Understanding empirical Bayes estimation using baseball statistics

But suppose you were a baseball recruiter, trying to decide which of two potential players is a better batter based on how many hits they get. One has achieved 4 hits in 10 chances, the other 300 hits cover tech21 iphone 7 plus in 1000 chances. While the first cover iphone 6 dybala player has a higher proportion of hits, it’s not a lot of evidence: a typical player tends to achieve a hit around 27% of the time, cover samsung s8 plus originale and this player’s could be due to luck. The second player, on the other hand, has a lot of evidence that he’s an above average batter.

This post isn’t really about baseball, I’m just using it as an illustrative example. (I actually know very little about sabermetrics. If you want a more technical huawei p10 lite hoesje version of this post, check out cover iphone 4s fashion this great paper). This post is, rather, about a very useful statistical method for estimating a large number of proportions, called empirical Bayes estimation. It’s to help you with data that looks like this:

A lot of data takes the form of these success/total counts, where you want to estimate a “proportion of success” for each instance. Each row might represent:

An ad you’re running: Which of your ads have higher clickthrough rates, and which have lower (Note that I’m not talking about running an A/B test comparing two options, but rather about ranking and analyzing a large list of choices.)

A user on your site: In my work at Stack Overflow, I might look at what fraction of a user’s visits are to Javascript questions, to guess whether they are a web developer. In another application, you might consider how often a user decides to read an article they browse over, or to purchase a product they’ve clicked on.

When you work with pairs of successes/totals like this, you tend samsung s6 hoesje to cover per iphone 5c amazon get tripped up by the uncertainty in low counts. does not mean the same thing as ; nor does mean the same thing as . One approach is to filter out all cases that don’t meet some minimum, but this isn’t always an option: you’re throwing away useful information.

I previously wrote a post about one approach to this problem, using the same analogy: Understanding the beta distribution (using baseball statistics). Using the beta distribution to represent your prior expectations, and updating based on the new evidence, can help make your estimate more accurate and practical. Now I’ll demonstrate the related method of empirical Bayes estimation, where the beta distribution is used to improve a large set of estimates. What’s great about this method is that as long as you have a lot of examples, cover iphone 5s marcelo burlon you don’t need to bring in prior expectations.

Here I’ll apply empirical Bayes estimation to a baseball dataset, with the cover kevlar iphone 6 goal of improving our estimate of each player’s batting average. I’ll focus on the intuition of this cover apple iphone 5s particolari approach, but will also show the R code for running this analysis yourself. (So cover samsung j6 2018 unicorno that the post doesn’t get cluttered, I don’t show the code for the graphs and tables, only the estimation itself. But you can custodia in pelle iphone 5 find all this post’s code here).

In my original post about the beta distribution, I made some vague guesses about the distribution of batting custodia rugged iphone 8 plus averages across history, but prezzo cover iphone 6 here we’ll work with real data. We’ll use the Batting dataset from the excellent cover battery iphone Lahman package. cover samsung a5 2016 adidas We’ll prepare and clean custodia cover iphone x xs the data a little first, using dplyr custodia cover samsung j3 and tidyr:career Batting %>%

filter(AB > 0) %>%

anti_join(Pitching, by = “playerID”) %>%summarize(H = sum(H), AB = sum(AB)) %>%

mutate(average = H / AB)

use names along with the player IDs

career Master %>%

tbl_df() %>%unite(name, nameFirst, nameLast, sep = ” “) %>%

inner_join(career, by = “playerID”) %>%Above, we filtered out pitchers (generally the weakest batters, who should be analyzed separately). We summarized each player across multiple years to get their career Hits (H) and At cover iphone 6 wolverine Bats (AB), and batting average. Finally, we added first and last names to the dataset, so we could work with them rather than an identifier:

(For the sake of estimating the prior distribution, I’ve filtered out all players that have fewer than 500 at bats, since we’ll get a better estimate from the less noisy cases. I show a more principled approach in the Appendix).

The custodia cover huawei p smart first step of empirical Bayes estimation is to estimate a beta prior using this data. Estimating priors from the data you’re currently analyzing is not the typical Bayesian approachusually you decide on your priors ahead of time. There’s a lot custodia silicone per iphone 5s of debate and discussion about when and where it’s appropriate to use empirical Bayesian methods, but it basically comes down to how many observations we have: if we have a lot, we can get a good estimate that doesn’t depend much on any one individual. Empirical Bayes is an approximation to more exact Bayesian methodsand with the amount of data we have, it’s a very good approximation.

So far, a beta distribution looks like a pretty appropriate choice based on the above histogram. (What would make it a bad choice Well, suppose the histogram had two peaks, or three, instead of one. Then we might need a mixture of Betas, or an even more complicated model). So we know we want to fit the following model:..