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Peyman
Peyman
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$$ H(p,q) \geq H(p) \xrightarrow[]{} \sum_{x}^{}-p_{x}\log(q_{x}) \geq \sum_{x}^{}-p_{x}\log(p_{x}) \\ \xrightarrow[]{-\log(x)=\log(\frac{1}{x})} \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \geq \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) \\ \xrightarrow[]{} \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) - \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \leq 0 \\ \xrightarrow[]{} \sum_{x}^{}p_{x}[\log(\frac{1}{p_{x}})- \log(\frac{1}{q_{x}})] \leq 0 \\ \xrightarrow[]{\log(x) - \log(y) = \log(\frac{x}{y})} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq 0 \\ \xrightarrow[\sum_{x}^{}p_{x} = 1, \text{So It Is Like $\color{red}\alpha$ In Concave Definition}]{{\log \text{is a Concave Function}}} \log[\sum_{x_{p_{x}\neq 0}}^{}p_{x}(\frac{q_{x}}{p_{x}})] \leq 0 \\ \xrightarrow[]{} \log[\sum_{x_{p_{x}\neq 0}}^{}\not{p_{x}}\frac{q_{x}}{\not{p_{x}}}] \leq 0 \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq 0 \\ \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq \log(\sum_{x}^{}q_{x}) = \log(1) = 0 \\ \xrightarrow[]{} \text{Proved} $$$$ H(p,q) \geq H(p) \xrightarrow[]{} \sum_{x}^{}-p_{x}\log(q_{x}) \geq \sum_{x}^{}-p_{x}\log(p_{x}) \\ \xrightarrow[]{-\log(x)=\log(\frac{1}{x})} \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \geq \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) \\ \xrightarrow[]{} \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) - \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \leq 0 \\ \xrightarrow[]{} \sum_{x}^{}p_{x}[\log(\frac{1}{p_{x}})- \log(\frac{1}{q_{x}})] \leq 0 \\ \xrightarrow[]{\log(x) - \log(y) = \log(\frac{x}{y})} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq 0 \\ \xrightarrow[\sum_{x}^{}p_{x} = 1, \text{So It Is Like $\color{red}\alpha$ In Concave Definition}]{{\log \text{is a Concave Function}}} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq \log[\sum_{}^{}p_{x}(\frac{q_{x}}{p_{x}})] \\ \log[\sum_{}^{}\not{p_{x}}\frac{q_{x}}{\not{p_{x}}}] = \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq \log(\sum_{x}^{}q_{x}) = \log(1) = 0 \\ \xrightarrow[]{} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq \log[\sum_{x_{p_{x}\neq 0}}^{}p_{x}(\frac{q_{x}}{p_{x}})] \leq 0 \\ \xrightarrow[]{} \text{Proved} $$

$$ H(p,q) \geq H(p) \xrightarrow[]{} \sum_{x}^{}-p_{x}\log(q_{x}) \geq \sum_{x}^{}-p_{x}\log(p_{x}) \\ \xrightarrow[]{-\log(x)=\log(\frac{1}{x})} \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \geq \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) \\ \xrightarrow[]{} \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) - \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \leq 0 \\ \xrightarrow[]{} \sum_{x}^{}p_{x}[\log(\frac{1}{p_{x}})- \log(\frac{1}{q_{x}})] \leq 0 \\ \xrightarrow[]{\log(x) - \log(y) = \log(\frac{x}{y})} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq 0 \\ \xrightarrow[\sum_{x}^{}p_{x} = 1, \text{So It Is Like $\color{red}\alpha$ In Concave Definition}]{{\log \text{is a Concave Function}}} \log[\sum_{x_{p_{x}\neq 0}}^{}p_{x}(\frac{q_{x}}{p_{x}})] \leq 0 \\ \xrightarrow[]{} \log[\sum_{x_{p_{x}\neq 0}}^{}\not{p_{x}}\frac{q_{x}}{\not{p_{x}}}] \leq 0 \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq 0 \\ \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq \log(\sum_{x}^{}q_{x}) = \log(1) = 0 \\ \xrightarrow[]{} \text{Proved} $$

$$ H(p,q) \geq H(p) \xrightarrow[]{} \sum_{x}^{}-p_{x}\log(q_{x}) \geq \sum_{x}^{}-p_{x}\log(p_{x}) \\ \xrightarrow[]{-\log(x)=\log(\frac{1}{x})} \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \geq \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) \\ \xrightarrow[]{} \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) - \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \leq 0 \\ \xrightarrow[]{} \sum_{x}^{}p_{x}[\log(\frac{1}{p_{x}})- \log(\frac{1}{q_{x}})] \leq 0 \\ \xrightarrow[]{\log(x) - \log(y) = \log(\frac{x}{y})} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq 0 \\ \xrightarrow[\sum_{x}^{}p_{x} = 1, \text{So It Is Like $\color{red}\alpha$ In Concave Definition}]{{\log \text{is a Concave Function}}} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq \log[\sum_{}^{}p_{x}(\frac{q_{x}}{p_{x}})] \\ \log[\sum_{}^{}\not{p_{x}}\frac{q_{x}}{\not{p_{x}}}] = \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq \log(\sum_{x}^{}q_{x}) = \log(1) = 0 \\ \xrightarrow[]{} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq \log[\sum_{x_{p_{x}\neq 0}}^{}p_{x}(\frac{q_{x}}{p_{x}})] \leq 0 \\ \xrightarrow[]{} \text{Proved} $$

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Arya McCarthy
Arya McCarthy
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$$ H(p,q) \geq H(p) \xrightarrow[]{} \sum_{x}^{}-p_{x}\log(q_{x}) \geq \sum_{x}^{}-p_{x}\log(p_{x}) \\ \xrightarrow[]{-\log(x)=\log(\frac{1}{x})} \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \geq \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) \\ \xrightarrow[]{} \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) - \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \leq 0 \\ \xrightarrow[]{} \sum_{x}^{}p_{x}[\log(\frac{1}{p_{x}})- \log(\frac{1}{q_{x}})] \leq 0 \\ \xrightarrow[]{\log(x) - \log(y) = \log(\frac{x}{y})} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq 0 \\ \xrightarrow[\sum_{x}^{}p_{x} = 1, So ItIsLike {\color{Red} \alpha} In Concave Definition]{{\log is a {\color{Red} Concave Function}}} \log[\sum_{x_{p_{x}\neq 0}}^{}p_{x}(\frac{q_{x}}{p_{x}})] \leq 0 \\ \xrightarrow[]{} \log[\sum_{x_{p_{x}\neq 0}}^{}\not{p_{x}}\frac{q_{x}}{\not{p_{x}}}] \leq 0 \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq 0 \\ \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq \log(\sum_{x}^{}q_{x}) = \log(1) = 0 \\ \xrightarrow[]{} Proved $$$$ H(p,q) \geq H(p) \xrightarrow[]{} \sum_{x}^{}-p_{x}\log(q_{x}) \geq \sum_{x}^{}-p_{x}\log(p_{x}) \\ \xrightarrow[]{-\log(x)=\log(\frac{1}{x})} \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \geq \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) \\ \xrightarrow[]{} \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) - \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \leq 0 \\ \xrightarrow[]{} \sum_{x}^{}p_{x}[\log(\frac{1}{p_{x}})- \log(\frac{1}{q_{x}})] \leq 0 \\ \xrightarrow[]{\log(x) - \log(y) = \log(\frac{x}{y})} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq 0 \\ \xrightarrow[\sum_{x}^{}p_{x} = 1, \text{So It Is Like $\color{red}\alpha$ In Concave Definition}]{{\log \text{is a Concave Function}}} \log[\sum_{x_{p_{x}\neq 0}}^{}p_{x}(\frac{q_{x}}{p_{x}})] \leq 0 \\ \xrightarrow[]{} \log[\sum_{x_{p_{x}\neq 0}}^{}\not{p_{x}}\frac{q_{x}}{\not{p_{x}}}] \leq 0 \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq 0 \\ \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq \log(\sum_{x}^{}q_{x}) = \log(1) = 0 \\ \xrightarrow[]{} \text{Proved} $$

$$ H(p,q) \geq H(p) \xrightarrow[]{} \sum_{x}^{}-p_{x}\log(q_{x}) \geq \sum_{x}^{}-p_{x}\log(p_{x}) \\ \xrightarrow[]{-\log(x)=\log(\frac{1}{x})} \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \geq \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) \\ \xrightarrow[]{} \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) - \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \leq 0 \\ \xrightarrow[]{} \sum_{x}^{}p_{x}[\log(\frac{1}{p_{x}})- \log(\frac{1}{q_{x}})] \leq 0 \\ \xrightarrow[]{\log(x) - \log(y) = \log(\frac{x}{y})} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq 0 \\ \xrightarrow[\sum_{x}^{}p_{x} = 1, So ItIsLike {\color{Red} \alpha} In Concave Definition]{{\log is a {\color{Red} Concave Function}}} \log[\sum_{x_{p_{x}\neq 0}}^{}p_{x}(\frac{q_{x}}{p_{x}})] \leq 0 \\ \xrightarrow[]{} \log[\sum_{x_{p_{x}\neq 0}}^{}\not{p_{x}}\frac{q_{x}}{\not{p_{x}}}] \leq 0 \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq 0 \\ \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq \log(\sum_{x}^{}q_{x}) = \log(1) = 0 \\ \xrightarrow[]{} Proved $$

$$ H(p,q) \geq H(p) \xrightarrow[]{} \sum_{x}^{}-p_{x}\log(q_{x}) \geq \sum_{x}^{}-p_{x}\log(p_{x}) \\ \xrightarrow[]{-\log(x)=\log(\frac{1}{x})} \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \geq \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) \\ \xrightarrow[]{} \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) - \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \leq 0 \\ \xrightarrow[]{} \sum_{x}^{}p_{x}[\log(\frac{1}{p_{x}})- \log(\frac{1}{q_{x}})] \leq 0 \\ \xrightarrow[]{\log(x) - \log(y) = \log(\frac{x}{y})} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq 0 \\ \xrightarrow[\sum_{x}^{}p_{x} = 1, \text{So It Is Like $\color{red}\alpha$ In Concave Definition}]{{\log \text{is a Concave Function}}} \log[\sum_{x_{p_{x}\neq 0}}^{}p_{x}(\frac{q_{x}}{p_{x}})] \leq 0 \\ \xrightarrow[]{} \log[\sum_{x_{p_{x}\neq 0}}^{}\not{p_{x}}\frac{q_{x}}{\not{p_{x}}}] \leq 0 \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq 0 \\ \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq \log(\sum_{x}^{}q_{x}) = \log(1) = 0 \\ \xrightarrow[]{} \text{Proved} $$

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Peyman
Peyman
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$$ H(p,q) \geq H(p) \xrightarrow[]{} \sum_{x}^{}-p_{x}\log(q_{x}) \geq \sum_{x}^{}-p_{x}\log(p_{x}) \\ \xrightarrow[]{-\log(x)=\log(\frac{1}{x})} \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \geq \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) \\ \xrightarrow[]{} \sum_{x}^{}p_{x}\log(\frac{1}{p_{x}}) - \sum_{x}^{}p_{x}\log(\frac{1}{q_{x}}) \leq 0 \\ \xrightarrow[]{} \sum_{x}^{}p_{x}[\log(\frac{1}{p_{x}})- \log(\frac{1}{q_{x}})] \leq 0 \\ \xrightarrow[]{\log(x) - \log(y) = \log(\frac{x}{y})} \sum_{x}^{}p_{x}\log(\frac{q_{x}}{p_{x}}) \leq 0 \\ \xrightarrow[\sum_{x}^{}p_{x} = 1, So ItIsLike {\color{Red} \alpha} In Concave Definition]{{\log is a {\color{Red} Concave Function}}} \log[\sum_{x_{p_{x}\neq 0}}^{}p_{x}(\frac{q_{x}}{p_{x}})] \leq 0 \\ \xrightarrow[]{} \log[\sum_{x_{p_{x}\neq 0}}^{}\not{p_{x}}\frac{q_{x}}{\not{p_{x}}}] \leq 0 \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq 0 \\ \xrightarrow[]{} \log(\sum_{x_{p_{x}\neq 0}}^{}q_{x}) \leq \log(\sum_{x}^{}q_{x}) = \log(1) = 0 \\ \xrightarrow[]{} Proved $$