Beta distribution and Dirichlet distribution

How is Gamma function discovered? It proves that \ begin {Align *} B (m, n) = \ int_0 ^ 1 x ^ {M-1} (32a) ^ {n-1} \ Text {d} X = \ frac {\ gamma (m) \ gamma (n)} {\ gamma (m + n )} \ end {Align *} So \ begin {Align *} f _ {m, n} (X) =\begin {cases} \ frac {x ^ {M-1} (32a) ^ {n-1 }}{ B (m, n )} = \ frac {\ gamma (m + n) }{\ gamma (m) \ gamma (n)} x ^ M-1) ^ {n-1} & 0 \ Leq x \ Leq 1 \ 0 & \ Text {Other cases} \ end {Align *}: $ F _ {M, n} (x) $ points are $1 $, that is$ F _ {m, n} (x) $ corresponds to a probability distribution.Because the denominator of this function is $ beta $, we generally call it the corresponding distribution$ Beta $ distribution with the parameter $ M and N $.

The following describes the Digital features of the distribution, yi Zhi's $ K $ moment of order is \ begin {Align *} e [x ^ K] = \ int_0 ^ 1 x ^ K f _ {m, n} (X) \ Text {d} X = \ int_0 ^ 1 \ frac {x ^ {M + k-1} (_x) ^ {n-1} {B (m + K, n)} \ frac {B (m + k, n)} {B (m, n)} \ Text {d} X = \ frac {\ gamma (m + k) \ gamma (m + n) }{\ gamma (m) \ gamma (m + K + n )} \ end {Align *} So \ begin {Align *} e [x] = \ frac {\ gamma (m + 1) \ gamma (m + n )} {\ gamma (m) \ gamma (m + 1 + n) }=\ frac {m} {m + n }, \ e [x ^ 2] = \ frac {\ gamma (m + 2) \ gamma (m + n)} {\ gamma (m) \ gamma (m + 2 + n) }=\ frac {(m + 1) m} {(m + n + 1) (m + n )} \ end {Align *} the mean and variance are \ begin {Align *} e [x] = \ frac {m} {m + n}, respectively }, \ D [x] = \ frac {(m + 1) m} {(m + n + 1) (m + n )} -\ left (\ frac {m} {m + n} \ right) ^ 2 = \ frac {Mn} {(m + n + 1) (m + n) ^ 2} \ end {Align *}

$ Beta $ functions are binary functions, which can be extended into the following $ k + 1 (k \ geq 2) $ yuan form: \ begin {Align} \ label {EQ: multivariate beta function} B (M_1, \ cdots, M _ {k + 1 }) = \ int_0 ^ 1 X_1 ^ {m_1-1} \ int_0 ^ {} X_2 ^ {m_2-1} \ cdots \ int_0 ^ {-\ cdots-X _ {k-1} X_k ^ {m_k-1} (1-X_1-\ cdots-X_k) ^ {M _ {k + 1}-1} \ Text {d} X_1 \ Text {d} X_2 \ cdots \ Text {d} X_k \ end {Align} attention (\ ref {EQ: multivariate beta function}) is a $ K $ multi-point, which is used to evaluate the core of $ X_k $, that is, \ begin {Align *} e_k (M_k, M _ {k + 1 }) = \ int_0 ^ {1-x_1-\ cdots-X _ {k-1} X_k ^ {m_k-1} (1-X_1-\ cdots-X_k) ^ {M _ {k + 1}-1} \ Text {d} X_k = \ int_0 ^ t X_k ^ {m_k-1} (t-X_k) ^ {M _ {k + 1}-1} \ Text {d} X_k \ end {Align *} where $ t =-\ cdots-X _ {k-1} $. The \ begin {Align *} e_k (m_k, M _ {k + 1}) & =\ int_0 ^ T (t-X_k) ^ {M _ {k + 1}-1} \ Text {d} \ frac {X_k ^ {m_k} \ & = (t-X_k) ^ {M _ {k + 1}-1} \ frac {X_k ^ {m_k} | _ 0 ^ t-\ int_0 ^ t \ frac {X_k ^ {m_k }}{ m_k} (M _ {k + 1}-1) (T-X_k) ^ {M _ {k + 1}-2} (-1) \ Text {d} X_k \ & =\ frac {M _ {k + 1}-1} {m_k} e_k (m_k + 1, M _ {k + 1}-1) \ end {Align *} So the recursion goes down with \ begin {Align *} e_k (m_k, M _ {k + 1 }) & =\ frac {M _ {k + 1}-1} {m_k} e_k (m_k + 1, M _ {k + 1}-1 )\ \ & =\ Frac {M _ {k + 1}-1} {m_k} \ frac {M _ {k + 1}-2} {m_k + 1} e_k (m_k + 2, M _ {k + 1}-2) \\& =\ cdots \\\& =\ frac {M _ {k + 1}-1} {m_k} \ cdots \ frac {1} {m_k + M _ {k + 1}-2} e_k (m_k + M _ {k + 1}-1, 1) \ end {Align *} And \ begin {Align *} e_k (m_k + M _ {k + 1}-1, 1) = \ int_0 ^ t X_k ^ {m_k + M _ {k + 1}-2} \ Text {d} X_k = \ frac {X_k ^ {m_k + M _ {k + 1}-1 }}{ m_k + M _ {k + 1}-1} | _ 0 ^ t = \ frac {t ^ {m_k + M _ {k + 1} -1 }}{ m_k + M _ {k + 1}-1} \ end {Align *} So \ begin {Align *} e_k (m_k, M _ {k + 1}) =\ frac {\ gamma (M _ {k + 1}) \ gamma (m_k )} {\ gamma (M _ {k + 1} + m_k)} (-\ cdots-X _ {k-1 }) ^ {m_k + M _ {k + 1}-1} \ end {Align *} replace it (\ ref {EQ: Multivariate beta function }), next we examine the points \ begin {Align *} e _ {k-1} (M _ {k-1 }, m_k + M _ {k + 1 }) & =\ int_0 ^ {1-x_1-\ cdots-X _ {K-2} X _ {k-1} ^ {M _ {k-1}-1} \ frac {\ gamma (M _ {k + 1 }) \ gamma (m_k)} {\ gamma (M _ {k + 1} + m_k)} (1-x_1-\ cdots- X _ {k-1 }) ^ {m_k + M _ {k + 1}-1} \ Text {d} X _ {k-1} \ & =\ frac {\ gamma (M _ {k + 1 }) \ gamma (m_k) }{\ gamma (M _ {k + 1} + m_k )} \ int_0 ^ t x _ {k-1} ^ {M _ {k-1}-1} (t-X _ {k-1 }) ^ {m_k + M _ {k + 1}-1} \ Text {d} X _ {k-1} \ end {Align *} where $ t =-\ cdots-x _ {K-2} $. So continue to follow the previous method (recurrence after the division points) can get \ begin {Align *} e _ {k-1} (M _ {k-1 }, m_k + M _ {k + 1}) & =\ frac {\ gamma (M _ {k + 1}) \ gamma (m_k )} {\ gamma (M _ {k + 1} + m_k)} \ frac {\ gamma (M _ {k + 1} + m_k) \ gamma (M _ {k-1})} {\ gamma (M _ {k + 1} + m_k + M _ {k-1 })} (-\ cdots-X _ {K-2 }) ^ {M _ {k + 1} + m_k + M _ {k-1}-1} \ & =\ frac {\ gamma (M _ {k + 1 }) \ gamma (m_k) \ gamma (M _ {k-1})} {\ gamma (M _ {k + 1} + m_k + M _ {k-1 })} (-\ cdots-X _ {K-2 }) ^ {M _ {k + 1} + m_k + M _ {k-1}-1} \ end {Align *} repeats this process and we can see \ begin {Align} \ label {EQ: e2} E_2 (M_2, M _ {k + 1} + m_k + \ cdots + M_3) = \ frac {\ gamma (M _ {k + 1 }) \ gamma (m_k) \ cdots \ gamma (M_2) }{\ gamma (M _ {k + 1} + m_k + \ cdots + M_2) ^ {M _ {k + 1} + m_k + \ cdots + M_2-1} \ end {Align} the final points for $ X_1 $ are \ begin {Align *} B (M_1, \ cdots, M _ {k + 1}) & =\ int_0 ^ 1 X_1 ^ {m_1-1} \ frac {\ gamma (M _ {k + 1 }) \ gamma (m_k) \ cdots \ gamma (M_2) }{\ gamma (M _ {k + 1} + m_k + \ cdots + M_2) ^ {M _ {k + 1} + m_k + \ cdots + M_2-1} \ Text {d} X_1 \\\&=\ frac {\ gamma (M _ {k + 1 }) \ gamma (m_k) \ cdots \ gamma (M_2) }{\ gamma (M _ {k + 1} + m_k + \ cdots + M_2 )} \ frac {\ gamma (M _ {k + 1} + m_k + \ cdots + M_2) \ gamma (M_1 )} {\ gamma (M _ {k + 1} + m_k + \ cdots + M_1 )} 1 ^ {M _ {k + 1} + m_k + \ cdots + M_1-1 }\\&=\ frac {\ gamma (M _ {k + 1 }) \ gamma (m_k) \ cdots \ gamma (M_1) }{\ gamma (M _ {k + 1} + m_k + \ cdots + M_1 )} \ end {Align *} order $ \ boldsymbol {m} = [M_1, \ cdots, M _ {k + 1}] $, $ \ boldsymbol {x} = [x_1, \ cdots, X _ {k + 1}] $ and define \ begin {Align *} f _ {\ boldsymbol {m} (\ boldsymbol {x }) =\begin {cases} \ frac {\ gamma (M _ {k + 1} + m_k + \ cdots + M_1) }{\ gamma (M _ {k + 1 }) \ gamma (m_k) \ cdots \ gamma (M_1 )} \ prod _ {I = 1} ^ {k + 1} X_ I ^ {m_ I-1} & \ sum _ {I = 1} ^ {k + 1} X_ I = 1 \ \ 0 & \ Text {Other cases} \ end {Align *} note that this is a $ K $ variable function (and the limit of $1 $ ), from the above derivation, we can see that $ K $ of $ F _ {\ boldsymbol {m} (\ boldsymbol {x}) $ is $1 $, therefore, $ F _ {\ boldsymbol {m} (\ boldsymbol {x}) $ also corresponds to a probability distribution, and our corresponding distribution is$ Dirichlet $ distribution of $ \ boldsymbol {m} $.

The following is a brief description of the Digital features of the distribution. It is easy to know \ begin {Align *} x_j ^ n f _ {\ boldsymbol {m} (\ boldsymbol {x }) & =\ frac {\ gamma (M _ {k + 1} + \ cdots + M_1)} {\ gamma (M _ {k + 1 }) \ cdots \ gamma (M_1 )} x_j ^ n \ prod _ {I = 1} ^ {k + 1} X_ I ^ {m_ I-1} \\&=\ frac {\ gamma (M _ {k + 1} + \ cdots + M_1 )} {\ gamma (M _ {k + 1} + \ cdots + m_j + N + \ cdots + M_1)} \ frac {\ gamma (m_j + n )} {\ gamma (m_j)} \ frac {\ gamma (M _ {k + 1} + \ cdots + m_j + N + \ cdots + M_1 )} {\ gamma (M _ {k + 1}) \ cdots \ gamma (m_j + n) \ cdots \ gamma (M_1 )} x_j ^ n \ prod _ {I = 1} ^ {k + 1} X_ I ^ {m_ I-1} \ end {Align *} Then
\ Begin {Align *} e [x_j] & =\ frac {\ gamma (M _ {k + 1} + \ cdots + M_1 )} {\ gamma (M _ {k + 1} + \ cdots + m_j + 1 + \ cdots + M_1)} \ frac {\ gamma (m_j + 1 )} {\ gamma (m_j )} = \ frac {m_j} {M _ {k + 1} + \ cdots + M_1} \ e [x_j ^ 2] & =\ frac {\ gamma (M _ {k + 1} + \ cdots + M_1 )} {\ gamma (M _ {k + 1} + \ cdots + m_j + 2 + \ cdots + M_1)} \ frac {\ gamma (m_j + 2 )} {\ gamma (m_j) }=\ frac {(m_j + 1) m_j} {(M _ {k + 1} + \ cdots + M_1 + 1) (M _ {k + 1} + \ cdots + M_1 )} \ end {Align *} the mean and variance are \ begin {Align *} e [x] & =\ frac {m_j} {M _ {k + 1} + \ cdots + M_1} \ D [x] & =\ frac {(m_j + 1) m_j} {(M _ {k + 1} + \ cdots + M_1 + 1) (M _ {k + 1} + \ cdots + M_1 )} -\ left (\ frac {m_j} {M _ {k + 1} + \ cdots + M_1} \ right) ^ 2 = \ frac {m_j (M _ {k + 1} + \ cdots + M_1-m_j )} {(M _ {k + 1} + \ cdots + M_1 + 1) (M _ {k + 1} + \ cdots + M_1) ^ 2} \ end {Align *} And \ begin {Align *} x_p x_q F _ {\ boldsymbol {m} (\ boldsymbol {x }) & =\ frac {\ gamma (M _ {k + 1} + \ cdots + M_1)} {\ gamma (M _ {k + 1 }) \ cdots \ gamma (M_1 )} x_p x_q \ prod _ {I = 1} ^ {k + 1} X_ I ^ {m_ I-1 }\\&=\ frac {\ gamma (M _ {k + 1} + \ cdots + M_1 )} {\ gamma (M _ {k + 1} + \ cdots + m_j + 2 + \ cdots + M_1)} \ frac {\ gamma (m_p + 1 )} {\ gamma (m_p)} \ frac {\ gamma (m_q + 1)} {\ gamma (m_q )} \ frac {\ gamma (M _ {k + 1} + \ cdots + m_j + 2 + \ cdots + M_1) }{\ gamma (M _ {k + 1 }) \ cdots \ gamma (m_p + 1) \ cdots \ gamma (m_q + 1) \ cdots \ gamma (M_1 )} x_p x_q \ prod _ {I = 1} ^ {k + 1} X_ I ^ {m_ I-1} \ end {Align *} So \ begin {Align *} e [x_p x_q] =\ frac {\ gamma (M _ {k + 1} + \ cdots + M_1 )} {\ gamma (M _ {k + 1} + \ cdots + m_j + 2 + \ cdots + M_1)} \ frac {\ gamma (m_p + 1 )} {\ gamma (m_p)} \ frac {\ gamma (m_q + 1)} {\ gamma (m_q )} = \ frac {m_p m_q} {(M _ {k + 1} + \ cdots + M_1 + 1) (M _ {k + 1} + \ cdots + M_1 )} \ end {Align *} the covariance is \ begin {Align *} Cov (x_p, x_q) & = E [x_p x_q]-E [x_p] E [x_q] \\\&=\ frac {m_p m_q} {(M _ {k + 1} + \ cdots + M_1 + 1) (M _ {k + 1} + \ cdots + M_1 )} -\ frac {m_p} {M _ {k + 1} + \ cdots + M_1} \ frac {m_q} {M _ {k + 1} + \ cdots + M_1 }\ \ & =\ frac {-m_p m_q} {(M _ {k + 1} + \ cdots + M_1 + 1) (M _ {k + 1} + \ cdots + M_1) ^ 2} \ end {Align *}

Formula (\ ref {EQ: E2}) zhi \ begin {Align *} p (X_1 = T) & = t ^ {M_1-1} \ frac {\ gamma (M _ {k + 1}) \ gamma (m_k) \ cdots \ gamma (M_2 )} {\ gamma (M _ {k + 1} + m_k + \ cdots + M_2)} (1-T) ^ {M _ {k + 1} + m_k + \ cdots + M_2-1 }\\&=\ frac {\ gamma (M _ {k + 1 }) \ gamma (m_k) \ cdots \ gamma (M_1) }{\ gamma (M_1) \ gamma (M _ {k + 1} + m_k + \ cdots + M_1-M_1 )} t ^ {M_1-1} (1-T) ^ {M _ {k + 1} + m_k + \ cdots + M_1-M_1-1} \ end {Align *} can be seen by symmetry
\ Begin {Align *} p (x_ I = T) = \ frac {\ gamma (M _ {k + 1}) \ gamma (m_k) \ cdots \ gamma (M_1 )} {\ gamma (m_ I) \ gamma (M _ {k + 1} + m_k + \ cdots + M_1-m_ I)} t ^ {m_ I-1} (1-T) ^ {M _ {k + 1} + m_k + \ cdots + M_1-m_ I-1} \ end {Align *} This means$ Dirichlet $ the marginal distribution of the variable $ X_ I $ is the $ beta $ distribution of the parameter $ m_ I, M _ {k + 1} + m_k + \ cdots + M_1-m_ I $.

Beta distribution and Dirichlet distribution