So in reality, many websites adopt Bayesian ranking in practice. For instance, the Internet Movie Database or IMDb, if you look at their top 250 movies, they follow the equation for Bayesian ranking directly in determining how to rank order those 250 movies. But they're some practical implications that we have to consider and there's some variations on it in different cases and we'll consider a few of those now. The first is to look at the equation for Bayesian ranking and to look and see whether that overall num value is always going to be exactly how we interpret it, whether it should be or not. So let's write out the equation here. For Bayesian ranking. Which if you recall was the overall average times the overall num plus the individual average times the individual num. Divided by the overall num, plus the individual num. So if you consider the case where we have a lot of products, right? And a lot of products that are gathering a lot more ratings over time. So, in the population we're considering there's a lot of products and they're gathering a lot of ratings over time. This overall num value. It's going to get really large, really fast. And as it starts to get much bigger, this term, the overall average times the overall num, is going to trump and dominate this equation, right? And so the individual num, the individual average are going to have much less of an effect. Especially in the case when we have a lot of products we're considering. And the products are gaining a lot more ratings a lot quickly. And in that case, we're just going to, the equation's just going to tend towards this term right here. And that's not really very useful because then we're just going to rank everything to be the mean value. Right? So, as the number of ratings increases over time. And, additional to that as we have a lot of products, this is going to be a problem. Because it can saturate the equation. The way this is done, the way this is dealt within practicing certain circumstances is by taking this overall num value and changing it to something else. So we still have the overall average here but we're just going to multiply it different value. If you go to beer advocate,. So, website http://beeradvocate.com/lists/popular. They rank a number of beers on their site, and what they do, for this is is they take the num value to be the minimum number of ratings necessary for a beer to actually appear on that list in the first place. So, that's used instead of the Overall Num, and that's a solution that's not going to just saturate as the number of ratings. Increase more and more, and that's how they deal with this problem. Another practical application is that we've assumed that there's a single true value that we're trying to estimate in our analysis. There's a single true mean value. But in reality the truth can vary from person to person so what I think about a product is could be different from what you think about it just because we have a different opinion on it all together. We're drawing from a different sample. So to speak. One example of this can be bipolar reactions. If you look at this customer review, for instance, of this DVD player on Amazon, you see, considering a good rating to be a four or a five star, a high rating, there's a lot of those. There's not very many two and three star which are kind of in the middle, but there's also then a lot of one stars so that. Amount of one star ratings is actually greater than the number of two and three star all together. So this has a lot of very positive reactions and also a lot of lower reactions too. And these types of products just all over the place. What we consider this is, this is actually a bi modal case, right? There's this kind of two modes or means, if you [INAUDIBLE] here. There can be one here. In the centerpoint, maybe, between these two. And then one, maybe, you know, down here, or something. This is just an approximation, but. This may be a better assumption about the distribution looks like, with the true distribution if there is a truth if we assume that there is some truth here, looks like. And in this case we have to make an extension. So what we've assumed in our analysis is that there's some single true value. Right, which is unimodal. Unimodal instead of multimodal. But what this is saying now is rather than there just being one value that we're trying to estimate based upon what people are guessing around it, there could be multiple values. So that brings upon a lot of practical implications, like how many values are there? How do we determine how many values there are from the sample? And how do we map a given user rating to one of these values, if we don't even know necessarily what they are in the first place? So it gets complicated and there's a lot of challenges there as well. But just know that there are multimodal extensions to Bayesian ranking. Final thing is that in practice you can only apply Bayesian ranking within a comparable family of products. So what we did before was. We took the population to be a set of DVD players. So the commonality was being a DVD player. Another thing would be, using a pair of shoes. Like, different brands of shoes. And so forth. You couldn't, for instance, combine DVD players and shoes and call that a population because then the average. The overall average, and the overall num are not going to be meaningful. So it has to be such that the average, is meaningful. [SOUND] Because otherwise the Bayesian ranking equation won't make sense. [BLANK_AUDIO]