Should Average Be Weighted or Kept As Is

Question

I am investigating the sentiment of television viewing. Armed with a matrix of sentiment scores like so.

#                       sad happy angry surprised disgusted
# A Word                  1     0     0         0         0
# Backstrom               0     1     0         0         1
# Good Witch Hallmark     0     0     0         0         0
# Shark Tank              0     0     1         1         0
# Above the Rim           0     0     1         0         0
# O'Reilly Factor         0     0     0         0         0
# Jack the Giant Slayer   0     1     1         1         0
# Late Night Snack        0     0     1         0         1
# Outlander               0     0     1         0         0
# Cake Wars               0     0     0         0         0

Each show has a 1 if it includes the sentiment, and a 0 if it doesn't. I can match this to an individual's viewing preferences, but a problem occurs. If the person did not view many shows, their averages can look very extreme.

In this example, an individual watches only one show, 'Shark Tank'. Of all the shows they watch, 100% indicate 'anger'. On average, it would appear that this person really loves shows with anger, but it is being skewed by the fact that they only watched one show.

x1 <- x[4,,drop=FALSE]
x1
           sad happy angry surprised disgusted
Shark Tank   0     0     1         1         0

Another individual watched all of the shows but one. This individual would get a 5/9 rating for anger. It would appear that they like angry shows less than the first.

x2 <- x[1:9,]
x2
#                       sad happy angry surprised disgusted
# A Word                  1     0     0         0         0
# Backstrom               0     1     0         0         1
# Good Witch Hallmark     0     0     0         0         0
# Shark Tank              0     0     1         1         0
# Above the Rim           0     0     1         0         0
# O'Reilly Factor         0     0     0         0         0
# Jack the Giant Slayer   0     1     1         1         0
# Late Night Snack        0     0     1         0         1
# Outlander               0     0     1         0         0

I researched how sites like Yelp and TripAdvisor do weighted restaurant ratings. If a restaurant gets one rating, it does not weight as much as a restaurant with 1000 ratings. But that Bayesian paradigm wouldn't work here because the analagous 'rating' is just a one or zero.

It reminds me of this famous beta distribution answer. Is it possible to use a beta distribution approach to the viewing just as a batting average?

Data

set.seed(45)
x <- matrix(sample(0:1, 50, replace=TRUE, prob=c(0.6, 0.4)), nrow=10)
dimnames(x) <- list(c("A Word", 
                      "Backstrom", "Good Witch Hallmark", "Shark Tank", "Above the Rim", 
                      "O'Reilly Factor", "Jack the Giant Slayer", "Late Night Snack", 
                      "Outlander", "Cake Wars"), c("sad", "happy", "angry", "surprised", 
                                                   "disgusted"))

Answer 1

"It reminds me of this famous beta distribution answer. Is it possible to use a beta distribution approach to the viewing just as a batting average?"

Definitely. Think of each of your categories as a batting average. Alpha + Beta should be the same for each category representing the total number of films viewed, alpha the number of films watch in that category. You should set alpha initially to reflect the average profile as a proportion of your chosen Alpha + Beta. You'll calculate a weighted score for each category which will only shift slightly as a film is watched that fits the category.

The trick here is to choose what an appropriate "head start" is. If you set the denominator (Alpha+Beta) too high your model won't be sensitive enough, too low and you'll have an oversensitive model again.

Hopefully you have a distribution like this of existing scores showing which film have been viewed by your entire population

#                           sad happy angry surprised disgusted TotalViews
# A Word                     1     0     0         0         0      23
# Backstrom                  0     1     0         0         1      30
# Good Witch Hallmark        0     0     0         0         0      10
# Shark Tank                 0     0     1         1         0      55
# Above the Rim              0     0     1         0         0      65
# O'Reilly Factor            0     0     0         0         0      20
# Jack the Giant Slayer      0     1     1         1         0      12
# Late Night Snack           0     0     1         0         1      35
# Outlander                  0     0     1         0         0      84

By multiplying total views by the flag you can get the population score for each category. So in the above example happy would get a score of 42 out of 334 (beta would be 334-42=292), angry a score of 251 out of 334. These scores can be scaled so your prior isn't too large for example divide by 10. You then have your alpha and beta for each category.