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# 数学代写|凸优化代写Convex Optimization代考|EE364a DETECTORS AND DETECTOR-BASED TESTS

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## 数学代写|凸优化代写Convex Optimization代考|Detectors and their risks

Let $\Omega$ be an observation space, and $\mathcal{P}{\chi}, \chi=1,2$, be two families of probability distributions on $\Omega$. By definition, a detector associated with $\Omega$ is a real-valued function $\phi(\omega)$ of $\Omega$. We associate with a detector $\phi$ and families $\mathcal{P}{\chi}, \chi=1,2$, risks defined as follows:
(a) $\operatorname{Risk}{-}\left[\phi \mid \mathcal{P}{1}\right]=\sup {P \in \mathcal{P}{1}} \int_{\Omega} \exp {-\phi(\omega)} P(d \omega)$
(b) $\operatorname{Risk}{+}\left[\phi \mid \mathcal{P}{2}\right]=\sup {P \in \mathcal{P}{2}} \int_{\Omega} \exp {\phi(\omega)} P(d \omega)$
(c) $\operatorname{Risk}\left[\phi \mid \mathcal{P}{1}, \mathcal{P}{2}\right]=\max \left[\operatorname{Risk}{-}\left[\phi \mid \mathcal{P}{1}\right], \operatorname{Risk}{+}\left[\phi \mid \mathcal{P}{2}\right]\right]$
Given a detector $\phi$, we can associate with it a simple test $\mathcal{T}{\phi}$ deciding via observation $\omega \sim P$ on the hypotheses $$H{1}: P \in \mathcal{P}{1}, H{2}: P \in \mathcal{P}{2} .$$ Namely, given observation $\omega \in \Omega$, the test $\mathcal{T}{\phi}$ accepts $H_{1}$ (and rejects $H_{2}$ ) whenever $\phi(\omega) \geq 0$, and accepts $H_{2}$ and rejects $H_{1}$ otherwise.
Let us make the following immediate observation:
Proposition 2.14. Let $\Omega$ be an observation space, $\mathcal{P}{\chi}, \chi=1,2$, be two families of probability distributions on $\Omega$, and $\phi$ be a detector. The risks of the test $\mathcal{T}{\phi}$ associated with this detector satisfy
\begin{aligned} \operatorname{Risk}{1}\left(\mathcal{T}{\phi} \mid H_{1}, H_{2}\right) & \leq \text { Risk }{-}\left[\phi \mid \mathcal{P}{1}\right] \ \operatorname{Risk}{2}\left(\mathcal{T}{\phi} \mid H_{1}, H_{2}\right) & \leq \operatorname{Risk}{+}\left[\phi \mid \mathcal{P}{2}\right] \end{aligned}
Proof. Let $\omega \sim P \in \mathcal{P}{1}$. Then the $P$-probability of the event ${\omega: \phi(\omega)<0}$ does not exceed Risk $\left[\phi \mid \mathcal{P}{1}\right]$, since on the set ${\omega: \phi(\omega)<0}$ the integrand in (2.45.a) is $>1$, and this integrand is nonnegative everywhere, so that the integral in (2.45.a) is $\geq P{\omega: \phi(\omega)<0}$. Recalling what $\mathcal{T}{\phi}$ is, we see that the $P$-probability to reject $H{1}$ is at most Risk $_{-}\left[\phi \mid \mathcal{P}{1}\right]$, implying the first relation in (2.47). By a similar argument, with (2.45.b) in the role of $(2.45 . a)$, when $\omega \sim P \in \mathcal{P}{2}$, the $P$-probability of the event ${\omega: \phi(\omega) \geq 0}$ is upper-bounded by Risk ${ }{+}\left[\phi \mid \mathcal{P}{2}\right]$, implying the second relation in (2.47).

## 数学代写|凸优化代写Convex Optimization代考|Detector-based tests

Observe that the fact that $\epsilon_{1}$ and $\epsilon_{2}$ are upper bounds on the risks of a detector are expressed by a system of convex constraints
\begin{aligned} &\sup {P \in \mathcal{P}{1}} \int_{\Omega} \exp {-\phi(\omega)} P(d \omega) \leq \epsilon_{1} \ &\sup {P \in \mathcal{P}{2}} \int_{\Omega} \exp {\phi(\omega)} P(d \omega) \leq \epsilon_{2} \end{aligned}
on $\epsilon_{1}, \epsilon_{2}$ and $\phi(\cdot)$. This observation is interesting, but not very useful, since the convex constraints in question usually are infinite-dimensional when $\phi(\cdot)$ is so, and are semi-infinite (suprema – over parameters ranging in an infinite set – of parametric families of convex constraints) provided $\mathcal{P}{1}$ or $\mathcal{P}{2}$ are of infinite cardinalities; constraints of this type can be intractable computationally.

Another important observation is that the distributions $P$ enter the constraints linearly; as a result, when passing from families of probability distributions $\mathcal{P}{1}, \mathcal{P}{2}$ to their convex hulls, the risks of a detector remain intact.
2.3.2.2 Renormalization
Let $\Omega, \mathcal{P}{1}$, and $\mathcal{P}{2}$ be the same as in Section 2.3.1, and let $\phi$ be a detector. When shifting this detector by a real $a$-passing from $\phi$ to the detector
$$\phi_{a}(\omega)=\phi(\omega)-a$$

• the risks clearly update according to:
\begin{aligned} \text { Risk }{-}\left[\phi{a} \mid \mathcal{P}{1}\right] &=\mathrm{e}^{a} \text { Risk }{-}\left[\phi \mid \mathcal{P}{1}\right] \ \text { Risk }{+}\left[\phi_{a} \mid \mathcal{P}{2}\right] &=\mathrm{e}^{-a} \text { Risk }{+}\left[\phi \mid \mathcal{P}_{2}\right] \end{aligned}

## 数学代写|凸优化代写Convex Optimization代考|Detectors and their risks

$\Omega$. 我们与检测器相关联 $\phi$ 和家人 $\mathcal{P} \chi, \chi=1,2$ ，风险定义如下:
(a) Risk $-[\phi \mid \mathcal{P} 1]=\sup P \in \mathcal{P} 1 \int_{\Omega} \exp -\phi(\omega) P(d \omega)$
(b) Risk $+[\phi \mid \mathcal{P} 2]=\sup P \in \mathcal{P} 2 \int_{\Omega} \exp \phi(\omega) P(d \omega)$
(C) $\operatorname{Risk}[\phi \mid \mathcal{P} 1, \mathcal{P} 2]=\max [\operatorname{Risk}-[\phi \mid \mathcal{P} 1]$, Risk $+[\phi \mid \mathcal{P} 2]]$

$$H 1: P \in \mathcal{P} 1, H 2: P \in \mathcal{P} 2 \text {. }$$

$\operatorname{Risk} 1\left(\mathcal{T} \phi \mid H_{1}, H_{2}\right) \leq \operatorname{Risk}-[\phi \mid \mathcal{P} 1]$ Risk $2\left(\mathcal{T} \phi \mid H_{1}, H_{2}\right) \quad \leq$ Risk $+[\phi \mid \mathcal{P} 2]$

$\geq P \omega: \phi(\omega)<0$. 回忆什么 $\mathcal{T} \phi$ 是，我们看到 $P$ – 拒绝概率 $H 1$ 最多是风险 $[\phi \mid \mathcal{P} 1]$ ，暗示 (2.47) 中的第 一个关系。通过类似的论点，具有 $(2.45 . b)$ 的作用(2.45. $a)$ ，什么时候 $\omega \sim P \in \mathcal{P} 2$ ，这 $P$ – 事件的概率 $\omega: \phi(\omega) \geq 0$ 以风险为上限 $+[\phi \mid \mathcal{P} 2]$ ，暗示 (2.47) 中的第二个关系。

## 数学代写|凸优化代写Convex Optimization代考|Detector-based tests

$$\sup P \in \mathcal{P} 1 \int_{\Omega} \exp -\phi(\omega) P(d \omega) \leq \epsilon_{1} \quad \sup P \in \mathcal{P} 2 \int_{\Omega} \exp \phi(\omega) P(d \omega) \leq \epsilon_{2}$$

2.3.2.2 重整化

$$\phi_{a}(\omega)=\phi(\omega)-a$$

• 风险根据以下情况明确更新:
$$\text { Risk }-[\phi a \mid \mathcal{P} 1]=\mathrm{e}^{a} \text { Risk }-[\phi \mid \mathcal{P} 1] \text { Risk }+\left[\phi_{a} \mid \mathcal{P} 2\right] \quad=\mathrm{e}^{-a} \text { Risk }+\left[\phi \mid \mathcal{P}_{2}\right]$$

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