Mark,

First, you have a good point, thank you for noticing it. Nothing I say below should detract from that!

I’ve only golfed a handful of times, but even that has been enough to see the ball ‘lip out’ sometimes: if you hit the ball a bit ‘too hard’ — harder than it needs to in order to reach the cup — then even if the center of the ball is inside the edge of the cup, it can (and sometimes does!) escape. It happens more often than you might expect…well, certainly more obvious than you do expect, given that it’s “obvious” to you that it can’t happen at all! There’s a nice example at https://www.youtube.com/watch?v=oWqGRlq_wDg , it’s low-resolution but it’s good because the video is shot from directly behind the club so you can see the path of the ball quite well.

Anyway, you’re right that the simple model that they’ve assumed is deficient — and I congratulate you for noticing! — but the model you propose also misses a substantial real-world effect.

]]>The golf ball is radius r. The hole has radius R. If the centre of the golf ball is within a distance of R from the centre of the hole, it is going to fall in. The Stan code above, and the description in Gelman and Nolan’s “Teaching Statistics” book (Section 9.3) assumes the ball falls in only if its centre is less than R-r from the centre of the hole. This views the golf ball and hole as two circles on the plane and requires that the smaller circle, with radius r, be fully enclosed by the larger one with radius R. But it seems obvious to me that all that is required is that the golf ball’s centre be within the larger circle, i.e. less than R from its centre.

If this is the case, the angle we are looking for is asin(R/x) and not asin((R-r)/x).

Of course, all of this just assumes the simple physical model, which of course could be expanded. But that’s another matter.

]]>I realized there was a ton of overdispersion. So I refit with Williams’ method to accommodate it. Much better I think, but still not great.

Justin

http://www.statisticool.com

Summary:

-golfers predicted to miss more than make putts if distance > ~12ft

-prob(making distance=0 putt) = 94%. Maybe not 100% for small distances because of the well-known “yips”

-model over-predicts success for short distances, and under-predicts success for large distances

Now I’ll do the same thing, but if a distance is 16, I’ll merge those counts in with the distance 16.43 cases, etc., instead of treating them like separate categories. For example:

16 201 27

16.43 35712 7196

16.43 35913(=35712+201) 7223(=7196+27)

I wouldn’t have merged like this if it was 16.67. I would have merged that in with the 17 category. Dichotomania.

Doing that I get..not much changes at all from what I did at first.

Justin

http://www.statisticool.com

Okay, you’re going to the idea of a low “relative weight” of the prior in the posterior (compared to the “high weight” of the data).

Doesn’t this legitimate a resurrection of the idea of using “part of” the data to choose a prior and the rest to estimate the posterior, or something like Zellner’s “g priors” ? The idea was enticing, but I don’t remember seeing used in a lot of applied papers…

ISTR that you used an idea close to that in the first edition of “Data Analysis Using Regression and Multilevel/Hierarchical Models” in order to show that some weakly informative priors “weighted” less than one observation (on having the book on hand, I can’t check that now).

BTW : if such “data-derived priors” stand to criticism, wouldn’t a tool more-or-less automating them be a valuable addition to Stan ?

]]>I imagine side-hill putts to be very difficult as well, you must putt somewhat uphill so that as the ball breaks downhill, it goes towards the hole, and the angle you have to putt is intimately connected with the speed you have to putt. Roughness of the green affects the speed, and that affects the direction as well. There are multiple solutions to the problem, just as for example you can throw a ball fast and hit a target on the ground, or you can lob it up in the air and hit a target on the ground, or anything in between if you keep the angle and the initial speed correct together. It gives you more options, but it can easily mean higher risk as well, if you miss your ball may roll downhill far from the hole. On a flat green if you miss but aren’t too far off, you’ll have a tap in at least.

It just seemed like an obviously interesting question to ask.

]]>Emmanuel:

Je pense que la distribution a priori, c’est la partie le moins important dans ce modèle. Plus généralement, je pense que tout le monde regarde la distribution a priori avec bien souci mais moins regarder sérieusement le modèle pour les données. En ce cas, c’est clair à moi que la distribution a priori est moins important : on peut examiner la distribution a priori de sigma et voir qu’elle est très précis en comparaison à toute distribution a priori qu’est raisonnable.

]]>> “Sure, default models are defensible.”

Of course. What I meant was “is this specific default model defensible in this particular case ?”.

>”Feel free to explain why you put “default prior” in quotes”

I wasn’t (and still am not) sure that this is the right term ; BTW, you use “default model” rather than “default prior”, and I’d nitpick that up to “default model part”…

>” and why you wrote “ahem.”

I thought that I might be underscoring an infelicious edit of the prior out of the model, hence apologizing for the inconvenience…

As for the notion of defaults : they are pertinent for a certain class of models and a certain class of problems. Using them in a given case entails to be able to defend their use in said case. That was what I meant.

My (non-)mastering of the goddamn’ English language might be responsible for the loss of meaning between what I intended and what you understood. Things might be easier if we both used, say, Latin, a non-mother tongue for al parties involvedf (and, BTW, one allowing for much more precision than the globish that passes for English in scientific communication…).

]]>Emmanuel:

Sure, default models are defensible. Applied statistics is full of default models. Indeed, it would be difficult to find a statistical model that does *not* have any default components. You can start with all the linear and logistic regressions out there, etc etc. For any default model, it’s the usual story that we should be aware of ways in which the model can be improved.

P.S. Feel free to explain why you put “default prior” in quotes and why you wrote “ahem.” I feel like there’s something more you want to say, or something you’re implying, but I’m not quite sure what. I doubt you think that every part of every model, to be defensible, needs to be constructed completely from first principles. So I don’t think you could be opposed to defaults in general. Are there some defaults that are more bothersome than others?

]]>I see it as just much better economy of research where most importantly getting less wrong is never blocked along with purposeful and continual attempts to maximize the expected rate of getting the less wrong. ]]>

The general consensus is that the proportion of putts that go into the hole from various distances will increase, since the flagstick will now serve as a backstop of sorts, allowing putts that would have previously skipped over the hole because they were going too fast, to go in. In fact, some professional golfers have specifically stated that they will now practice and judge putting speed based upon the ball going past the hole some distance, since they will now have an increased level of comfort that the flagstick will help in directing the ball into the hole.

]]>While the ranking of players may not change when you use a more sophisticated model, I don’t think that is the question here. The question is whether the frequency of puts made is affected by break and other factors. I would be very surprised if break was not an important factor. I also strongly suspect downhill puts are much more frequently missed.

]]>What’s the prior for sigma ? Stan’s “default prior” ? Is this defensible ?

]]>It’s a modern version of Lakatos/Kuhn normal science and scientific revolution, and we can do this in so many different application areas. Just amazing.

I feel lucky to have been an active researcher in the period during which these ideas became fully realized and part of statistical workflow rather than just implicitly visible in a few special examples (such as the Mosteller/Wallace Federalist Papers study, and whatever Turing et al. did to crack those codes).

]]>That’s a fabulous example of what I’m thinking about. On a flat green, there is essentially only one direction to putt in, but on a contoured green direction is intimately connected to initial speed and the contour along the whole path. Strangely it can be the case that there are many ways to hit the ball into a contoured hole. That’s very obvious if you think about a drain at the bottom of a shower stall…

]]>Sure, but with a flat green the question is how well can you estimate the angle at which the hole lies… a question of kinematics. With a sloped green it’s how well can you estimate the angle at which you should putt the ball so that dynamically it arcs into the hole… I think these are distinct skills, one is essentially measurement of angles the other is estimation of dynamical forces etc

]]>Presumably good golfers have more shorter putts than longer putts, bad golfers typically have more equal numbers of long putts and short putts.

]]>I thought about that briefly and decided that the break was what a (good) golfer would take into account when deciding on the angle, so is already incorporated in the model, to a first approximation. Perhaps the standard deviation on the angle should increase with distance since the golfer has to decide on the angle rather than aim for the hole.

]]>Curious if you interacted slope with length, it’s clear here that slope has to interact, it doesn’t really have a non-interacted effect

]]>I made a nice post (imo) with quotes/link from ET Jaynes and a reference to Fisher talking about it but looks like it got spambinned. Anyway, I find making the distinction helpful, but not important enough to attempt a repost.

]]>Yes, you are correct. I was being very loose with the term ‘probability’. I should have used ‘one-putt frequency’.

]]>The best available data does not have (as far as I know) information about green contours. Just distances to the nearest inch or thereabouts.

]]>Did you just logically equate “probability” and “frequency”?

You’re waving a red flag in front of a bull, right there…

]]>The fundamental, inescapable distinction between probability and frequency lies in this relativity principle: probabilities change when we change our state of knowledge, frequencies do not. It follows tha the probability p(E) that we assign to an event E can be equal to its frequency f(E) only for certain particular states of knowledge. Intuitively, one would expect this to be the case when the only information we have about E consists of its observed frequency.

[…]

Then why should there by such unending conflict, unresolved after over a Century of bitter debate? Why cannot both exist in peace? What we have never been able to comprehend is this: If Mr. A wants to talk about frequencies, then why can’t he just use the word “frequency”? Why does he insist on appropriating the word probability and using it in a sense that flies in the face of both historical precedent and the common colloquial meaning of that word? By this practice he guarantees that his meaning will be misunderstood by almost every reader who does not belong to his inner circle clique.

http://omega.math.albany.edu:8008/JaynesBook.html (Chapter 9)

For example, I can know someone is about to blow a foghorn just as the golfer is about to put. Then I would say the probability of success should be lower than the frequency. Even with no data on the effects of well-timed foghorns on putting (no idea of the frequency for that subset), we can give success a lower probability than the frequencies shown in the chart. Fisher makes a similar argument here: Fisher, R N (1958). “The Nature of Probability”. Centennial Review. 2: 261–274. http://www.york.ac.uk/depts/maths/histstat/fisher272.pdf

Also:

> set.seed(1234)

> rbinom(5, 1, .05)

[1] 0 1 1 1 1

Probability next number is a “success” = 0.5

Frequency of “success” = 0.8

“One-putt probability” is the same as “Frequency of success”

]]>