Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) detector to detect $C$ from $X$. Consider a determinstic mapping $f: \mathbb{R}^d \mapsto \mathbb{R}^k$ for $k > d$, and define $Y = f(X)$. Now define a new ML based on $Y$ to detect $T$. So,

\begin{align} &\text{ML1}: ~C \rightarrow X \rightarrow \hat{C} \\ &\text{ML2}: ~C \rightarrow X \rightarrow Y \rightarrow \hat{C} \end{align}

Is it possible to have a lower detection error probability, namely $\Pr(C\neq \hat{C})$, in the second case? Seems not.

• Assume that all the required probabilities are available, no learning setup.

Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) detector to detect $C$ from $X$. Consider a deterministic mapping $f: \mathbb{R}^d \mapsto \mathbb{R}^k$ for $k > d$, and define $Y = f(X)$. Now define a new ML based on $Y$ to detect $C$. So,

\begin{align} &\text{ML1}: ~C \rightarrow X \rightarrow \hat{C} \\ &\text{ML2}: ~C \rightarrow X \rightarrow Y \rightarrow \hat{C} \end{align}

Is it possible to have a lower detection error probability, namely $\Pr(C\neq \hat{C})$, in the second case? Seems not.

• Assume that all the required probabilities are available, no learning setup.

Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) detector to detect $C$ from $X$. Consider a determinstic mapping $f: \mathbb{R}^d \mapsto \mathbb{R}^k$ for $k > d$, and define $Y = f(X)$. Now define a new ML based on $Y$ to detect $T$. So,

\begin{align} &\text{ML1}: ~C \rightarrow X \rightarrow \hat{C} \\ &\text{ML2}: ~C \rightarrow X \rightarrow Y \rightarrow \hat{C} \end{align}

Is it possible to have a lower detection error probability, namely $\Pr(C\neq \hat{C})$, in the second case? Seems not.

Consider a binary variable $C$ with $\Pr(C=0)=\Pr(C=1)=0.5$. Consider a random vector $X \in \mathbb{R}^d$, characterized by $C$, with PDF $p_m(x)$, $m\in\{0,1\}$. Define a maximum likelihood (ML) detector to detect $C$ from $X$. Consider a determinstic mapping $f: \mathbb{R}^d \mapsto \mathbb{R}^k$ for $k > d$, and define $Y = f(X)$. Now define a new ML based on $Y$ to detect $T$. So,

\begin{align} &\text{ML1}: ~C \rightarrow X \rightarrow \hat{C} \\ &\text{ML2}: ~C \rightarrow X \rightarrow Y \rightarrow \hat{C} \end{align}

Is it possible to have a lower detection error probability, namely $\Pr(C\neq \hat{C})$, in the second case? Seems not.

• Assume that all the required probabilities are available, no learning setup.