Notes on Chapter 2 Probability

$\def\bra#1{\mathinner{\langle{#1}|}} \def\ket#1{\mathinner{|{#1}\rangle}} \def\braket#1{\mathinner{\langle{#1}\rangle}} \newcommand{\mdef}{\overset{\mathrm{def}}{=}} \newcommand{\bm}{\mathbf} \DeclareMathOperator{\Var}{Var} \DeclareMathOperator{\det}{det} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\E}{E} \DeclareMathOperator{\Cov}{Cov} \DeclareMathOperator{\Beta}{B} \DeclareMathOperator{\Bdist}{Beta} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\adj}{adj} \DeclareMathOperator{\ii}{i} \DeclareMathOperator{\dd}{d} \newcommand{\pp}{\partial} \DeclareMathOperator{\rhs}{RHS} \DeclareMathOperator{\lhs}{LHS} \DeclareMathOperator{\Beta}{B}$

# Multinomial Integral

For a group of variables $x_i>0$ with constraint $\sum_{i=1}^{n} x_i$. We define tail sum of exponents $p_i>0$ to be $S_k=\sum_{i=k}^n p_i$, and $T_k=\sum_{i=k}^{n} x_i$. Define Integral operator $$\hat \int_k=\prod_{i=1}^k\int_0^{1-S_{i-1}} x_i^{p_i-1}dx_i$$ we can write multi-integral:

\begin{align} \Beta(\mathbf{p}) &=\hat\int_n \delta\left(1-S_n\right)=\hat \int_{n-1}(1-S_{n-1})^{T_n-1}\\ &=\Beta(T_n,p_{n-1})\hat \int_{n-2} (1-S_{n-2})^{T_{n-1}-1}\\ &=\prod_{i=1}^{n-1}\Beta(T_{i+1},p_i)=\prod_{i=1}^{n-1}\frac{\Gamma(T_{i+1})}{\Gamma(T_i)}\Gamma(p_i)\\ &=\frac{\Gamma(T_n)}{\Gamma(T_1)}\prod_{i=1}^{n-1}\Gamma(p_i)=\frac{\prod_i\Gamma(p_i)}{\Gamma(\sum_i p_i)} \end{align}

If we have $\mathbf{p'}=\mathbf{p}+(1,0,\ldots,0)$, then $$\Beta(\mathbf{p'})=\frac{p_0}{\sum p_i}\Beta(\mathbf{p})$$