Sequential Sampling by Bayes’ Theorem
In class, we discussed about they Bayes’ Theorem. Bayes’ Theorem stated how a subjective degree of belief should rationally change to account for evidence. Bayes’ Theorem is also widely used in sequential sampling. The paper stated several different kind of sequential sampling problem.
The basic problem is The Sampling Selection Problem. For example, someone wants to implement one out of k alternatives whole rewards X_i are random with a particular prior distribution such as the normal distribution or beta distribution. He also has the choice to implement a know option whose expected reward is a known value m. Before selecting an alternative to implement, he/she can choose to sequentially sample one or more of the k different options to get evidence for the unknown means. After each observation, he gets a signal Y for the alternative. With the signals, he can calculate the posterior distribution of the unknown option, updating his belief for the new option. The calculation for posterior distribution is based on Bayes’ Theorem.
P(X_i | Y_1 … Y_n) = P(Y_1 … Y_n|X_i) / P(Y_1 … Y_n).
With the new belief about X_i, he can make decision between X_i and m by selecting the larger value or he could choose to take another observation and get more information about the alternative. Observing sequentially is essential here.
After reading the paper, I found this application could be used restaurant selection. For example, I want to dine out at a fancy restaurant and go on Yelp to check the review for the restaurant. There are forty review of the restaurant on the first page, some are short and some are lengthy. So how many of them should I read before I make my decision. Instead of reading all of them, I could apply the sequential sampling method, each time I read a review, then I update my decision score for the restaurant and stop reading when the score reaches a point where I am sure of going or not.
Sequential sampling is also widely used in treatment design. If a pharmaceutical company want to release a new treatment, how many people should they observe to ensure that the new treatment is valid. Rather than putting out large numbers of observations at the same time and have them all failed at the end, observing sequentially often save time and money for the company. If the number of negatives get large at the beginning of the observations, then the company could ban the treatment immediately to avoid future cost. If they had very good outcomes, mostly positive, then they could also pause the observations ahead of time and save money rather than wait for the outcome when putting out all observations at the same time.
Reference:
http://mansci.journal.informs.org/content/58/3/550.full.pdf
http://en.wikipedia.org/wiki/Bayes’_theorem
http://www.yelp.com/biz/zazas-cucina-ithaca
-kchan