Bayesian Averaging in SAS

Hypothetical situation, lets say you've got a list of movies that you want to rank in a website or a report or something, and you have user-submitted ratings for them, but some are more popular than others, so your data looks like this:

data ratings;
  input name $ rating;
  datalines;
Lincoln 9
Lincoln 8
Lincoln 9
Amour 9
Argo 5
Argo 10
;
run;

The easiest thing to do would be to calculate an average rating for each movie like this:

proc sql;
  select distinct name, avg(rating) as average
    from ratings
    group by name
    order by average desc;
run;

But hey! That's not cool. It looks like Amour wins, because its average rating is 9. Maybe we want to consider Lincoln as better because 3 people think it's very high. A good way to deal with this is by instead taking a Bayesian Average .

This means we're going to add in some "dummy" votes for each movie, who give each movie the average rating a movie gets. How many (C) is a judgement call, the more we add, the harder we make it for an obscure movie to be near the top. Likewise, if a movie's first rating is low, it keeps it from suddenly dropping to the bottom of the list. If we expect thousands of ratings for each movie, a C=1000 might be appropriate. In this example, I use a small C of 10.

proc sql;
  select avg(rating) into :average
    from ratings;
  select distinct
      name,
      (sum(rating) + &average * 10) / (count(*) + 10) as b_average
    from ratings
    group by name
    order by b_average;
quit;

And look! Lincoln is back on top, since its bayesian average more closely reflects a product of the number of ratings it has and what those ratings are.