\(
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\newcommand{\real}{\mathbb{R}}
\newcommand{\wvec}{\mathbf{w}}
\newcommand{\xvec}{\mathbf{x}}
\)

$$ S = \{ f: \int(f''(x))^2 dx < \infty \} $$

is known as

1 If \( Y \) is a collider, then \( X \) and \( Z \) are d-connected, but they are d-separated given \( Y \)

2 If \( Y \) is not a collider, then \( X \) and \( Z \) are d-separated, but they are d-connected given \( Y \)

3 Conditioning on the descendant of a collider has the same effect as conditioning on the collider.

Are these rules correct?

$$ 1: \min_{\mathbf{w},b} || \mathbf{w} || $$

subject to

$$ 2: y_i( \mathbf{w} \cdot \mathbf{x}_i + b) \ge 1 - \xi_i, \forall i $$

$$ 3: \xi_i \le 0, \forall i $$

$$ 4: \sum_i \xi_i \le C $$

Do not fret if the questions seemed too hard. Over time these concepts will get etched in your brain as you practice more often on related problems.

Do not get complacent if they were too easy either. If those were easy, wait till you get to the harder ones.

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