Suppose Two Class $C_1$ and $C_2$ has an attribute $x$ and has distribution $ cal{N} (0, 0.5)$ and $ cal{N} (1, 0.5)$. if we have equal prior $P(C_1)=P(C_2)=0.5$ for following cost matrix:

$L= begin{bmatrix} 0 & 0.5 \ 1 & 0 end{bmatrix}$

why, $x_0 < 0.5$ is the threshold for minimum risk (cost) classifier?

This is my note example that I misunderstand, (i.e, how this threshold is reached? )

Edit 1: I think for thresholds of likelihood ratio we can use P(C1) / P(C2).

Edit 2: I add from Duda Book on Pattern some text about threshold.

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#### Best Answer

For a cost matrix $$L= begin{bmatrix} 0 & 0.5 \ 1 & 0 end{bmatrix} begin{matrix} c_1 \ c_2 end{matrix} ;text{prediction} \ hspace{-1.9cm} begin{matrix} c_1 & c_2 end{matrix} \ hspace{-1.9cm}text{truth}$$

the loss of predicting class $c_1$ when the truth is class $c_2$ is $L_{12} = 0.5$, and the cost of predicting class $c_2$ when the truth is class $c_1$ is $L_{21} = 1$. There is no cost for correct predictions, $L_{11} = L_{22} = 0$. The conditional risk $R$ for predicting either class $k$ is then

$$ begin{align} R(c_1|x) &= L_{11} Pr (c_1|x) + L_{12} Pr (c_2|x) = L_{12} Pr (c_2|x) \ R(c_2|x) &= L_{22} Pr (c_2|x) + L_{21} Pr (c_1|x) = L_{21} Pr (c_1|x) end{align} $$ For a reference see these notes on page 15.

In order to minimize the risk/loss you predict $c_1$ if the cost from the mistake of doing so (that's the loss of the wrong prediction times the posterior probability that the prediction is wrong $L_{12} Pr (c_2|x)$) is smaller than the cost of wrongfully predicting the alternative,

$$ begin{align} L_{12} Pr (c_2|x) &< L_{21} Pr (c_1|x) \ L_{12} Pr (x|c_2) Pr (c_2) &< L_{21} Pr (x|c_1) Pr (c_1) \ frac{L_{12} Pr (c_2)}{L_{21} Pr (c_1)} &< frac{Pr (x|c_1)}{ Pr (x|c_2)} end{align} $$ where the second line uses Bayes' rule $Pr (c_2|x) propto Pr (x|c_2) Pr (c_2)$. Given equal prior probabilities $Pr (c_1) = Pr (c_2) = 0.5$ you get $$frac{1}{2} < frac{Pr (x|c_1)}{ Pr (x|c_2)}$$

so you choose to classify an observation as $c_1$ is the likelihood ratio exceeds this threshold. Now it is not clear to me whether you wanted to know the "best threshold" in terms of the likelihood ratios or in terms of the attribute $x$. The answer changes according to the cost function. Using the Gaussian in the inequality with $sigma_1 = sigma_2 = sigma$ and $mu_1 = 0$, $mu_2 = 1$, $$ begin{align} frac{1}{2} &< frac{frac{1}{sqrt{2pi}sigma}exp left[ -frac{1}{2sigma^2}(x-mu_1)^2 right]}{frac{1}{sqrt{2pi}sigma}exp left[ -frac{1}{2sigma^2}(x-mu_2)^2 right]} \ log left(frac{1}{2}right) &< log left(frac{1}{sqrt{2pi}sigma}right) -frac{1}{2sigma^2}(x-0)^2 – left[ log left(frac{1}{sqrt{2pi}sigma}right) -frac{1}{2sigma^2}(x-1)^2 right] \ log left(frac{1}{2}right) &< -frac{x^2}{2sigma^2} + frac{x^2}{2sigma^2} – frac{2x}{2sigma^2} + frac{1}{2sigma^2} \ frac{x}{sigma^2} &< frac{1}{2sigma^2} – log left(frac{1}{2}right) \ x &< frac{1}{2} – log left(frac{1}{2}right) sigma^2 end{align} $$ so a prediction threshold in terms of $x$ as you search for can only be achieved if the losses from false predictions are the same, i.e. $L_{12} = L_{21}$ because only then can you have $log left( frac{L_{12}}{L_{21}} right) = log (1) = 0$ and you get the $x_0 < frac{1}{2}$.