with positive correlation, thus providing the necessary flexibility to model probability forecasts. Magnussen’s distribution is a special case of a more general 8-parameter bivariate beta distribution introduced by Arnold and Ng [14] and reviewed in Arnold and Ghosh [1], which even allows for positive and negative correlations. However, in many applications, it is enough to model positive correlations, for which the less complex 6-parameter distribution is sufficient. For example, if Xand Yare probability estimates elicited from two skilled forecasters, we do not expect negative correlations. But we do want to allow for the possibility that their marginal forecasts have different distributions that should not be tied together by parameter constraints on the marginals. 123 166 S. Trick et al. Hence, the bivariate beta distribution proposed by Magnussen [5], which can model arbitrary beta-distributed marginals with a positive correlation, is an appropriate distribution for modeling correlated probability forecasts. However, just like any other distribution, the bivariate beta distribution can only be used if its parameters can be estimated correctly. While Magnussen [5] proposes a moment matching approach for fitting the distribution’s parameters, this approach relies on a rough and sometimes inaccurate approximation for the covariance. Also, Magnussen [5] did not discuss the fact that very similar data can be generated with different parameter values, which makes it hard to statistically infer the parameters of the bivariate beta distribution from data without constraining the distribution. Therefore, in this work we introduce an alternative approach for estimating this bivariate beta distribution’s parameters. First, we will derive the full joint distribution, which is missing in the work of Magnussen [5], probably because it is intractable. We will then clarify the relationship between Magnussen’s distribution and the Olkin-Liu distribution [10]. Using this relationship with the Olkin-Liu distribution we derive all product moments and in particular the exact covariance function (and in passing we correct a small mistake in the product moments from Olkin and Liu [10]). For parameter inference, we propose to match moments numerically using the exact covariance we derived. While other estimation methods such as Bayesian inference could be used [22], here we focus on moment matching due to its simplicity and efficiency. In order to make parameter inference unambiguous, we additionally show how to reasonably constrain the distribution’s parameters. We evaluate the proposed parameter estimation method in a simulation study and demonstrate its practical use on a real data set consisting of predictions from two correlated forecasters. We discuss the relationship between the distribution’s parameters and the correlation and then extend the bivariate beta distribution to a correlated Dirichlet distribution, for which the proposed parameter estimation method can be used analogously. The remainder of the paper is structured as follows. In Sect.2we discuss the bivariate beta distribution with arbitrary beta marginals, including its joint distribution in Sect.2.1, its moments in Sect.2.2, correlation and covariance in Sect.2.3, and parameter inference in Sect.2.4. Section3shows how to generalize the bivariate beta distribution to a correlated Dirichlet distribution. 2 A bivariate beta distribution with arbitrary beta marginals Magnussen [5] uses six independent gamma-distributed random variables A1,A2,B1,B2, D1,D2that are distributed according to Ai∼Gamma(αi,1)i=1,2 Bi∼Gamma(βi,1)i=1,2 Di∼Gamma(δi,1)i=1,2, (5) to construct two bivariate-beta-distributed random variables X=A1+D1 A1+A2+D1+D2 and Y=B1+D1 B1+B2+D1+D2 .(6) The resulting marginal distributions of Xand Yare Beta(a1,a2)andBeta(b1,b2) with a1=α1+δ1a2=α2+δ2b1=β1+δ1b2=β2+δ2.(7) 123 Parameter estimation for a bivariate... 167 The marginals follow immediately from the definition because the sum of gamma random variables of the same scale is gamma-distributed with the same scale but with the original shape parameters summed. In contrast to other constructions that were discussed above [10, 14,18], this construction allows for arbitrary marginal distributions. In particular, when δ1 and δ2tend to zero, we can model arbitrary independent marginal distributions Beta(α1,α2) and Beta(β1,β2). Since all parameters α1,α 2,β 1,β 2,δ 1,δ 2need to be positive by definition, for fixed marginal distributions Beta(a1,a2)forXand Beta(b1,b2)forYit must hold that δ1< δmax 1=min(a1,b1)and δ2<δ max 2=min(a2,b2). Therefore, for most marginal distributions the maximum correlation that can be generated is below 1. The higher the difference between two marginal distributions, the lower the possible maximum correlation. A perfect correlation approaching 1 can, of course, only be generated for equal marginal distributions, i.e. if a1=b1and a2=b2and α1,α2,β1,andβ2tend to 0, as also noted by Magnussen [5]. Note that this limitation applies to other bivariate distributions that do not allow for arbitrary marginal beta distributions as well [e.g. 18]. The construction of this bivariate beta distribution can also be seen as a pairwise combination of three beta distributions. First transform the six independent gamma-distributed random variables (5) into three independent gammaand three independent beta-distributed random variables, U1=A1+A2,U1∼Gamma(υ1,1) U2=B1+B2,U2∼Gamma(υ2,1) U3=D1+D2,U3∼Gamma(υ3,1) W1=A1 A1+A2 ,W1∼Beta(α1,α 2) W2=B1 B1+B2 ,W2∼Beta(β1,β 2) W3=D1 D1+D2 ,W3∼Beta(δ1,δ 2) (8) with υ1=α1+α2υ2=β1+β2υ3=δ1+δ2.(9) With these definitions we can then rewrite construction (6)as X=U1 U1+U3·W1+U3 U1+U3·W3=XW1+(1−X)W3 Y=U2 U2+U3·W2+U3 U2+U3·W3=YW2+(1−Y)W3, (10) where Xand Yare defined as in (1) but with υ1,υ2,andυ3as in (9). Furthermore, X and Yare independent of W1,W2,W3. If parameters δ1and δ2and with them U3tend to 0, X≈W1and Y≈W2are independent with marginal distributions Beta(α1,α 2)forX and Beta(β1,β 2)forY. Mixing in the shared component W3by increasing the values of parameters δ1and δ2increases the correlation between Xand Y.IfU1and U2are negligible compared to U3because δ1and δ2dominate the parameters, the correlation will be close to 1 with X≈W3≈Yand hence Xand Yhave the same marginal distribution Beta(δ1,δ2), as mentioned before. 123 168 S. Trick et al. 2.1 Joint distribution Xand Yin (10) are linear transformations of Xand Y.GivenW1,W2,W3it is easy to recover Xand Yfrom observed Xand Y, X=X−W3 W1−W3=|X−W3| |W1−W3|=f1(X) Y=Y−W3 W2−W3=|Y−W3| |W2−W3|=f2(Y). (11) Note that we can ignore the sign because according to (10)Xis always between W1and W3 and Ybetween W2and W3, so that the numerator and denominator always have the same sign. As Xand Yjointly follow the Olkin-Liu distribution [10], p(x,y)=xυ1−11−xυ2+υ3−1yυ2−11−yυ1+υ3−1 B(υ1,υ 2,υ 3)(1−xy)υ1+υ2+υ3,(12) where B(υ1,υ 2,υ 3)=(υ1)(υ2)(υ3) (υ1+υ2+υ3), the joint distribution of Xand Ygiven W1,W2,W3 is p(x,y|w1,w 2,w 3)= df 1(x) dx df 2(y) dy  p(f1(x), f2(y)) =1 |w1−w3||w2−w3| |x−w3| |w1−w3|υ1−11−|x−w3| |w1−w3|υ2+υ3−1|y−w3| |w2−w3|υ2−11−|y−w3| |w2−w3|υ1+υ3−1 B(υ1,υ 2,υ 3)1−|x−w3| |w1−w3||y−w3| |w2−w3|υ1+υ2+υ3 =|w1−w3||w2−w3| B(υ1,υ 2,υ 3) ·|x−w3|υ1−1|x−w1|υ2+υ3−1|y−w3|υ2−1|y−w2|υ1+υ3−1 (|w1−w3||w2−w3|−|x−w3||y−w3|)υ1+υ2+υ3 (13) with xbetween w1and w3and ybetween w2and w3according to (10). We have not been able to integrate out w1,w 2,w 3from their joint distribution with xand y.However,wesuspect that even if the joint density for Xand Ycould be expressed in terms of special functions, computing those might not be efficient enough for parameter inference for which we will resort to moment matching. Example plots with smoothed samples for the joint density are showninFig.1for several parameter settings showing different marginal distributions for Xand Yand different correlations between Xand Y. Sampling from the bivariate beta distribution is realized with JAGS [23]. 2.2 Moments As the marginal distributions for Xand Yare beta-distributed, their moments are readily available, even in closed form. Computation of the product moments E(XkYl)is more challenging but can be realized with help of the work of Olkin and Liu [10]. Looking at construction (10), Xand Yare a linear combination of independent beta-distributed random variables W1,W2,W3with weights Xand Y. Thus, we can express the product moments 123 Parameter estimation for a bivariate... 169 Fig. 1 Joint densities of bivariate beta distributions with selected parameter values. The plots were created with kernel density estimation based on 10 million samples of the respective distributions. Note that the smoothing is inaccurate at the borders and produces artifacts close to zero and one as a consequence of smoothing with a symmetric kernel 123 170 S. Trick et al. as EXkYl=EXW1+(1−X)W3kYW2+(1−Y)W3l.(14) Since Xand Yare independent of W1,W2,W3, it is possible to compute the expectation if the moments of W1,W2,W3,X,Yand the product moments of Xand Yare known. W1,W2,W3as well as the marginals of Xand Yare beta-distributed, so their moments can be computed straightforwardly in closed form. Furthermore, Xand Yare jointly OlkinLiu distributed according to (12), and Olkin and Liu [10] have shown how to compute their product moments. However, note that the derivation of E((X)k(Y)l)in equation (2.2) in the work of Olkin and Liu [10] is incorrect and should read E(X)k(Y)l=∞  j=0 dA(j)B(υ1+k+j,υ 2+υ3) B(υ1+j,υ 2+υ3) B(υ2+l+j,υ 1+υ3) B(υ2+j,υ 1+υ3)(15) =(υ1+υ3)(υ2+υ3)(υ1+k)(υ2+l)(ϒ) (υ1)(υ2)(υ3)(ϒ +k)(ϒ +l) ∞  j=0 (υ1+k)j(υ2+l)j(ϒ) j (ϒ +k)j(ϒ +l)j 1 j!(16) =h·3F2(υ1+k,υ 2+l,ϒ;ϒ+k,ϒ +l;1), (17) with d=(υ1+υ3)(υ2+υ3) (υ3)(ϒ) (18) A(j)=(υ1+j) (υ1) (υ2+j) (υ2) (ϒ) (ϒ +j) 1 j!(19) h=(υ1+υ3)(υ2+υ3)(υ1+k)(υ2+l)(ϒ) (υ1)(υ2)(υ3)(ϒ +k)(ϒ +l),(20) where ϒ=υ1+υ2+υ3and pFqis the generalized hypergeometric function. Equations (15), (18), and (19) are taken directly from Olkin and Liu [10] with a=υ1,b=υ2,c=υ3. Equations (16), (17), and (20) are our corrections of their equations. 2.3 Correlation and covariance The correlation rbetween Xand Yis r=Cov(X,Y) √Var(X)Var(Y)(21) with the known variances of the beta marginals Var(X)=a1a2 (a1+a2)2(a1+a2+1) =(α1+δ1)(α2+δ2) (α1+δ1+α2+δ2)2(α1+δ1+α2+δ2+1) Var(Y)=b1b2 (b1+b2)2(b1+b2+1) =(β1+δ1)(β2+δ2) (β1+δ1+β2+δ2)2(β1+δ1+β2+δ2+1). (22) 123 Parameter estimation for a bivariate... 177 Fig. 6 The exact correlation computed as in Sect.2.3 compared to the approximate correlation computed with Eq. (33) for all parameter configurations used in the simulation study in Sect.2.4.2 2.4.4 Relationship between ı1and the correlation We found empirically that over a large range of parameters the following approximate relationship holds: r≈δ1 δmax 1 rmax,(33) while δmax 1=min(a1,b1)and rmax is the maximum possible correlation for the given marginals. While this relationship could be used for approximate inference of δ1, we still recommend the moment matching approach proposed in Sect.2.4.1 that gives exact results. However, the shown relationship allows interpreting the correlation parameter δ1. Particularly for equal marginal distribution with a1=b1and a2=b2, for which rmax =1, this interpretation of δ1is very simple: The fraction of δ1 δmax 1approximately matches the generated correlation. For example, if a1=b1=2anda2=b2=4, for δ1=1 we generate a correlation of r=0.468 ≈1 2rmax =0.5 with rmax =1. For differing marginal distributions, interpreting δ1is more difficult because rmax must be computed numerically using the formulas given above. If, e.g., a1=a2=2andb1=1,b2=4, with δ1=0.5, we generate a correlation of 0.3≈0.5 1rmax =0.313 with rmax =0.627. In Fig. 6, we plot the exact correlation and the approximated correlation computed with (33) for all parameter configurations used in the simulation study in Sect.2.4.2. As can be seen, the relationship in (33) holds for all parameter values. The plateaus seen for approximate correlations of 0.25, 0.5, and 0.75 are a consequence of choosing δ1=p·δmax 1with p=0.25,0.5,0.75 in the simulation study (Sect.2.4.2) leading to approximate correlations of 0.25, 0.5, 0.75 for all simulations with equal marginals. We leave it as an open problem to show when approximation (33) holds to what accuracy. 3 Generalization to the correlated Dirichlet distribution The bivariate beta distribution can be generalized to a correlated Dirichlet distribution [27] in order to model two positively correlated random vectors X=(X1,...,Xk)and Y=(Y1,...,Yk)with the two marginal vectors being Dirichlet-distributed. A k-dimensional correlated Dirichlet distribution can be constructed from 3kgamma-distributed random 123 178 S. Trick et al. variables A1,...,Ak,B1,...,Bk,D1,...,Dkwith 3kparameters α1,...,α k,β1,...β k, δ1,...δ kdistributed according to Ai∼Gamma(αi,1)i=1,...,k Bi∼Gamma(βi,1)i=1,...,k Di∼Gamma(δi,1)i=1,...,k. (34) These random variables are used to construct the correlated Dirichlet-distributed random variables X=(X1,...,Xk)and Y=(Y1,...,Yk)with Xi=Ai+Di k i=1Ai+k i=1Di and Yi=Bi+Di k i=1Bi+k i=1Di .(35) The two resulting marginal distributions are Dirichlet(X;α1+δ1,...,α k+δk)and Dirichlet(Y;β1+δ1,...,β k+δk). Analogous to the example for the bivariate beta distribution in Sect.2.4.3, this correlated Dirichlet distribution can be used for modeling non-binary probabilistic predictions of experts, sensors, or classifiers. This is particularly useful for Bayesian approaches to classifier or expert fusion, which are the reason why we started working on this distribution in the first place. For example, in a companion paper we apply it to classifier fusion, but instead of using moment matching—as developed here—we use rather inefficient Markovchain methods to sample from the posterior distribution over the parameters [27]. Being able to explicitly model the correlation between probabilistic classifiers or probability estimates given by human experts with the correlated Dirichlet distribution allows Bayes optimal fusion of classifiers or experts, avoids overconfidence of the ensemble and thereby improves its performance. Applications of classifier fusion are widespread. Popular examples are intrusion detection, fake news detection, detection of diseases in medicine, or recognition of human states such as emotions [31]. Another application of the correlated Dirichlet distribution is the generation of stochastic matrices with individual rows or columns being Dirichletdistributed and correlated, which can be beneficial for Markov processes, in optimal control, or reinforcement learning. The derivations of the product moments and the exact covariance of the correlated Dirichlet distribution are analogous to the derivations for the bivariate beta distribution shown in this work. Thus, the parameters of the correlated Dirichlet distribution can also be estimated using the proposed moment matching approach, extended to the higher dimensionality of the Dirichlet distribution. Funding This work was supported by the German Federal Ministry of Education and Research (BMBF) under Grant 16SV7984. Open Access funding enabled and organized by Projekt DEAL. 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Arnold, B.C., Ghosh, I.: Bivariate beta and Kumaraswamy models developed using the Arnold-Ng bivariate beta distribution. REVSTAT–Stat. J. 15(2), 223–250 (2017). https://doi.org/10.57805/revstat.v15i2. 211 2. Samanthi, R.G., Sepanski, J.: A bivariate extension of the beta generated distribution derived from copulas. Commun. Stat.-Theory Methods 48(5), 1043–1059 (2019). https://doi.org/10.1080/03610926.2018. 1429626 3. Pirs, G., Strumbelj, E.: Bayesian combination of probabilistic classifiers using multivariate normal mixtures. J. Mach. Learn. Res. 20, 51–1 (2019) 4. Johnson, N.L., Kotz, S., Balakrishnan, N.: Continuous Univariate Distributions, Volume 2 vol. 289. John Wiley & Sons, New York, USA (1995) 5. Magnussen, S.: An algorithm for generating positively correlated beta-distributed random variables with known marginal distributions and a specified correlation. Comput. Stat. Data Anal. 46(2), 397–406 (2004). https://doi.org/10.1016/S0167-9473(03)00169-5 6. Libby, D.L., Novick, M.R.: Multivariate generalized beta distributions with applications to utility assessment. J. Educ. Stat. 7(4), 271–294 (1982). https://doi.org/10.3102/10769986007004271 7. Gupta, A.K., Wong, C.: On three and five parameter bivariate beta distributions. Metrika 32(1), 85–91 (1985). https://doi.org/10.1007/BF01897803 8. Ting Lee, M.-L.: Properties and applications of the Sarmanov family of bivariate distributions. Commun. Stat. Theory Methods 25(6), 1207–1222 (1996). https://doi.org/10.1080/03610929608831759 9. Jones, M.: Multivariate t and beta distributions associated with the multivariate F distribution. Metrika 54(3), 215–231 (2002). https://doi.org/10.1007/s184-002-8365-4 10. Olkin, I., Liu, R.: A bivariate beta distribution. Stat. Probability Lett. 62(4), 407–412 (2003). https://doi. org/10.1016/S0167-7152(03)00048-8 11. Nadarajah, S., Kotz, S.: Some bivariate beta distributions. Statistics 39(5), 457–466 (2005). https://doi. org/10.1080/02331880500286902 12. Sarabia, J.M., Castillo, E.: Bivariate distributions based on the generalized three-parameter beta distribution. In: Advances in Distribution Theory, Order Statistics, and Inference, pp. 85–110. Birkhäuser Boston, Boston (2006). https://doi.org/10.1007/0-8176-4487-3_6 13. El-Bassiouny, A., Jones, M.: A bivariate F distribution with marginals on arbitrary numerator and denominator degrees of freedom, and related bivariate beta and t distributions. Stat. Methods Appl. 18(4), 465 (2009). https://doi.org/10.1007/s10260-008-0103-y 14. Arnold, B.C., Ng, H.K.T.: Flexible bivariate beta distributions. J. Multivariate Anal. 102(8), 1194–1202 (2011). https://doi.org/10.1016/j.jmva.2011.04.001 15. Bran-Cardona, P.A., Orozco-Castañeda, J., Nagar, D.K.: Bivariate generalization of the Kummer-beta distribution. Revista Colombiana de Estadística 34(3), 497–512 (2011) 16. Gupta, A.K., Orozco-Castañeda, J.M., Nagar, D.K.: Non-central bivariate beta distribution. Stat. Papers 52(1), 139–152 (2011). https://doi.org/10.1007/s00362-009-0215-y 17. Orozco-Castañeda, J.M., Nagar, D.K., Gupta, A.K.: Generalized bivariate beta distributions involving Appell’s hypergeometric function of the second kind. Comput. Math. Appl. 64(8), 2507–2519 (2012). https://doi.org/10.1016/j.camwa.2012.06.006 18. Olkin, I., Trikalinos, T.A.: Constructions for a bivariate beta distribution. Stat. Probability Lett. 96, 54–60 (2015). https://doi.org/10.1016/j.spl.2014.09.013 19. Nadarajah, S., Shih, S.H., Nagar, D.K.: A new bivariate beta distribution. Statistics 51(2), 455–474 (2017). https://doi.org/10.1080/02331888.2016.1240681 20. Koutoumanou, E., Wade, A., Cortina-Borja, M.: Local dependence in bivariate copulae with beta marginals. Revista Colombiana de Estadística 40(2), 281–296 (2017). https://doi.org/10.15446/rce. v40n2.59404 21. David Sam Jayakumar, G., Sulthan, A., Samuel, W.: A new bivariate beta distribution of Kind-1 of Type-A. J. Stat. Manag. Syst. 22(1), 141–158 (2019). https://doi.org/10.1080/09720510.2018.1537593 22. Crackel, R., Flegal, J.: Bayesian inference for a flexible class of bivariate beta distributions. J. Stat. Comput. Simul. 87(2), 295–312 (2017). https://doi.org/10.1080/00949655.2016.1208202 23. Plummer, M.: JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. In: Proceedings of the 3rd International Workshop on Distributed Statistical Computing, vol. 124, pp. 1–10 . Vienna, Austria (2003) 24. Johansson, F., et al.: Mpmath: a Python Library for Arbitrary-precision Floating-point Arithmetic (version 0.18). (2013). http://mpmath.org/ 25. Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S.J., Brett, M., Wilson, J., Millman, K.J., Mayorov, 123 180 S. Trick et al. N., Nelson, A.R.J., Jones, E., Kern, R., Larson, E., Carey, C.J., Polat, ˙ I., Feng, Y., Moore, E.W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E.A., Harris, C.R., Archibald, A.M., Ribeiro, A.H., Pedregosa, F., van Mulbregt, P., SciPy 1.0 Contributors: SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods 17, 261–272 (2020). https://doi.org/10.1038/s41592019-0686-2 26. Jacobs, R.A.: Methods for combining experts’ probability assessments. Neural Comput. 7(5), 867–888 (1995). https://doi.org/10.1162/neco.1995.7.5.867 27. Trick, S., Rothkopf, C.: Bayesian classifier fusion with an explicit model of correlation. In: International Conference on Artificial Intelligence and Statistics. PMLR, pp 2282–2310 (2022) 28. Ueda, N., Nakano, R.: Generalization error of ensemble estimators. In: Proceedings of International Conference on Neural Networks (ICNN’96), vol. 1, pp. 90–95 . https://doi.org/10.1109/ICNN.1996. 548872. IEEE (1996) 29. Shapiro, A.D.: Structured Induction in Expert Systems. Addison-Wesley, Boston (1987) 30. Dua, D., Graff, C.: UCI Machine Learning Repository (2017). http://archive.ics.uci.edu/ml 31. Sesmero, M.P., Iglesias, J.A., Magán, E., Ledezma, A., Sanchis, A.: Impact of the learners diversity and combination method on the generation of heterogeneous classifier ensembles. Appl. Soft Comput. 111, 107689 (2021). https://doi.org/10.1016/j.asoc.2021.107689 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 123