# Reverse engineer a Bayesian estimate?

My apologies if this is a basic question because I am no mathematician. Struck on my work on this, so came here to get some help.

I am working on this bayesian estimate explained here.
This is a bayesian estimator used to find out the Top 250 list of all the movies in Imdb.com with the following factors.

where,
W= weighted rating
R= average for the movie as a number from 0 to 10 (mean) = (Rating)
m= minimum votes required to be listed in the Top 250 (currently 25000)
C= the mean vote across the whole report (currently 7.1)

Here what I am trying to do is to find the required number of votes and rating for a normal movies to get into the Top 250 list.
This is how I do it:

1. Find the Weighted Rating(W) of 250th movie in the list.
2. subtract 0.0001 to the W to get newW.
3. With the new weighted rating I have to calculate the required number of votes and the rating.

This 3rd step is where I got struck. Can you simplify me an expression to calculate
2. Average Rating

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It seems that by "WR" you don't mean W times R as you had introduced the variables; you introduced a second name "WR" for the variable you had previously named "W"? That's rather confusing. – joriki Aug 22 '12 at 6:09
@joriki sorry.. made the correction now. – vivek_jonam Aug 22 '12 at 7:10

This is not Bayesian in my view. No matter.

Looking at the table itself, it seems to report $W$ rather than $R$ and it looks as if both $W$ and $C$ are rounded to 1 decimal place.

So you rewrite your formula as $$R=\frac{W(v+m)-Cm}{v}.$$

The film at place 250 has $W\approx 8$ so you need $W \gt\approx 8$ for your new film which, because of the rounding, makes it probably better to aim for $W \ge 8.05$. You have a $-$ before $C$ so for safety you should probably look at $C=7.05$ rather than $C \approx 7.1$. So your formula becomes $$R \ge \frac{8.05(v+25000)-7.05 \times 25000}{v}=8.05+\frac{25000}{v}.$$

So if you have 25,000 votes for the new film then you need $R \ge 9.05$, while if you have 250,000 votes you need $R \ge 8.15$.

The rounding is probably not quite as extreme as that so you might get away with slightly lower ratings. If you looking at the rounding as extreme in the opposite direction you would find that $R \le 8.75$ with 25,000 votes or $R \le 8.03$ with 250,000 votes would ensure you do not get into the top 250.

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Thanks. This gives total requirement as 25000 votes of 9.05 rating. But if the film already has 7.7 of 31,400 votes then how do i calculate the "remaining requirement" to meet top250? – vivek_jonam Aug 22 '12 at 11:31
@vivek_jonam: That depends on how many extra voters you have. For example, assuming your $7.7$ is exactly $R$ rather than a rounded $W$, to reach the $8.15$ at 250,000 votes you need an additional 218,600 votes giving an average of at least $8.215$ between them. Or you could instead have 18,457 additional voters giving $10$ each. – Henry Aug 22 '12 at 11:55
How did you find its 218600 of 8.215 votes or 18,457 of 10 votes? Can you update an expression for this? – vivek_jonam Aug 22 '12 at 12:01
@vivek_jonam: It just depends on finding the right average to fit. For example $31400+218600=250000$ and $7.7 \times 31400 + 8.215 \times 218600 \approx 8.15 \times 250000$ – Henry Aug 22 '12 at 17:28
so, you saying given the current votes and total requirement, we have use above way to get remaining? – vivek_jonam Aug 23 '12 at 4:45