Maximum entropy probability distribution

\begin{array}{lcl} \int_{-\infty}^\infty p(x) dx &=& 1 \\ E(f_j(x)) &=& \int_{-\infty}^\infty f_j(x) p(x) dx=a_j \\ H(x) &=& - \int_{-\infty}^\infty p(x) \ln p(x) dx \\ J(p(x)) &=& - \int_{-\infty}^\infty p(x) \ln p(x) dx + \lambda_0 (\int_{-\infty}^\infty p(x) dx -1) + \sum_j \lambda_j (\int_{-\infty}^\infty f_j(x) p(x) dx- a_j) \\ \frac {\partial J}{\partial p(x)} &=& -\ln p(x) -1 + \lambda_0 + \sum_j \lambda_j f_j(x) = 0 \\ p(x) &=& e^{-1+\lambda_0 } e^{\sum_j \lambda_j f_j(x)} \end{array}

\begin{array}{lcl} \Gamma(x) &=& \int_0^{+\infty} e^{-t} t^{x-1} dt \quad (x \gt 0) = (x-1)! \qquad \text{这是Gamma函数，不是Gamma分布} \\ \psi(x) &=& \frac {d}{d x} \ln \Gamma(x) = \frac {\Gamma^\prime (x)} {\Gamma(x)} \qquad \text{digamma function} \\ B(p,q) &=& \frac {\Gamma(p) \Gamma(q)}{\Gamma(p+q)} \qquad \text{beta function} \\ \gamma_E \qquad \text{Euler's constant} \\ \end{array}

\int_{-\infty}^\infty p(x) dx = 1 \\ E(x) = k \theta \\ E(\ln(x)) = \psi(k) + \ln(\theta) \\

\begin{align} J(p(x)) = & - \int_{-\infty}^\infty p(x) \ln p(x) dx \\ & + \lambda_0 (\int_{-\infty}^\infty p(x) dx - 1) \\ & + \lambda_1 (\int_{-\infty}^\infty p(x)x dx - k) \\ & + \lambda_2 (\int_{-\infty}^\infty p(x) \ln(x) dx- \psi(k) - \ln(\theta)) \\ \end{align} \\ \frac {\partial J}{\partial p(x)} = -\ln p(x) -1 + \lambda_0 + \lambda_1 x + \lambda_2 \ln(x) = 0 \\ p(x) = \exp(\lambda_0 -1 + \lambda_1 x + \lambda_2 \ln(x)) = e^{\lambda_0 -1} x ^ {\lambda_2} e^{\lambda_1 x} \\

\delta (H(p(x))) + \delta \lambda_0 (\int_{-\infty}^\infty p(x) dx ) + \delta \lambda_1 (\int_{-\infty}^\infty p(x)x dx ) + \delta \lambda_2 (\int_{-\infty}^\infty p(x) \ln(x) dx ) = 0 \\ p(x) = \exp(\lambda_0 -1 + \lambda_1 x + \lambda_2 \ln(x)) = e^{\lambda_0 -1} x ^ {\lambda_2} e^{\lambda_1 x} \\

\int_0^\infty x^n e^{-ax} dx = \frac {n!}{a^{n+1}} \\ \int_{-\infty}^\infty p(x) dx = 1 \\ \Rightarrow e^{\lambda_0 -1} = \frac {(\lambda_0 -1)^{\lambda_2+1}}{\Gamma(\lambda_2+1)} \\

p(x) = \frac {1}{\theta^k \Gamma(k)} x^{k-1} e^{- \frac {x}{\theta}} \\ \text{实在是凑不出来}\theta

1 = \int_{x_{min}}^\infty p(x) dx = \int_{x_{min}}^\infty x^{-a} dx = \frac {C}{1-a} [x^{-a+1}]_{x_{min}}^\infty \qquad a \gt 1 \\ C = (a-1) x_{x_{min}}^\infty \\ p(x) = \frac {a-1}{x_{min}} (\frac {x}{x_{min}})^{-a} \\

m^n = x_1^{n_1} \cdot x_2^{n_2} \cdot \ldots \cdot x_n^{n_n}\\ N \ln m = n_1 \ln x_1 + n_2 \ln x_2 + \ldots + n_n \ln x_n = \sum n_j \ln x_j

\ln m = \int_a^b p(x) \ln x dx

F = -\int p(x) \ln p(x) d x + \lambda_0 (\int p(x) dx - 1) + \lambda_1 (\int p(x) \ln x dx - \ln m) \\ \frac {\partial J}{\partial p(x)} = -\ln p(x) -1 + \lambda_0 + \lambda_1 \ln(x) = 0 \\ p(x) = e^{-1 + \lambda_0} x^{\lambda_1}

正态分布