arrow_back 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 mathematical notation.

Question

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 mathematical notation.

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.

Answer 1

Now we have another hypothesis for the things you observed:

$\mathrm{H}_{2}=$ child accidentally broke your window and took the laptop for safekeeping
We can express this as:
$$
P\left(D \mid H_{2}\right) \gg P\left(D \mid H_{1}\right)
$$

And we would expect:
$$
\frac{P\left(D \mid H_{2}\right)}{P\left(D \mid H_{1}\right)}=\text { a large number }
$$

Of course, you might think that this child is untrustworthy and notorious for causing trouble, which might change your mind about how likely their explanation is and lead you to hypothesize that they have robbed you! As you move through this book, you'll learn more about how you can reflect that mathematically.