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There are many ways to define bivariate beta distributions, that is, bivariate distributions on the square $[0,1]\times [0,1]$ with beta marginals.One way is to start with the usual stochastic representation of the beta distribution using gamma variates, let $X\sim\mathcal{Gamma}(\alpha,\theta),~~ Y\sim\mathcal{Gamma}(\beta,\theta)$ (independent), then  \...

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The intuitive way to interpret f(x) is through its integral. If you integrate f(x) over the interval x=a to x=b, then the result is the probability of x falling into [a,b]. So its not quite the probability that X=x, but its very closely related. Specifically, you can interpret f(x) as being a relative probability. So for example if f(1) = 0.8 and f(2) = 1.6 ...

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Every conjugate prior distribution to an exponential family has some set of parameters $\eta_0$ that result in a uniform distribution over the space. You can see this here by considering what happens to the natural parameters $\eta'$ of the conjugate prior when the number of pseudo-observations $n$ equals $0$. Unless I'm mistaken, you can interpret a and b ...

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In Bayesian statistics one uses a 'flat' prior distribution for a parameter in the absence of knowledge or opinion about about the parameter value. When the parameter is binomial success probability $p$ it may seem natural to use either a uniform prior $\mathsf{Beta}(\alpha=1,\beta=1)\equiv\mathsf{Unif}(0,1)$ or even the "bathtub shaped" prior $\... 1 The cumulants can be obtained from the cumulant generating function (cgf), which is the logarithm of the moment generating function. The cgf can be used to approximate the density function via the saddlepoint approximation, see How does saddlepoint approximation work?. Other ideas can be found at Constructing a continuous distribution to match$m\$ moments ...

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