Mixing
Is actions per minute (APM) Poisson-distributed or log-normal distributed?
$begingroup$
Yesterday DeepMind's AlphaStar beat a high-ranked professional player at StarCraft II.
My co-workers and I were chatting about whether the AlphaStar has an unfair advantage over the human player because as a machine, it can perform many more actions per minute than any human can possible do. Then a colleague shared the following image showing that AlphaStar has leveled the playing field a bit by restricting its APM to be at human-level, in fact, performing far few actions per minute than the human player on average.
Now, another colleague who is more mathematically-minded shouted out that "The distribution of the APMs follows a nice Poisson distribution!". This got me thinking, I feel although the random variable defined by APM is discrete, it does not satisfy the two hypotheses of a Poisson-distributed random variable, namely:
- The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
- The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals.
Instead, I argued that APM is log-normal distributed. Because the actor (human or AI) has a mean APM, and depending on the situation, may be acting at some multiplier of this mean APM. For example, if there is not much to do, the actor may be doing $0.1 times textrm{mean APM}$; but if there is a fight, then the actor may be doing $10 times textrm{mean APM}$. What's more, this multiplier is a random variable. And each minute can be viewed as an experiment, where we draw a value for this multiplier random variable. Thus by the Central Limit Theorem, the drawn samples of the multipliers approximate a normal distribution, hence the samples of APMs approximate a log-normal distribution, and so APM is log-normal distributed.
But I can't get over the fact that APM is discrete-valued.
So am I right or wrong? What is the most appropriate distribution of APM (viewed as a random variable)?
probability probability-distributions
asked Jan 26 at 1:35
motivicmotivic
1012
$endgroup$
add a comment |
$begingroup$
Yesterday DeepMind's AlphaStar beat a high-ranked professional player at StarCraft II.
My co-workers and I were chatting about whether the AlphaStar has an unfair advantage over the human player because as a machine, it can perform many more actions per minute than any human can possible do. Then a colleague shared the following image showing that AlphaStar has leveled the playing field a bit by restricting its APM to be at human-level, in fact, performing far few actions per minute than the human player on average.
Now, another colleague who is more mathematically-minded shouted out that "The distribution of the APMs follows a nice Poisson distribution!". This got me thinking, I feel although the random variable defined by APM is discrete, it does not satisfy the two hypotheses of a Poisson-distributed random variable, namely:
- The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
- The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals.
Instead, I argued that APM is log-normal distributed. Because the actor (human or AI) has a mean APM, and depending on the situation, may be acting at some multiplier of this mean APM. For example, if there is not much to do, the actor may be doing $0.1 times textrm{mean APM}$; but if there is a fight, then the actor may be doing $10 times textrm{mean APM}$. What's more, this multiplier is a random variable. And each minute can be viewed as an experiment, where we draw a value for this multiplier random variable. Thus by the Central Limit Theorem, the drawn samples of the multipliers approximate a normal distribution, hence the samples of APMs approximate a log-normal distribution, and so APM is log-normal distributed.
But I can't get over the fact that APM is discrete-valued.
So am I right or wrong? What is the most appropriate distribution of APM (viewed as a random variable)?
probability probability-distributions
asked Jan 26 at 1:35
motivicmotivic
1012
$endgroup$
1
$begingroup$
I won't write a full answer, but note that "Because the actor (human or AI) has a mean APM, and depending on the situation, may be acting at some multiplier of this mean APM" applies to any situation where there is a non-zero mean - it can't be used as an indicator of something being log-normal.
$endgroup$
– Milo Brandt
Jan 26 at 1:43
$begingroup$
Having trouble understanding exactly what this is the distribution of. Is it like, in the first minute of the game (0:00-0:01), the player did 125 actions, in the second minute (0:01-0:02) they did 313, in the third minute they did 75... and you histogram 125, 313, 75, etc?
$endgroup$
– spaceisdarkgreen
Jan 26 at 1:46
add a comment |
$begingroup$
Yesterday DeepMind's AlphaStar beat a high-ranked professional player at StarCraft II.
My co-workers and I were chatting about whether the AlphaStar has an unfair advantage over the human player because as a machine, it can perform many more actions per minute than any human can possible do. Then a colleague shared the following image showing that AlphaStar has leveled the playing field a bit by restricting its APM to be at human-level, in fact, performing far few actions per minute than the human player on average.
Now, another colleague who is more mathematically-minded shouted out that "The distribution of the APMs follows a nice Poisson distribution!". This got me thinking, I feel although the random variable defined by APM is discrete, it does not satisfy the two hypotheses of a Poisson-distributed random variable, namely:
- The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
- The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals.
Instead, I argued that APM is log-normal distributed. Because the actor (human or AI) has a mean APM, and depending on the situation, may be acting at some multiplier of this mean APM. For example, if there is not much to do, the actor may be doing $0.1 times textrm{mean APM}$; but if there is a fight, then the actor may be doing $10 times textrm{mean APM}$. What's more, this multiplier is a random variable. And each minute can be viewed as an experiment, where we draw a value for this multiplier random variable. Thus by the Central Limit Theorem, the drawn samples of the multipliers approximate a normal distribution, hence the samples of APMs approximate a log-normal distribution, and so APM is log-normal distributed.
But I can't get over the fact that APM is discrete-valued.
So am I right or wrong? What is the most appropriate distribution of APM (viewed as a random variable)?
probability probability-distributions
asked Jan 26 at 1:35
motivicmotivic
1012
$endgroup$
Yesterday DeepMind's AlphaStar beat a high-ranked professional player at StarCraft II.
My co-workers and I were chatting about whether the AlphaStar has an unfair advantage over the human player because as a machine, it can perform many more actions per minute than any human can possible do. Then a colleague shared the following image showing that AlphaStar has leveled the playing field a bit by restricting its APM to be at human-level, in fact, performing far few actions per minute than the human player on average.
Now, another colleague who is more mathematically-minded shouted out that "The distribution of the APMs follows a nice Poisson distribution!". This got me thinking, I feel although the random variable defined by APM is discrete, it does not satisfy the two hypotheses of a Poisson-distributed random variable, namely:
- The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
- The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals.
Instead, I argued that APM is log-normal distributed. Because the actor (human or AI) has a mean APM, and depending on the situation, may be acting at some multiplier of this mean APM. For example, if there is not much to do, the actor may be doing $0.1 times textrm{mean APM}$; but if there is a fight, then the actor may be doing $10 times textrm{mean APM}$. What's more, this multiplier is a random variable. And each minute can be viewed as an experiment, where we draw a value for this multiplier random variable. Thus by the Central Limit Theorem, the drawn samples of the multipliers approximate a normal distribution, hence the samples of APMs approximate a log-normal distribution, and so APM is log-normal distributed.
But I can't get over the fact that APM is discrete-valued.
So am I right or wrong? What is the most appropriate distribution of APM (viewed as a random variable)?
probability probability-distributions
probability probability-distributions
asked Jan 26 at 1:35
motivicmotivic
1012
asked Jan 26 at 1:35
motivicmotivic
1012
asked Jan 26 at 1:35
motivicmotivic
1012
asked Jan 26 at 1:35
motivicmotivic
1012
asked Jan 26 at 1:35
motivicmotivic
1012
1012
1
$begingroup$
I won't write a full answer, but note that "Because the actor (human or AI) has a mean APM, and depending on the situation, may be acting at some multiplier of this mean APM" applies to any situation where there is a non-zero mean - it can't be used as an indicator of something being log-normal.
$endgroup$
– Milo Brandt
Jan 26 at 1:43
$begingroup$
Having trouble understanding exactly what this is the distribution of. Is it like, in the first minute of the game (0:00-0:01), the player did 125 actions, in the second minute (0:01-0:02) they did 313, in the third minute they did 75... and you histogram 125, 313, 75, etc?
$endgroup$
– spaceisdarkgreen
Jan 26 at 1:46
add a comment |
1
$begingroup$
I won't write a full answer, but note that "Because the actor (human or AI) has a mean APM, and depending on the situation, may be acting at some multiplier of this mean APM" applies to any situation where there is a non-zero mean - it can't be used as an indicator of something being log-normal.
$endgroup$
– Milo Brandt
Jan 26 at 1:43
$begingroup$
Having trouble understanding exactly what this is the distribution of. Is it like, in the first minute of the game (0:00-0:01), the player did 125 actions, in the second minute (0:01-0:02) they did 313, in the third minute they did 75... and you histogram 125, 313, 75, etc?
$endgroup$
– spaceisdarkgreen
Jan 26 at 1:46
1
1
$begingroup$
I won't write a full answer, but note that "Because the actor (human or AI) has a mean APM, and depending on the situation, may be acting at some multiplier of this mean APM" applies to any situation where there is a non-zero mean - it can't be used as an indicator of something being log-normal.
$endgroup$
– Milo Brandt
Jan 26 at 1:43
$begingroup$
I won't write a full answer, but note that "Because the actor (human or AI) has a mean APM, and depending on the situation, may be acting at some multiplier of this mean APM" applies to any situation where there is a non-zero mean - it can't be used as an indicator of something being log-normal.
$endgroup$
– Milo Brandt
Jan 26 at 1:43
$begingroup$
Having trouble understanding exactly what this is the distribution of. Is it like, in the first minute of the game (0:00-0:01), the player did 125 actions, in the second minute (0:01-0:02) they did 313, in the third minute they did 75... and you histogram 125, 313, 75, etc?
$endgroup$
– spaceisdarkgreen
Jan 26 at 1:46
$begingroup$
Having trouble understanding exactly what this is the distribution of. Is it like, in the first minute of the game (0:00-0:01), the player did 125 actions, in the second minute (0:01-0:02) they did 313, in the third minute they did 75... and you histogram 125, 313, 75, etc?
$endgroup$
– spaceisdarkgreen
Jan 26 at 1:46
add a comment |
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$begingroup$
I won't write a full answer, but note that "Because the actor (human or AI) has a mean APM, and depending on the situation, may be acting at some multiplier of this mean APM" applies to any situation where there is a non-zero mean - it can't be used as an indicator of something being log-normal.
$endgroup$
– Milo Brandt
Jan 26 at 1:43
$begingroup$
Having trouble understanding exactly what this is the distribution of. Is it like, in the first minute of the game (0:00-0:01), the player did 125 actions, in the second minute (0:01-0:02) they did 313, in the third minute they did 75... and you histogram 125, 313, 75, etc?
$endgroup$
– spaceisdarkgreen
Jan 26 at 1:46