Suppose a set of 2-dimensional points are selected from the unit square following a distribution D that assigns the proba- bilities to point-label pairs as follows. First, the chance of any point from any colored region is equal to the area covered by that color in the unit square. For example, the chance of a point being drawn from the blue region is 1/2, and the chance of a point being drawn from the green region is 0.25. Suppose the chance of a blue point getting a '+' label is 0.65 (and thus the chance of a blue point getting a '-' label is 0.35). Similarly, the chance of a green/red/orange/yellow point getting a '+' label is 0.6/0.4/0.8/0.9, respectively (and thus their chance of getting a '-' label is 0.4/0.6/0.2/0.1, respectively). (a) Describe the best possible hypothesis for the above distribution and compute its error. (b) Consider a hypothesis that labels all points as '+'. Compute the error of this hypothesis. (c) Consider a hypothesis that labels all points as '-'. Compute the error of this hypothesis. (d) Consider a hypothesis that flips a fair coin and randomly chooses a label for each point. Compute the error of this hypothesis.
Suppose a set of 2-dimensional points are selected from the unit square following a distribution D that assigns the proba- bilities to point-label pairs as follows. First, the chance of any point from any colored region is equal to the area covered by that color in the unit square. For example, the chance of a point being drawn from the blue region is 1/2, and the chance of a point being drawn from the green region is 0.25. Suppose the chance of a blue point getting a '+' label is 0.65 (and thus the chance of a blue point getting a '-' label is 0.35). Similarly, the chance of a green/red/orange/yellow point getting a '+' label is 0.6/0.4/0.8/0.9, respectively (and thus their chance of getting a '-' label is 0.4/0.6/0.2/0.1, respectively). (a) Describe the best possible hypothesis for the above distribution and compute its error. (b) Consider a hypothesis that labels all points as '+'. Compute the error of this hypothesis. (c) Consider a hypothesis that labels all points as '-'. Compute the error of this hypothesis. (d) Consider a hypothesis that flips a fair coin and randomly chooses a label for each point. Compute the error of this hypothesis.
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