Often we think of our prior beliefs in terms of a central tendency and certainty about that central tendency. You can see in Figure 6.1 that when a = 4 and b = 4, the beta distribution is peaked at θ = 0.5, but higher or lower values of θ are moderately probable too. As another example, if we think that the coin is probably fair but we're not very sure, then we can imagine that the previously observed data had, say, a = 4 heads and b = 4 tails. You can see in Figure 6.1 that when a = 1 and b = 1, the beta distribution is uniform: All values of θ are equally probable. For example, if we have no prior knowledge other than the knowledge that the coin has a head side and a tail side, that's tantamount to having previously observed one head and one tail, which corresponds to a = 1 and b = 1. You can think of a and b in the prior as if they were previously observed data, in which there were a heads and b tails in a total of n = a + b flips. We would like to specify a beta distribution that describes our prior beliefs about θ. The shape parameter a increases from left to right across the columns, while the shape parameter b increases from top to bottom across the rows. Examples of the beta distribution (Equation 6.1). Therefore, to express the prior beliefs over θ, we seek a probability density function involving θ a(1 ‒ θ) b.įigure 6.1. If you think about it a minute, you'll notice that if the prior is of the form θ a (1 - θ) b, then when you multiply the Bernoulli likelihood with the prior, you'll again get a function of the same form, namely θ ( y+ a)(1 ‒ θ) (1 ‒ y+ b). In the present situation, we are seeking a functional form for a prior density over θ that is conjugate to the Bernoulli likelihood function in Equation 6.1. Notice that the prior is conjugate only with respect to a particular likelihood function. When the forms of p( y| θ) and p( θ) combine so that the posterior distribution has the same form as the prior distribution, then p( θ) is called a conjugate prior for p( y| θ). This quality also depends on how the form of the function p( θ) relates to the form of the function p( y| θ). 107), namely ∫ d θ p ( y | θ ) p ( θ ), to be solvable analytically. Second, we desire the denominator of Bayes' rule (Equation 5.9, p. Therefore, no matter how much data we include, we always get a posterior of the same functional form.

This quality allows us to include subsequent additional data and derive another posterior distribution, again of the same form as the prior. When this is the case, the prior and posterior beliefs are described using the same form of function. First, it would be convenient if the product of p( y| θ) and p( θ), which is in the numerator of Bayes' rule, results in a function of the same form as p( θ). 106), however, there are two desiderata for mathematical tractability. When we intend to apply Bayes' rule (Equation 5.7, p. In principle, we could use any probability density function supported on the interval. That is, we need a mathematical formula that describes the prior probability for each value of the parameter θ on the interval. To do this, we need a mathematical description of the prior allocation of credibilities. In this chapter, we use purely mathematical analysis, with no numerical approximation, to derive the mathematical form of the posterior credibilities of parameter values.

Kruschke, in Doing Bayesian Data Analysis (Second Edition), 2015 6.2 A description of credibilities: The beta distribution In addition, in some applications of the Markov chain Monte Carlo (MCMC) algorithm to fit item response theory (IRT) models, the prior distribution for the guessing parameter is sometimes assumed to be the beta distribution. Alternatively, if Y 1, Y 2, …, Y n are independent random variables following the U(0, 1) (i.e., the continuous uniform distribution between 0 and 1) distribution, then the sth order statistic Y s ′ follows a beta distribution with parameters s and n − s + 1.Ī well-known application of the beta distribution (actually, that of a more general version of the distribution that has, in addition to the a and b parameters, two more parameters specifying the bounds of the distribution) in education can be found in Lord (1965), where the true test score was modeled using the beta distribution. If V 1 and V 2 are independent and follow the χ 2 distribution (the section on the χ 2 distribution that follows will discuss generating random numbers following that distribution) with parameters c 1 and c 2, respectively, then V 1 V 1 + V 2 follows a beta distribution with parameters c 1/2 and c 2/2. Random numbers following the beta distribution can be generated in several ways. It is also possible to estimate a and b using maximum likelihood, but the method requires an iterative procedure and hence is more computationally intensive.

Beta distribution cdf