How not to sort by a complicated frequentist formula

Well, you have to make somewhat of a secret sauce, because what you show on top should depend on what you want the top things and the people who see them to do. If you're a site like reddit, putting highly-rated stuff on the front page is good, but what you really want is highly rated new stuff. But if you're urbandictionary, you don't really care when something was submitted, you want to put the best answer at the top - except that if a word has multiple definitions, you want to have variety in your top definitions. Or maybe you're amazon, and you want people to see stuff they've already bought so it's easy for them to rate it. Etc.

This is interesting, especially considering that it favors low-data items, as opposed to both the confidence-interval-lower-bound and the notability adjustment factor, which penalize low-data items.

You can try to optimize it in an explore-vs-exploit framework, but there would be a lot of modeling parameters, and additional kinds of data will need to be considered. Specifically, a measure of how many of those who viewed the item bothered to vote at all. Some comments will not get any votes simply because they are not that interesting; so if you keep placing them on top hoping to learn more about them, you'll end up with very few total votes because you show people things they don't care about.

Something of interest: Jeffery's interval. Using the lower bound of a credible interval based on that distribution (which is the same as yours) will probably give better results than just using the mean: it is handles small sample sizes more gracefully. (I think, but I'm certainly willing to be corrected.)

I recently did a similar thing for ranking vendors by feedback, using both a Jeffreys interval and a Wilson interval; even on the vendors with little feedback, they were overall pretty similar. IIRC, I don't think they differed by more than 10% anywhere.

(Link to How Not To Sort By Average Rating.)

I forgot to link in the OP. Then remembered, and forgot again.

Something of interest: Jeffery's interval. Using the lower bound of a credible interval based on that distribution (which is the same as yours) will probably give better results than just using the mean: it handles small sample sizes more gracefully. (I think, but I'm certainly willing to be corrected.)

This seems to use specific parameters for the beta distribution. In the model I describe, the parameters are tailored per domain. This is actually an important distinction.

I think using the lower bound of an interval makes every item "guilty until proven innocent" - with no data we assume the item is of low quality. In my method we give the mean quality of all items (and it is important we calibrate the parameters for the domain). Which is better is debatable.

But I fear that it would cause irreparable damage if the world settles on this solution.

This is probably vastly exaggerating the possible consequences; it's just a method of sorting, and either the Wilson's interval method and a Bayesian method are definitely far better than the naive methods.

I just feel that it will place this low-hanging fruit out of reach. e.g.,

Me: Hey Reddit, I have this cool new sorting method for you to try!

Reddit: What do you mean? We've already moved beyond the naive methods into the correct method. Here, see Miller's paper. No further changes are needed.

Maybe I'm exaggerating - I mean, things can be improved again after being improved once - but I just feel that if the world had a "naive rating method" itch to scratch, and something like Miller's method became the go-to method, something is wrong.

It means that the model used per item doesn't have enough parameters to encode what we know about the specific domain (where domain is "Reddit comments", "Urban dictionary definitions", etc.)

The formulas discussed define a certain mapping between pairs (positive votes, negative votes) to a quality score. In Miller's model, the same mapping is used everywhere without consideration of the characteristics of the specific domain. In my model, there are parameters a and b (or alternatively, a/(a+b) and a+b) that we first train per-domain, and then apply per item.

For example, let's say you want to decide the order of a (5, 0) item and a (40, 10) item. Miller's model just gives one answer. My model gives different answers depending on:

The average quality - if the overall item quality is high (say, most items have 100% positive votes), the (5,0) item should be higher because it's likely one of those 100% items, while (40,10) has proven itself to be of lower quality. If, however, most items have low quality, (40,10) will be higher because it has proven itself to be one of the rare high-quality items, while (5,0) is more likely to be a low-quality item which lucked out.

The variance in quality - say the average quality is 50%. If the variance in quality is low, (5,0) will be lower because it is likely to be an average item which lucked out, while (40, 10) has proven to be of high quality. If the variance is high (with most items being either 100% or 0%), (5,0) will be higher because in all likelihood it is one of the 100% items, while (40, 10) has proven to be only 80%.

In short, using a cookie-cutter model without any domain-specific parameters doesn't make the most efficient use of the data possible.

Specifically, the option can be found above the "Show help" button. "Sort by: Best" is what you want.

Given that a and b are arbitrary, I think the differences can be large. Whether they actually are large for typical datasets I can't readily answer.

In any case the advantages are:

1. Simplicity. Tuning the parameters is a bit involved, but once you do the formula to apply for each item is very simple. In many (not all) cases, a complicated formula reflects insufficient understanding of the problem.

2. Motivation. Taking the lower bound of a confidence/credible interval makes some sense but it's not that obvious. The need for it arises because we don't model the prior mean, so we don't want to take risk on unproven items. A posterior mean of the quality is more natural, and won't cause much problems because items default to the true population mean.

3. Parametrization. The interval methods has a parameter for the probability to take for the size of the interval, but it's not at all clear how to choose it. My method has parameters for mean and variance which are based on the data.

4. Generalization. This framework makes it easier to clearly think about what we want, and replace the posterior mean of p with a posterior mean of some other quantity of interest. e.g., the suggested "explore vs. exploit" tends to give something closer to an interval upper bound than lower bound, and other methods have been suggested.

Why a Poisson distribution? It seems fairly clear we are looking at Bernoulli trials (people who look either upvote, or not). I doubt its a rare enough event (though it depends on the site, I suppose) that a poisson is a better approximation than a normal.

I think some factor for decreasing votes over time should be included. Exponentially decaying rates seem reasonable, and the decay time constant can be calibrated with the overall data in the domain (assuming we have data on voting times available).

Guilty as charged.

Showing upvotes and downvotes separately would stop me (and, probably, other people) from doing that, but I'm kind of getting tired of beating this dead horse.

True. This is a problem since the current net vote count is mutable, while an individual vote, once cast, is not. You could try fitting a much more complicated model that can reproduce this behavior, calibrate it with A/B testing, etc. Or maybe try to prevent it by sorting according to quality, but not actually displaying the metrics.

The beta distribution is a conjugate prior for Bernoulli trials, so if you start with such a prior the posterior is also beta, which greatly simplifies the calculations. It also converges to normal for large alpha and beta, and in any case can be fit into any mean and variance, so it's a good choice.

Whatever your target function is, you'll want the item with the greatest posterior mean for this target. To do this generally you'll need the posterior distribution of p rather than the mean of p itself. But the distribution just describes what you know about p, it doesn't itself encode properties such as "controversial".