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For instance, to preview a future post, much of the appeal and popularity of structural equation models (SEMs) that they let researchers take causal diagrams (variables connected by arrows indicating which ones causally affect which others) and turn them directly into fitted statistical models. When a foul is committed, the offender's turn ends and the referee announces the penalty. A player (or team) continues to shoot until committing a foul or failing to legally pocket an object ball (whether intentionally or not); thereupon it is the turn of the opposing players. Because of this, it is possible for a game to end with only one of the players having shot, which is known as "running the table" or a "denial"; conversely, it's also possible to win a game without taking a shot; such a scenario may occur if the opposing player illegally pockets the 8 ball on any shot other than the break (such as sinking the 8 ball in an uncalled pocket, knocking the 8 ball off the table, sinking the 8 ball when a player is not yet on the black ball, or sinking both the 8 ball and the cue ball off a single shot).

One example of an inelastic collision in billiards is when the player hits the cue ball with the pool stick. It can make you a better billiards player. The player must then play away from that ball without moving it or else the player will concede penalty points. However, it is only a loss if the 8 ball is no longer in play. Well, here is an answer, you can always stay fit while playing games as well, all you have to do is play and have fun and you will stay fit. Also be sure to read the revised manual for version 0.4 for more information on playing and configuring Billiards. Which I think makes them positively misleading in many circumstances (as I say, much more on SEMs in a future post). You cannot think about this dynamical system in terms of sequences of causal events. This is a case where it’s sooo tempting to think in terms of sequences of events; I know because my undergrad students do it every year.

You cannot think about equilibria in terms of sequences of causal events, it’s like trying to think about smells in terms of their colors, or bricks in terms of their love of Mozart. There are no sequences of events here. You’ve got some prey that reproduce and die, and some of those deaths are due to predators. And again for the sake of simplicity, let’s say it’s a constant environment and there’s no particular time at which organisms reproduce or die (e.g., there’s no "mating season"), so reproduction and mortality are always happening, albeit at per-capita and total rates that may vary over time as prey and predator abundances vary. Purely for the sake of simplicity (because it doesn’t affect my argument at all), let’s say it’s a closed, deterministic, well-mixed system with no population structure or evolution or anything like that, so we can describe the dynamics with just two coupled equations, what is billiards one for prey dynamics and one for predator dynamics.

This system is reversible, and you can approximate it efficiently (by fixed or floating point arithmetic). Can you approximate it reversibly and efficiently? Now I can hear some of you saying, ok, that’s true of the math we use to describe the world, but it’s not literally true of the real world. To learn to use English properly, you must first understand what it is. It is most common in traditional billiards to use only three balls. That is, SEMs mesh with and reinforce our natural tendency to think about causality in terms of colliding billiard balls. "The prey go up, which causes the predators to go up, which causes the prey to crash, which causes the predators to crash." In lecture, even I’ve been known to slip and fall back on talking this way, and when I do the students’ eyes light up because it "clicks" with them, they feel like they "get" it, they find it natural to think that way. No. What that increase in prey abundance did was slightly change the expected time until the next birth or death event, by increasing prey abundance and (in any reasonable model) feeding back to slightly change the per-capita probabilities per unit time of giving birth and dying.