Dart shots are not evenly distributed. You are more likely to get the height wrong then the width - this is because of how the dart travels.
Darts dip a lot in flight - they do not move side to side as much. As a dart player, I guesstimate that after only a few hours practice most people will be off 1.5cm or so in the vertical but only 0.75cm in the horizontal
Professional dart players are always aiming at 20. If they block the triple-twenty with their first or second dart, they then aim at triple-19.with some players preferring triple-18 since height-wise it is in similar range
To adjust the model for this effect one needs to scale the axis of the multivariate Gaussian noise differently.
But does the variation in the two different axis result from "how the dart travels", i.e. the physics ? To me it seems to be a matter of relative consistency of the angles with which one throws the dart. I remember reading an article that analyzed trajectories of basket ball free throws. They found for a given player their velocity of projection remained surprisingly consistent. Maybe that is true for throwing darts as well.
Three statisticians played a game of darts. The first statistician shot, but missed, by a meter to the left. The second statistician shot, but also missed, by a meter to the right. The third statistician yelled "bullseye!"
Gaussians? Why Gaussians? Why not Cauchy for example (it has fatter tail)? And why identity matrix for the Gaussian covariance?
I really like the way you are using your technical skills for computation but I think you are using it on the wrong problem.
In reality, even if your dart throws are Gaussians they need not be independent of each other.
I'm coming from math background and was observing math people(physicists as a proxy) to the approach of making unreasonable assumptions if they will simplify the problem and make it solvable.
It reminds me of the joke:
A person is walking around a street light. Another asks him
There's only one more thing this needs to be perfect. I'd love to be able to calibrate it to me, by aiming for the bullseye 10 times and telling it what I hit. That way, it could assess my expertise, and suggest what I should personally be aiming for.
back in college the first problem set in my intro AI class was something very similar to this, except instead of maximizing the points scored on the current throw, it looked ahead and solved for the best "path" to get to zero points given your current score (in darts you win by reducing your score down to zero). Like the OP, I also remember being surprised by the fact that the bullseye wasn't the highest scoring point on the dartboard, but things get even more interesting when you look at the optimal strategy for actually winning the game.
This would make it much more complicated, but did your algorithm take into account the difficulty of hitting each scoring area too? That would be pretty interesting to have as an iPhone app or something.
Double 16 is most professional players favourite close out double because darts players tend to "line up" darts. Basically, they have a much better chance of hitting it on the second go if the first one was closer.
Double 16 is most forgiving because double 8 is above it. If they hit single 16 they can go to double 8 above it and if they hit double 8 above it, they go to double 8 again (requiring minimal or no correction)
There are some players who like double tops (20) but for the most part double 16 is the conventionally wise way to check out.
This reminds me of two things. The first is the intro to statistics exercise on how you can't have uniform probability on a continuous domain, since you will almost surely miss any particular point on the board. It is often illustrated with dart boards, and caused me great consternation once.
The second is that in my experience, dart throws aren't normally distributed: they're skewed towards the bottom, because I don't throw very hard. I wonder if that effect goes away with practise.
Darts dip a lot in flight - they do not move side to side as much. As a dart player, I guesstimate that after only a few hours practice most people will be off 1.5cm or so in the vertical but only 0.75cm in the horizontal
Professional dart players are always aiming at 20. If they block the triple-twenty with their first or second dart, they then aim at triple-19.with some players preferring triple-18 since height-wise it is in similar range