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Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that $$ \frac{y\left( x \right)}{p\left( x \right)}=\frac{\left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta}}{\mathbb{E} _{x\sim p}\left[ \left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta} \right]}, $$ or equivalently $$ y\left( x \right) =\frac{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}{\sum_x{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}}. $$ Under these conditions, can this inequality $$ \mathrm{KL}\left( y\parallel q \right) \ge (1-\beta)\, \mathrm{KL}\left( p\parallel q \right) $$ hold? If not, what's the counter-example?

Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that $$ \frac{y\left( x \right)}{p\left( x \right)}=\frac{\left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta}}{\mathbb{E} _{x\sim p}\left[ \left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta} \right]}, $$ or equivalently $$ y\left( x \right) =\frac{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}{\sum_x{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}}. $$ Under these conditions, can this inequality $$ \mathrm{KL}\left( y\parallel q \right) \ge \beta\, \mathrm{KL}\left( p\parallel q \right) $$ hold? If not, what's the counter-example?

Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that $$ \frac{y\left( x \right)}{p\left( x \right)}=\frac{\left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta}}{\mathbb{E} _{x\sim p}\left[ \left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta} \right]}, $$ or equivalently $$ y\left( x \right) =\frac{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}{\sum_x{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}}. $$ Under these conditions, can this inequality $$ \mathrm{KL}\left( y\parallel q \right) \ge \beta\, \mathrm{KL}\left( p\parallel q \right) $$ hold? If not, what's the counter-example?

Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that $$ \frac{y\left( x \right)}{p\left( x \right)}=\frac{\left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta}}{\mathbb{E} _{x\sim p}\left[ \left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta} \right]}, $$ or equivalently $$ y\left( x \right) =\frac{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}{\sum_x{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}}. $$ Under these conditions, can this inequality $$ \mathrm{KL}\left( y\parallel q \right) \ge (1-\beta)\, \mathrm{KL}\left( p\parallel q \right) $$ hold? If not, what's the counter-example?

Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that $$ \frac{y\left( x \right)}{p\left( x \right)}=\frac{\left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta}}{\mathbb{E} _{x\sim p}\left[ \left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta} \right]}, $$ or equivalently $$ y\left( x \right) =\frac{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}{\sum_x{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}}. $$ Under these conditions, can this inequality $$ \mathrm{KL}\left( y||q \right) \ge \beta\, \mathrm{KL}\left( p||q \right) $$ hold? If not, what's the counter-example?

Suppose $p$ and $q$ are two discrete distributions. Given a positive constant $\beta\in(0,1)$, we create a new discrete distribution $y$ such that $$ \frac{y\left( x \right)}{p\left( x \right)}=\frac{\left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta}}{\mathbb{E} _{x\sim p}\left[ \left( \frac{q\left( x \right)}{p\left( x \right)} \right) ^{1-\beta} \right]}, $$ or equivalently $$ y\left( x \right) =\frac{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}{\sum_x{q\left( x \right) ^{1-\beta}p\left( x \right) ^{\beta}}}. $$ Under these conditions, can this inequality $$ \mathrm{KL}\left( y\parallel q \right) \ge \beta\, \mathrm{KL}\left( p\parallel q \right) $$ hold? If not, what's the counter-example?