# Why no "extra coefficient K" in the formula of Laplace's law of succession?

Below is an example about the calculation of Laplace's law of succession:

Suppose we observe $$y$$ responses out of $$n$$ binomial trials. Assuming the trials are indenpendent, with unknown response probabilty $$\theta$$. The event it will happen next time is $$\widetilde{Y}$$.

The binomial sampling distribution is:

$$p(y|n,\theta)=\left(\begin{array}{c}n\\ y\end{array}\right)\theta^{y}(1-\theta)^{n-y}$$ (Eq. 1)

Suppose a $$Beta(1,1)$$ (i.e., uniform) prior on $$\theta$$, and consider the case $$y=n$$, that is, the event has happended at every opportunity. The posterior distribution is:

$$p(\theta|y,n)\propto p(y|n,\theta)*p(\theta) \propto Beta(y+1,1)$$ (Eq. 2)

NOTE: in Eq. 2, there is "proportion symbol $$\propto$$", not a "$$=$$".

So, I give it a "coefficient": $$K$$.

$$p(\theta|y,n)=K*Beta(y+1,1)$$ (Eq. 3)

Then Laplace's law of succession (i.e., posterior-predictive expectation) is:

$$\begin{split} E[\widetilde{Y}|y,n]&=p(\widetilde{Y}=1|y,n)\\ &=\int_{}^{} \theta*p(\theta|y,n)d\theta\\ &=\int_{}^{} \theta*K*\frac{\Gamma(y+2)}{\Gamma(y+1)\Gamma(1)}*\theta^{y}(1-\theta)^{0}d\theta\\ &=K*\frac{y+1}{y+2}=K*\frac{n+1}{n+2} \end{split}$$

But the correct answer should be:$$\frac{n+1}{n+2}$$.

Why do I have an extra coefficient "$$K$$"?

Well, $$p(\theta|y,n)$$ is actually equal to $$\text{Beta}(y+1,1)$$, not just proportional. Yes, they're both proportional to the middle term, $$p(y|n,\theta)p(\theta)$$, however they're equal, which means $$K=1$$ and your result is correct. The proportionality argument is generally used to describe the format, and therefore type, of the PDF; where you actually found that $$p(\theta|y,n)$$ is of Beta form.
Also, from pure mathematical view, you can easily check that they're equal if you integrate both sides, because since both are PDFs, they must integrate to $$1$$. $$1=\int_0^1 p(\theta|y,n)d\theta=\int_0^1 K\text{Beta}(y+1,1)d\theta=1$$ which leaves us with $$1=K$$.
• The mathematical view helps me understand better! $K$ is truly 1