# on 29-Jan-2021 (Fri)

#### Annotation 4769617677580

 #MLBook #linear-regression #machine-learning #problem-statement We have a collection of labeled examples $$\{ ( \mathbf x_i , y_i ) \}^N_{i=1}$$ , where $$N$$ is the size of the collection, $$\mathbf x_i$$ is the $$D$$-dimensional feature vector of example $$i = 1 , . . . , N$$ , $$y_i$$ is a real-valued target and every feature $$x^{(j)}_i , j = 1, \ldots , D$$, is also a real number. We want to build a model $$f_{\mathbf w,b} (\mathbf x)$$ as a linear combination of features of example $$\mathbf x$$: $$f_{\mathbf w,b} (\mathbf x) = \mathbf w \mathbf x + b$$, where $$\mathbf w$$ is a $$D$$-dimensional vector of parameters and $$b$$ is a real number. The notation $$f_{\mathbf w,b} (\mathbf x)$$ means that the model $$f$$ is parametrized by two values: $$\mathbf w$$ and $$\mathbf b$$. We will use the model to predict the unknown $$y$$ for a given $$\mathbf x$$ like this: $$y \leftarrow f_{\mathbf w,b} ( \mathbf{x} )$$. Two models parametrized by two different pairs $$( \mathbf w, b )$$ will likely produce two different predictions when applied to the same example. We want to find the optimal values $$( \mathbf w^\ast, b^\ast )$$. Obviously, the optimal values of parameters define the model that makes the most accurate predictions.

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#### Flashcard 4789243874572

Tags
#MLBook #linear-regression #machine-learning #problem-statement
Question
State the problem of linear regression.

We have a collection of labeled examples $$\{ ( \mathbf x_i , y_i ) \}^N_{i=1}$$ , where $$N$$ is the size of the collection, $$\mathbf x_i$$ is the $$D$$-dimensional feature vector of example $$i = 1 , . . . , N$$ , $$y_i$$ is a real-valued target and every feature $$x^{(j)}_i , j = 1, \ldots , D$$, is also a real number. We want to build a model $$f_{\mathbf w,b} (\mathbf x)$$ as a linear combination of features of example $$\mathbf x$$:

$$f_{\mathbf w,b} (\mathbf x) = \mathbf w \mathbf x + b$$,

where $$\mathbf w$$ is a $$D$$-dimensional vector of parameters and $$b$$ is a real number. The notation $$f_{\mathbf w,b} (\mathbf x)$$ means that the model $$f$$ is parametrized by two values: $$\mathbf w$$ and $$\mathbf b$$.

We will use the model to predict the unknown $$y$$ for a given $$\mathbf x$$ like this: $$y \leftarrow f_{\mathbf w,b} ( \mathbf{x} )$$. Two models parametrized by two different pairs $$( \mathbf w, b )$$ will likely produce two different predictions when applied to the same example. We want to find the optimal values $$( \mathbf w^\ast, b^\ast )$$. Obviously, the optimal values of parameters define the model that makes the most accurate predictions.

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#### Parent (intermediate) annotation

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We have a collection of labeled examples $$\{ ( \mathbf x_i , y_i ) \}^N_{i=1}$$ , where $$N$$ is the size of the collection, $$\mathbf x_i$$ is the $$D$$-dimensional feature vector of example $$i = 1 , . . . , N$$ , $$y_i$$ is a real-valued target and every feature $$x^{(j)}_i , j = 1, \ldots , D$$, is also a real number. We want to build a model $$f_{\mathbf w,b} (\mathbf x)$$ as a linear combination of features of example $$\mathbf x$$: $$f_{\mathbf w,b} (\mathbf x) = \mathbf w \mathbf x + b$$, where $$\mathbf w$$ is a $$D$$-dimensional vector of parameters and $$b$$ is a real number. The notation $$f_{\mathbf w,b} (\mathbf x)$$ means that the model $$f$$ is parametrized by two values: $$\mathbf w$$ and $$\mathbf b$$. We will use the model to predict the unknown $$y$$ for a given $$\mathbf x$$ like this: $$y \leftarrow f_{\mathbf w,b} ( x )$$. Two models parametrized by two different pairs $$( \mathbf w, b )$$ will likely produce two different predictions when applied to the same example. We want to find the optimal values $$( \mathbf w^\ast, b^\ast )$$. Obviously, the optimal values of parameters define the model that makes the most accurate predictions.

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