On p38 of Lee and Wagenmakers (2012) "Bayesian Cognitive Modeling: A Practical Course" the following passage appears:
"One of the nice properties of using the θ ~ Beta (α,β) prior distribution for a rate is that it has a natural interpretation. The α and β values can be thought of as counts of, respectively, “prior successes” and “prior failures.” This means that using a θ ~ Beta (3,1) prior corresponds to having the prior information that 4 previous observations have been made, and 3 of them were successes. Or, more elaborately, starting with a θ ~ Beta (3,1) is the same as starting with a θ ~ Beta (1,1), and then seeing data giving two more successes (i.e., the posterior distribution in the second scenario will be same as the prior distribution in the first)."
I'm finding it hard to match that explanation with the following diagram, which I generated using MATLAB. In all instances in the figure the 'b' parameter is three times the size of 'a'.
I understand why the distribution becomes more peaked around 0.25 in the Beta (100,300) condition than in the (10,30) condition - there's stronger evidence for θ being 0.25.
However, I don't understand what's going on in the (0.25,0.75) and (1,3) conditions. would have thought they'd both be fat-tailed distributions centred on 0.25. I don't understand why the mode of both distributions seems to be around 0.