User:Tchanders/sandbox

$$P(A\mid B) = \frac{P(B \mid A)\, P(A)}{P(B)}$$

$$P(X=\mathrm{Not\ Different}\mid Y=y) = \frac{f(y\mid X=\mathrm{Not\ Different})\,P(X=\mathrm{Not\ Different})}{f(y)}$$

$$P(X=x\mid Y=y) = \frac{f(y\mid X=x)\,P(X=x)}{f(y)}$$

$$P(X=\mathrm{Not\ Different}\mid Y=y) = \frac{f_0(y)\,P(X=\mathrm{Not\ Different})}{f(y)}$$

$$P(X=\mathrm{Not\ Different}\mid Y=y) = \frac{f_0(y)\,p_0}{f(y)}$$

$$P(X=\mathrm{Different}\mid Y=y) = 1 - \frac{f_0(y)\,p_0}{\widehat{f}(y)}$$

$$\begin{array}{lcl} Y & & t\mathrm{-statistic} \\ X & & \mathrm{Genes\ different} \\ p_0 & & P(X') \\ p_1 & & P(X) \end{array}$$

$$\begin{array}{lcl} Y & & t\mathrm{-statistic} \\ X & & \mathrm{Genes\ different} \\ p_0 & & P(X') \\ p_1 & & P(X) \\ f_0(y) & & \mathrm{Density\ of\ } Y | X' \\ f_1(y) & & \mathrm{Density\ of\ } Y | X \\ f(y) & & \mathrm{Density\ of\ } Y = p_0f_0(y) + p_1f_1(y) \\ \end{array}$$

$$\begin{array}{lcl} x_{Aaron} & & \lambda_{Aaron} \\ \ldots & & \ldots \\ x_{zounds} & & \lambda_{zounds} \end{array}$$

$$\begin{array}{lcl} x_{lorem} = 0 & & \lambda_{lorem} = \lambda_0 \\ x_{ipsum} = 0 & & \lambda_{ipsum} = \lambda_0 \\ \ldots & & \ldots \end{array}$$

$$G(\lambda)$$

$$\sim Po(\lambda)$$

$$H(X) = \sum_{i=1}^n {{P}(x_i)\,{I}(x_i)} = -\sum_{i=1}^n {{P}(x_i) \log_b {P}(x_i)}$$

$$I(X;Y) = H(X) + H(Y) - H(X,Y)$$

$$I_{spec}(y; X) = \sum_{x \in X} p(x|y)log \frac{p(y|x)}{p(y)}$$

$$Redundancy(X_1, X_2) = I_{min}(Y ; X_1, X_2) = \sum_{y \in Y} p(y) min_{X_{i}}I_{spec}(y; X_i)$$

$$I(X_1, X_2; Y) = Synergy(X_1, X_2; Y) \ + \ Unique(X_1; Y) \ + \ Unique(X_2; Y)\ + \ Redundancy(X_1, X_2; Y)$$

$$\begin{align} I(X_1; Y) = Unique(X_1; Y) \ + \ Redundancy(X_1, X_2; Y) \\ I(X_2; Y) = Unique(X_2; Y) \ + \ Redundancy(X_1, X_2; Y) \end{align} $$