The “hot hand” has been mostly dead for the last half-decade — the province of irrational believers in “the zone” and “clutch hitting” and other amorphous concepts that haven’t withstood killjoy statistical analysis. Almost every search for the alleged “hot hand,” from the mid-1980s and onward, has either come up empty or found the opposite phenomenon — that players who make a basket are less likely to make their next shot than they are after a miss. David Brooks mocked the hot hand in the New York Times, and Larry Summers, who has been both U.S. Treasury secretary and president of Harvard, chastised the university’s basketball team for believing in the concept.

And, really, how disappointing is that? Few things in basketball are as fun as the burbling excitement that rumbles down from the crowd when a player hits one shot, then another, and gets the ball in position to try an even more difficult shot — and makes it. How could Stephen Curry not have been hot in raining 54 insane points on the Knicks at Madison Square Garden last season? Was it really just a random streak — basketball’s version of a run of “heads” in a series of coin flips?

But the hot hand is quietly enjoying a bit of a renaissance, and that revival might peak this weekend in Boston, when a group of authors with access to new optical tracking data present a paper showing that the hot hand might actually exist. The paper comes on the heels of one study that showed a possible hot-hand effect on free throws, and a series of papers that essentially concluded it might be impossible to debunk the concept. Most players and coaches, including shooting coaches, have kept the faith in something like the “hot hand” even as the statistical proof against it has piled up.

The authors of the new paper, to be presented this weekend at the New England Symposium on Statistics in Sports, actually find themselves in agreement with much of the hot-hand debunking that came before them. But the fancy SportVU motion-tracking cameras allowed them to look for things previous researchers couldn’t — the exact location of each shot, the identity of the closest defender, how close that defender actually was to the shooter, the trajectory of the ball in the air, and lots of other things. All of that stuff allowed them to measure the difficulty of a shot in ways that were once impossible.

And that gets at two points where the authors, John Ezekowitz (a former consultant for the Suns), Andrew Bocskocsky, and Carolyn Stein (Harvard grads all), find common ground with the earlier research:

• Players are indisputably more likely to take their team’s next shot if they have made their previous shot, or the last two or three shots they attempted. Here’s a graph showing how much more likely J.R. Smith was to shoot on any New York possession that came after Smith had made a certain number of shots in a row within that game:

The numbers inside the circles represent the number of instances in which Smith could have attempted a shot under the listed criteria; the “9” above the little circle over at the right indicates Smith had only nine instances in which he was on the floor while having made his last five shots.

Here’s the same graph for Curry:

Again: Everyone agrees on this. Sandy Weil, whose presentation at the MIT Sloan Sports Analytics Conference in 2009 argued very persuasively against the hot hand, found a similar tendency among shooters who think they are hot.

• Players are more likely to take increasingly difficult shots after a make or two. Weil found that players were significantly more likely to take jump shots after having made a jumper than they were after misses or non-jumper makes. Ezekowitz and his coauthors found the same effect, in part by charting the 83,000 camera-recorded shots into 2-by-2-foot squares on the floor:

But this is where the new study begins to break some new ground using that camera data. “We’re not saying those prior studies were wrong,” Ezekowitz says. “Just that they didn’t have enough access to information.” The three also found that defenders tended to creep a bit closer to guys who had made an unusually high percentage of their last five shots. The effect was often small; a shooter making one more shot than expected in a string of five resulted in defenders chopping only about half an inch of the usual distance between shooter and defender. But the effect increased as shooters hit more shots, and the effect was consistent; defenders on average were within four feet of shooters, so even a few inches might matter a lot.

Measuring difficulty was the key to the authors finding some kind of hot-hand effect. Previous studies either treated every shot equally, so that making a layup and missing a 3-pointer were basically coded as 1s and 0s, or could estimate shot difficulty only in a rough way — so rough a lot of studies couldn’t do much with it.

The camera data allows for increased precision. In addition to plotting shots and defenders, Ezekowitz, Bocskocsky, and Stein were able to look at where defenders were in relation to a shooter and factor that into their measure of shot difficulty. A defender directly between the shooter and the rim probably represents more of a distraction than a defender exactly as far from the shooter but positioned to the shooter’s side. They controlled for the height of the specific defenders and shooters, so they could account for the difference between having Nate Robinson and LeBron James closing out on Steve Novak. (The camera data do not record appendages, so that the authors could not suss out whether defenders had their arms up in challenging shots. This is a limitation to their study, they say.)

In a basic sense, the authors were able to estimate the probability of making each specific shot based upon all these variables. From there, they could take any string of five, or six, or eight shots by an individual player and figure out how likely it would be for that player to make a certain number of those shots. If their shooting percentage toward the end of some of those strings trends upward when the player begins them with makes, well, that might be evidence of something like the hot hand.

And that’s what Ezekowitz, Bocskocsky, and Stein found. Take a string of five given shots, and control for the difficulty of each shot. If the player in question has hit even just one more of those five shots than we’d expect, his shooting percentage on shot no. 6 in the string goes up by something between 1 and 3 percent, the study found. In other words: If shot no. 6 is on average, given its difficulty level, a 40 percent proposition, having even a single extra make in the recent rearview appears to nudge that shooting percentage up to 42 or 43 percent.

That’s not a huge effect, something the authors readily concede. “It’s not like NBA Jam, with players having flames shoot out of them,” Ezekowitz says. But it’s also not nothing, and it’s certainly not evidence that players shoot worse after making a shot or two.

But here’s the thing: In raw terms, they actually do shoot worse after makes. Remember, the authors of this study controlled for shot difficulty in very sophisticated ways. They found that “hot” players are indeed more likely to make the shots they take, but that those shots are very, very tough — low-percentage looks under any condition. Heat checks, perhaps. Their raw field-goal percentage declines slightly in the “hot” state, a finding that jibes with Weil’s work and other prior studies.

In other words, players might really get “hot” and shoot better than we’d expect, but they could also be hurting their teams by taking crazy shots.

“That was almost our conclusion, in a way,” says Weil, author of the 2009 study presented at the Sloan conference. “That the hot hand probably doesn’t exist, but even if it does, it’s trumped by counter-effects — by players thinking they are hot and getting overconfident.”

Finding implications here is hard. On a simple level, the hot hand is cool, and it would be affirming in some ways to know it might exist in some form. But it is very hard to find, and even if it exists, it might be both minuscule in scope and detrimental to team goals. It’s hard to read any of these studies, even with the conclusions veering in slightly different directions, and conclude that heat checks are a good thing for offenses. This could in theory impact play-calling, with coaches gently steering shots away from players who have just made a few attempts; I know a lot of New York fans who react to Smith hitting step-back 20-footers by applauding and politely asking Smith to please not read too much into that particular result going forward.

But such on-the-fly adjustments are hard. If Smith gets the ball and he feels good, he’s shooting the damn thing. And if his teammates believe in the hot hand — and just about every player seems to — they are going to enable him by passing Smith the ball in tougher and tougher spots. Smart teams might eventually rejigger their approach to these kinds of situations, especially if a demanding coach has lectured them about it for months. Maybe defenses will one day encourage “hot” players to test their heat levels.

But what is Mike Woodson supposed to do? Call a timeout after that first made step-back 20-footer and remove Smith so that he cools down before going berzerk? “That might affect his mind-set,” Weil says, “knowing he’s going to get yanked every time he makes a shot.”

The same implication issue arises when we consider work by Jeremy Arkes, a professor at the Naval Postgraduate School in Monterey, who found over a giant sample size that players are about 3 percent more likely to hit their second free throw on a two-shot trip to the line if they also hit the first one. That’s fascinating, if it holds over multiple seasons. But how do coaches and players adjust to that kind of information?

Incorporating all this research is easier during timeouts, when a coach can design plays to minimize the chances of a bad heat check, as Henry Abbott has written before at TrueHoop.

Believing or not believing in the hot hand might change some things about the way a game flows, but even proponents of the hot hand’s existence claim it’s a relatively small effect that doesn’t emerge very often. And that’s part of the challenge in the data, even apart from trying to explain the factors that might lead a player into a better rhythm on a particular day. What is “hot,” statistically? Making two in three shots? Eight in 12? How do we know when to start the streak and when to stop it? How many times do players really get “hot” in a given season? Five? Ten? Two?

“It’s very hard to define,” Ezekowitz says.

But the hot hand is certainly fun to talk about as a mysterious, ephemeral thing that opens up debate about human nature and abilities. It’s a cool concept. And even for that reason alone, it’s fun to see the debate about it heat up again.