Chapter 1: Bayesian Thinking and Everyday Reasoning

Author

Zane Billings

Published

2022-12-06

Notes

I already have a math degree where I focused on stats, so I kind of rushed through this chapter.

Solutions

Q1

Rewrite the following statements as equations using the mathematical notation you learned in the chapter:

  • The probability of rain is low

  • The probabiliy of rain given that it is cloudy is high

  • The probability of you having an umbrella given it is raining is much greater than the probability of you having an umbrella in general.

Nothing too out of the ordinary here!

  • \(\displaystyle P\left( \text{rain} \right) = \text{low}.\)

  • \(P \left( \text{rain} \mid \text{cloudy} \right) = \text{high}.\)

  • \(P \left( \text{umbrella} \mid \text{raining} \right) \gg P \left(\text{umbrella}\right).\)

Q2

Organize the data you observe in the following scenario into a mathematical notation, using the techniques we’ve covered in this chapter. Then come up with a hypothesis to explain this data:

You come home from work and notice that your front door is open and the side window is broken. As you walk inside, you immediately notice that your laptop is missing.

Let \(D\) represent the data we’ve observed. In order words, \[D = \{\text{door open, window broken, laptop missing}\}.\]

Our hypothesis might be \[H_1: \text{My house was robbed and they took my laptop!}\]

So we might conjecture that \[P(D \mid H_1) = \mathrm{high}.\]

Q3

The following scenario adds data to the previous one. Demonstrate how this new information changes your beliefs and come up with a second hypothesis to explain the data, using the notation you’ve learned in this chapter.

A neighborhood child runs up to you and apologizes profusely for accidentally throwing a rock through your window. They claim that they saw the laptop and didn’t want it stolen so they opened the front door to grab it, and your laptop is safe at their house.

Now we’ll let \(D_U\) represent our updated data (augmented with the new information that we learned from this child). Then, \[D_U = \{D, \text{ child's explanation}\}\]

and our new hypothesis could be \[H_2: \text{the child has my laptop.}\]

If the child appears to be telling the truth, we could then conjecture that \[P(D_U \mid H_2) \gg P(D_U \mid H_1),\] although we shouldn’t rule out the possibility that the child is being coerced or is a part of the robbery.