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Bayes Rule: How Much is too Much?

P(B|A) = P(A|B)P(B)/P(A)

 

That’s Bayes’ Rule. One of the most important concepts in probability and likely the bane of many Cornell students’ existence, Bayes’ Rule gives the probability of an event conditional on the occurrence of another event A (i.e., assuming has already occurred). It’s an unassuming equality on the surface, as the proof of it only requires three or four lines and is basically immediate from the simplest definitions of conditional probability. However, Bayes’ Rule is an immensely useful tool; it’s applications range from medicine to gambling, from predicting outcomes of the World Series to outcomes of presidential elections. Above all, Bayes’ Rule is simple. It makes it easy for us to compute probabilities that are not obvious from data that we record, and it has the power to either solidify or dismantle our gut feelings about events that we experience every day. Except when it doesn’t.

Enter Gary Marcus and Ernest Davis, two professors at NYU. In their article entitled “What Nate Silver Gets Wrong”, Marcus and Davis criticize the well-known statistician Nate Silver for his unwavering support of Bayes’ Rule in his recent book, The Signal and the Noise. In his book Silver argues that scientists and statisticians alike make too little use of Bayes’ Rule in their findings, and that this has led to a number of misleading findings in a variety of fields. Silver then encourages anyone reading the book to take a Bayesian viewpoint when calculating probabilities of interest because Bayes’ Rule provides valuable information that frequentist statistics cannot (frequent statistics refers to the “other” branch of statistics that often competes with Bayesian statistics). According to Silver, Bayes’ Rule always applies.

Marcus and Davis disagree. In this article, the two professors argue that Silver takes an overtly biased stance on the issue of Bayesian statistics and that he completely fails to acknowledge its pitfalls. For example, in The Signal and the Noise Silver analyzes a 2011 study by NASA linking a decrease in toad populations in Italy to an increase in earthquake activity in the same region. The study, which was performed using frequentist methods, reveals a positive correlation between declining toad populations and increasing numbers of earthquakes in a given year. Silver claims that the findings are unbelievable. After all, how could toads and earthquakes possibly be correlated? The reason the results are so ridiculous, he argues, is because NASA did not perform a Bayesian analysis of the data; if they did, it would be clear that the probability of more earthquakes given fewer toads is negligible. Marcus and Davis take the opposite stance, complementing the study. They then argue that real issue lies in the numbers. In particular, when NASA samples a large number of data points, they are bound to get at least a few false positives. It has nothing to do with Bayes’ Rule vs. Frequentist principles; it’s simply a matter of data collection.

Bayesian Statistics is a booming field right now and it is only going to increase in popularity as machine learning and artificial intelligence become more sophisticated. It is therefore important to consider the pros and cons of Bayesian statistics because it plays a role in our daily lives. Like anything in life, moderation is key. Considering Bayes’ Rule when analyzing probabilities certainly provides us with valuable information, but we can’t solely rely on it. It’s crucial that we find a balance. Our favorite CNN and ESPN commentators’ livelihood depends on it.

 

 

 

 

 

 

Link: http://www.newyorker.com/books/page-turner/what-nate-silver-gets-wrong

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