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<html> |
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<title>MathJax Example</title> |
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<script> |
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MathJax = { |
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tex: { |
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inlineMath: [['$', '$'], ['\(', '\)']] |
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}, |
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svg: { |
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fontCache: 'global' |
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} |
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}; |
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</script> |
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<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> |
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</head> |
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<body> |
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<p> $x_{t+1},\ldots x_{t+\ell}$ </p> |
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<p> $O(|{\cal X}|^{2L})$ </p> |
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<p> $\forall i=1,\ldots,L$ </p> |
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<p> $1000$ </p> |
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<p> ${\cal M}_{b}$ </p> |
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<p> $(X^{\tau},Z^{L-\tau})\sim{\cal M}_{b}^{L}.$ </p> |
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<p> $(x^{i},y^{j})$ </p> |
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<p> $x^{L}$ </p> |
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<p> $\forall x\in{\cal X}$ </p> |
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<p> $\forall x^{i_{0}}\in{\cal X}^{i_{0}}$ </p> |
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<p> $X^{L}\sim{\cal M}_{s}^{L}$ </p> |
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<p> $\mathbb{E}_{X^{L}\sim{\cal M}_{s}^{L}}[\tau]$ </p> |
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<p> $(X^{L},Y^{L})$ </p> |
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<p> $\ell=L$ </p> |
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<p> $Z^{L-\tau}$ </p> |
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<p> $x^{i}\in{\cal X}^{i}$ </p> |
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<p> ${\cal M}_{b}(\cdot|x^{n})=\text{Ber}(q),$ </p> |
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<p> $\forall\ell\leq L$ </p> |
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<p> $\tau,{\cal M}_{b,{\rm next}}$ </p> |
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<p> $L-1$ </p> |
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<p> $\beta(X^{L},Y^{L})=\tau$ </p> |
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<p> $\displaystyle=\sum_{\ell=0}^{L}\ell\cdot\Pr{\left({\tau=\ell}\right)}=\sum_{% |
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\ell=1}^{L}\sum_{x^{\ell}\in{\cal X}^{\ell}}\prod_{i=1}^{\ell}\min\{{\cal M}_{% |
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b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\},$ </p> |
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<p> $\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\sum_{\ell\leq L}\sum_{x^{% |
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\ell}}\min\left\{\text{Pr}\left(X^{1:\ell}=x^{\ell}\right),\text{Pr}\left(Y^{1% |
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:\ell}=x^{\ell}\right)\right\}$ </p> |
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<p> $\displaystyle=\Pr{\left({X^{i_{0}+1}=x^{i_{0}+1},\tau\geq i_{0}+1}\right)}+\Pr% |
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{\left({Y^{i_{0}}=x^{i_{0}},\tau\leq i_{0}}\right)}\cdot p_{\rm res}^{x^{i_{0}% |
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}}(x_{i_{0}+1})$ </p> |
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<p> $Y_{\tau+1}\neq X_{\tau+1}$ </p> |
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<p> $\displaystyle=\sum_{x^{\ell+1}\in{\cal X}^{\ell+1}}\prod_{i=1}^{\ell}\min\{{% |
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\cal M}_{b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\}({\cal M}_{s}(% |
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x_{\ell+1}\mid x^{\ell})-\min\{{\cal M}_{b}(x_{\ell+1}\mid x^{\ell}),{\cal M}_% |
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{s}(x_{\ell+1}\mid x^{\ell})\})$ </p> |
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<p> ${\cal M}_{s}(\cdot\mid X^{i-1}),{\cal M}_{b}(\cdot\mid X^{i-1}),\forall i=0,% |
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\ldots,L$ </p> |
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<p> $L=8$ </p> |
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<p> $\displaystyle\min\{{\cal M}_{s}(x^{\ell}),{\cal M}_{b}(x^{\ell})\}=\min\{\prod% |
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_{i=1}^{\ell}{\cal M}_{s}(x_{i}\mid x^{i-1}),{\cal M}_{b}(x_{i}\mid x^{i-1})\},$ </p> |
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<p> ${\cal M}^{*}(\cdot\mid x^{t})$ </p> |
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<p> $L-\beta(X^{L},Y^{L})$ </p> |
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<p> $X_{1},\ldots,X_{L}\sim{\cal M}_{s}^{L}(\cdot)$ </p> |
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<p> $\forall y^{L-\tau}\in{\cal X}^{{L-\tau}}$ </p> |
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<p> $\Pr{\left({\tau=L}\right)}=\sum_{x^{L}\in{\cal X}^{L}}\prod_{i=1}^{L}\min\{{% |
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\cal M}_{b}(x_{i}\mid x^{i-1}),{\cal M}_{s}(x_{i}\mid x^{i-1})\}.$ </p> |
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<p> $\eta_{i}\sim U(0,1)$ </p> |
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<p> $\Pi({\cal M}_{s}^{L},{\cal M}_{b}^{L})$ </p> |
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<p> $Y\sim{\cal M}_{b}$ </p> |
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<p> $L=4$ </p> |
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<p> $\displaystyle=\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau\geq\ell_{0}}\right% |
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)}+\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau=\ell_{0}-1}\right)}$ </p> |
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<p> $\displaystyle=\Pr{\left({Y^{i_{0}+1}=x^{i_{0}+1},\tau\geq i_{0}+1}\right)}+\Pr% |
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{\left({Y^{i_{0}+1}=x^{i_{0}+1},\tau\leq i_{0}}\right)}$ </p> |
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<p> $L=1$ </p> |
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<p> $\ell=\ell_{0}-1$ </p> |
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<p> $\displaystyle p_{\rm rej}(x^{i})$ </p> |
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<p> $\displaystyle{:=}\sum_{x}{\left({{\cal M}_{b}(x^{i},x)-{\cal M}_{s}(x^{i},x)}% |
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\right)}_{+},$ </p> |
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<p> $\displaystyle\;\;\;\Pr{\left({\tau=\ell}\right)}$ </p> |
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<p> $\displaystyle=\sum_{\ell=1}^{L}\Pr{\left({\tau\geq\ell}\right)}=\sum_{\ell=1}^% |
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{L}\sum_{x^{\ell}\in{\cal X}^{\ell}}\Pr{\left({X^{\ell}=x^{\ell},\tau\geq\ell}% |
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\right)}$ </p> |
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<p> $\displaystyle=\min\{\sum_{x\in{\cal X}}\min\{{\cal M}_{b}(x^{\ell_{0}-1},x),{% |
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\cal M}_{s}(x^{\ell_{0}-1},x)\}+p_{\rm remain}(x^{\ell_{0}-1}),$ </p> |
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<p> $\ell>1$ </p> |
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<p> $\forall\ell\leq L-1,x^{\ell}\in{\cal X}^{\ell}$ </p> |
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<p> $x^{t}\in{\cal X}^{t}$ </p> |
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<p> $\Pr{\left({X^{\ell}=x^{\ell},\tau\geq\ell}\right)}=\min\{{\cal M}_{b}(x^{\ell}% |
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),{\cal M}_{s}(x^{\ell})\}.$ </p> |
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<p> $\displaystyle{:=}\sum_{x}{\left({{\cal M}_{s}(x^{i},x)-{\cal M}_{b}(x^{i},x)}% |
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\right)}_{+}.$ </p> |
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<p> $Y^{L}=(X^{\tau},Z^{L-\tau})$ </p> |
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<p> $\displaystyle\sum_{y^{L}}\pi(x^{L},y^{L})$ </p> |
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<p> $(x)_{+}=\max\{x,0\}.$ </p> |
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<p> ${\cal M}_{b}(\cdot\mid X^{\tau},Y)$ </p> |
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<p> $\displaystyle={\cal M}_{s}(x^{L})\cdot\min\left\{1,\frac{{\cal M}_{b}(x^{L})}{% |
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{\cal M}_{s}(x^{L})}\right\}$ </p> |
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<p> ${\cal M}^{L}_{s}(x^{L})$ </p> |
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<p> $(x_{i},\ldots,x_{j})$ </p> |
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<p> $0\leq\tau\leq L$ </p> |
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<p> $\pi\in\Pi({\cal M}^{L}_{s},{\cal M}^{L}_{b})$ </p> |
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<p> $\displaystyle=\sum_{\ell\leq L}\sum_{x^{\ell}}\text{Pr}_{X^{L},Y^{L}}\left(X^{% |
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1:\ell}=Y^{1:\ell}=x^{\ell},\beta(X^{L},Y^{L})\geq\ell\right)$ </p> |
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<p> $X^{L}$ </p> |
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<p> $Y_{\tau+1}$ </p> |
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<p> $x^{L}\in{\cal X}^{L}$ </p> |
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<p> $\textsc{Verify}_{\pi}$ </p> |
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<p> ${\cal M}_{b,{\rm next}}^{L-\tau}$ </p> |
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<p> $\tau\leq L$ </p> |
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<p> $\min\{{\cal M}_{b}(x^{L}),{\cal M}_{s}(x^{L})\}$ </p> |
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<p> $\displaystyle=\prod_{i=1}^{\ell}\min\{1,\frac{{\cal M}_{b}(x_{i}\mid x^{i-1})}% |
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{{\cal M}_{s}(x_{i}\mid x^{i-1})}\}{\left({1-\min\{1,\frac{{\cal M}_{b}(x_{% |
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\ell+1}\mid x^{\ell})}{{\cal M}_{s}(x_{\ell+1}\mid x^{\ell})}\}}\right)}.$ </p> |
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<p> $x^{t}$ </p> |
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<p> $X_{i}\sim{\cal M}_{s}(\cdot\mid X^{i-1}).$ </p> |
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<p> $\tau=L-i$ </p> |
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<p> ${\rm E.O.S}\in X^{\tau}$ </p> |
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<p> $\displaystyle=\sum_{x^{L}\in{\cal X}^{L}}\Pr{\left({\tau=\ell,X^{L}=x^{L}}% |
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\right)}$ </p> |
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<p> $\pi_{\rm token}$ </p> |
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<p> $\displaystyle=\sum_{x^{\ell+1}\in{\cal X}^{\ell+1}}\Pr{\left({\tau=\ell,X^{% |
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\ell+1}=x^{\ell+1}}\right)}$ </p> |
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<p> $Z\sim{\cal M}_{b,{\rm next}}(\cdot)$ </p> |
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<p> $X^{\tau}$ </p> |
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<p> $(a)$ </p> |
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<p> $\ell<L$ </p> |
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<p> $\displaystyle=\Pr{\left({X^{\ell_{0}-1}=x^{\ell_{0}-1},\tau\leq\ell_{0}-1}% |
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\right)}\cdot\Pr{\left({X^{\ell_{0}-1}\text{ is accepted.}}\right)}$ </p> |
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<p> $Y_{1}=X_{1}$ </p> |
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<p> $\displaystyle=\min\left\{{\cal M}_{b}(x^{\ell_{0}-1}),{\cal M}_{s}(x^{\ell_{0}% |
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-1})\right\},$ </p> |
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<p> $\displaystyle=\text{Pr}_{\pi}{\left({\beta{{\color[rgb]{0,0,0}=\ell}}\mid X^{L% |
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}}\right)}{:=}{{\color[rgb]{0,0,0}\frac{\sum_{\ell:\beta(X^{L},Y^{L})=\ell}\pi% |
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(X^{L},Y^{L})}{\pi(X^{L})}}},$ </p> |
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<p> $Y_{2}$ </p> |
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<p> $\displaystyle\max_{\pi\in\Pi({\cal M}_{s}^{L},{\cal M}_{b}^{L})}\mathbb{E}_{X^% |
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{L},Y^{L}\sim\pi}\left[\beta(X^{L},Y^{L})\right]\leq\sum_{\tau=1}^{L}\sum_{x^{% |
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\tau}\in{\cal X}^{\tau}}\min\{{\cal M}_{s}(x^{\ell}),{\cal M}_{b}(x^{\ell})\}$ </p> |
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<p> $\forall x^{n}$ </p> |
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<p> $\eta_{0}\leq\min\left\{\frac{{\cal M}_{b}(X^{L})}{{\cal M}_{s}(X^{L})},1\right\}$ </p> |
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<p> $O(|{\cal X}|^{L})$ </p> |
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<p> $Z^{L-\tau}\sim{\cal M}_{b,{\rm next}}$ </p> |
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<p> $L=6$ </p> |
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<p> ${\cal M}_{b}(Y_{3}\mid Y_{1},y)$ </p> |
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<p> $Y^{L}=(X^{\tau},Z^{i-\tau})$ </p> |
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<p> $\displaystyle=\sum_{\ell\leq L}\sum_{x^{\ell}}\min\{{\cal M}_{s}^{\ell}(x^{% |
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\ell}),{\cal M}_{b}^{\ell}(x^{\ell})\}.$ </p> |
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<p> $\displaystyle=\Pr{\left({X^{L}=x^{L}}\right)}\Pr{\left({\tau=L\mid X^{L}=x^{L}% |
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}\right)}$ </p> |
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<p> $p_{\rm res}(x)=\frac{{\left({{\cal M}_{b}(x\mid X^{\tau})-{\cal M}_{s}(x\mid X% |
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^{\tau})}\right)}_{+}}{\sum_{x^{\prime}}{\left({{\cal M}_{b}(x^{\prime}\mid X^% |
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{\tau})-{\cal M}_{s}(x^{\prime}\mid X^{\tau})}\right)}_{+}},$ </p> |
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<p> ${\cal M}_{s}=\text{Ber}(1)$ </p> |
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<p> $\frac{{\left({{\cal M}_{b}(x^{L})-{\cal M}_{s}(x^{L})}\right)}_{+}}{\sum_{x^{L% |
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}}{\left({{\cal M}_{b}(x^{L})-{\cal M}_{s}(x^{L})}\right)}_{+}}.$ </p> |
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<p> $i=0,1,\ldots,L$ </p> |
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</body> |
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</html> |
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