Broken Fire Alarm with Bayes’ Rule
Website Link:
http://allendowney.blogspot.com/2015/08/bayes-theorem-in-real-life.html
We studied about Bayes’ Rule in class as the way to understand information cascade. This article talks about a situation in which Bayes’ Rule may not be as applicable in real life as we thought. The writer of the article talks about how the Carbon Monoxide alarm in his house went off. At first he, “hypotheses came to [his] mind: (1) there is a dangerous amount of CO in [his] house, (2) it’s a false alarm.” Initially he thought option (2) was more likely since he. couldn’t find anything leaking in his house.
After some research, he found out that the probability for the alarm to be faulty is low, and also the fact that “a smoldering fire inside a wall” is possible, his expectations now changed to (1). He called the firefighter and they went to check the level of Carbon Monoxide in the house, they did not find CO in a dangerous level, or in any level, in the house. In the end, it was deemed that the alarm was probably too old and needed replacement.
The writer then concluded the article by talking about Bayes’ Rule and how he did not think it is applicable in this situation. For Bayes’ Rule, the equation is:
Pr [A | B] = Pr [A] · Pr [B | A] / Pr[B]
which Pr[A] and Pr[B] are prior probability of each while Pr[B | A] is the posterior probability of A given that B is true. The writer’s initial estimate for the probability of the alarm being a faulty was high at first, however the probability of the CO actually being leaked got higher and higher as he realized more things. However, the probability didn’t get as high as when he came up with a new hypothesis that there may be a smoldering fire. The writer talks about how he should use this as the grounds for posterior probability OR the revised prior probability and start over.
The writer finished the article of like that. I believe that this is a very complicated model just like the writer said due to the fact that although the probability of it being a faulty and the CO being leaked is definitely connected, the change in the thinking by the writer contributed to a lot of the statistics. So I also don’t really know where the change in thinking, or the trigger, should be accounted for in the equation, but it is an interesting idea to delve into.