The following link describes the Bayes weighted average with an example.
$$S = wR + (1-w)C$$
$S$ = score of the restaurant
$R$ = average of user ratings for the restaurant
$C$ = average of user ratings for all restaurants
$w$ = weight assigned to $R$ and computed as $v/(v+m)$, where $v$ is the number of user ratings for that restaurant, and $m$ is average number of reviews for all restaurants.
The example mentioned is
Restaurant A has 100 opinions with an average rating of 4.5
Restaurant B has 1 opinion with an average rating of 5
The grades are recalculated using the above formula. However I have the following question:
Q1) How is the value of $m = 40$ and $c = 4.2$ derived? As a follow up of the above question: Shouldn't the value of $m$ be $(100+1)/2 = 50.5$ and the value of $C$ be $(4.5+5)/2 = 4.75$?
Q2) Are the final weighted scores on a scale of 1-5 ?
Update as of 06/12/2016 As a follow up to the link provided by tim and my understanding of the bayes weighted average, i have proposed an example with calculations. Please find attached the image.I have 2 questions A&B with respect to the image attached.
There are 5 products only each with their mean score and no of ratings
I have used the following formula to calculate the scores. Its the same formula posted earlier but i found this easier to implement. b(r) = [ W(a) * a + W(r) * r ] / (W(a) + W(r)]
where r = average rating for an item W(r) = weight of that rating, which is the number of ratings a = average rating for the entire data set of 5 products. W(a) = weight of that average, which is the average number of ratings for all the 5 products, b(r) = new bayesian rating
QA)The average rating for all products SUM(WX)/SUM(W) = 251.78/64 = 3.93. I took a weighted mean instead of an arithmetic mean. The average number of ratings for all W(a)= 64/5 = 12.80. I have set m = 12.80 on the observation that 70 % of the products(data points with regards to the number of ratings fall above this value) .
Is this value optimal ? Does it need to change dynamically each time the total avg number of ratings changes ?
QB)Based on the bayes score calculations,
Why is BR2 (4.01) which has only 1 rating with a mean score of 5/5 still ranked higher than BR1 (4.00) which has 25 ratings with a mean score of 4.03/5 and BR4 (3.68) which has 21 ratings with a mean score of 3.52 ?