Update rule for beta distribution with fixed K/confidence/sample size Normally you have a beta distribution with shape parameters $a$ and $b$.
The mean of this distribution is $a / (a + b)$ and the sample size, or the confidence (or K) is $a + b$.
Now, if you do some trials, with let's say N positive outcomes and M negative outcomes, you end up with a posterior distribution that is $\text{Beta}(a + N,~b + M)$.
So, now your mean is $\frac{a + N}{a + N + b + M}$ and the sample size / confidence / K is $a + N + b + M$.
Now, my question is: what if you want to keep the K / confidence level fixed? So let's say K should always be 10. So $\text{Beta}(5,5)$ is fine, as is $\text{Beta}(9,1)$, or $\text{Beta}(1.23, 8.77)$.  
In other words: in this case I would like the mean of the posterior to be able change to reflect the evidence found in the new data, but the confidence level should remain the same (rather than increasing all the time).
Is there a simple update rule for this scenario as well?
 A: I've taken a look at the book. It seems to me that the rationale for this "prior sample size" teminology is the following. We have the usual model with $X_1,\dots,X_n$ conditionaly independent and identically distributed, given $\Theta=\theta$, with distribution $X_1\mid\Theta=\theta\sim\mathrm{Ber}(\theta)$. Suppose that a priori $\Theta\sim\mathrm{Beta}(a,b)$. The prior mean is just $\mathbb{E}[\Theta]=a/(a+b)=:\mu$. If $a$ and $b$ are integers bigger than $1$, one way to interpret this prior is to suppose that we started with a $\mathrm{U}[0,1]$ prior and observed $a-1$ successes and $b-1$ failures in a Bernoulli experiment. By Bayes's theorem, the "posterior" of $\Theta$ for this gedanken experiment is exaclty $\mathrm{Beta}(a,b)$. Hence, we may suggestively define the prior sample size $\nu:=a+b-2$. We know from the properties of the beta distribution that a bigger $\nu$ will give us a more concentrated distribution. Now, to answer your question, what you want to do seems impossible: the more data you observe, the smaller will be your posterior uncertainty about $\Theta$. The posterior of $\Theta$ is $\mathrm{Beta}(c,d)$, with $c=a+\sum_{i=1}^n x_i$, and $d=b+n-\sum_{i=1}^n x_i$. Hence, $c+d$ grows linearly with $n$.
A: I like Zen's answer and want to add to it in order to clarify some misconceptions in the question.  "Confidence" or "sample size" cannot be meaningfully applied to a posterior distribution.  Only a likelihood can have a notion of "sample size".  That's the motivation behind Zen's point that the confidence should be $a + b - 2$.
In general, the only way to map the likelihood of an exponential family distribution into a confidence is to linearly transform its natural parameters (for the Beta distribution these are $a-1, b-1$) to the positive real line.
(For example, a light that turns on when two heads are flipped, another light turns on when two tails are flipped.  So, a single sample might induce a likelihood such that $a$ or $b$ goes up by more than 1.)
To answer your question: do the whole Bayesian inference to arrive at the natural parameters of the induced likelihood ($\Delta a, \Delta b$ in your case), and then scale these to your desired "confidence" if it exceeds it.  After scaling, you can combine your likelihood with your prior.
To me, this scaling is meaningful:  You're saying that the evidence you have saturates after a certain point.  (I'm interested in this subject if anyone has any references.)
