Riemann surface
matlab
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Exercise 2.1. [Purpose: To get you to think about what beliefs can be altered by inference from data.] Suppose I believe that exactly 47 angels can dance on my head. (These angels cannot be seen or felt in any way.) Can you provide any evidence that would change my belief?
No. By assumption, the belief has no observable consequences, and therefore no observable data can affect the belief.
Suppose I believe that exactly 47 anglers can dance on the floor of the bait shop. Is there any evidence you could provide that would change my belief?
Yes. Because dancing anglers and bait-shop floors have measurable spatial extents, data from observed anglers and floors can influence the belief.
Exercise 2.2. [Purpose: To get you to actively manipulate mathematical models of probabilities. Notice, however, that these models have no parameters.] Suppose we have a four-sided die from a board game. (On a tetrahedral die, each face is an equilateral triangle. When you roll the die, it lands with one face down and the other three visible as the faces of a three-sided pyramid. To read the value of the roll, you pick up the die and see what landed face down.) One side has one dot, the second side has two dots, the third side has three dots, and the fourth side has four dots. Denote the value of the bottom face as $\mathbf{x}$. Consider the following three mathematical descriptions of the probabilities of $\mathbf{x}$. Model A: $p(x)=1 / 4$. Model B: $p(x)=x / 10$. Model C: $p(x)=12 /(25 x)$. For each model, determine the value of $p(x)$ for each value of $x$. Describe in words what kind of bias (or lack of bias) is expressed by each model.
Model A: $\mathrm{p}(\mathrm{x}=1)=1 / 4, \mathrm{p}(\mathrm{x}=2)=1 / 4, \mathrm{p}(\mathrm{x}=3)=1 / 4, \mathrm{p}(\mathrm{x}=4)=1 / 4$. This model is unbiased, in that every value has the same probability.
Model B: $\mathrm{p}(\mathrm{x}=1)=1 / 10, \mathrm{p}(\mathrm{x}=2)=2 / 10, \mathrm{p}(\mathrm{x}=3)=3 / 10, \mathrm{p}(\mathrm{x}=4)=4 / 10$. This model is biased toward higher values of $x$.
Model C: $\mathrm{p}(\mathrm{x}=1)=12 / 25, \mathrm{p}(\mathrm{x}=2)=12 / 50, \mathrm{p}(\mathrm{x}=3)=12 / 75, \mathrm{p}(\mathrm{x}=4)=12 / 100$. (Notice that the probabilities sum to 1.) This model is biased toward lower values of $x$.
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