If you have two options, A and B, 50% odds is maximal ignorance; you aren’t saying they have equivalent odds of being true, you’re saying you have no information by which to make an inference which is true.
If you then say we can split A into A1 and A2, you have added information to the problem. Like the Monty Hall problem, information can change the odds in unexpected ways!
There’s no contradiction here—you have more information than when you originally assigned odds of 50⁄50. And the information you have added should, in real situations, inform how to distribute the odds. If A1 and A2 are sufficiently distinct (independent), it is possible that a 33/33/33 split is appropriate; if they aren’t, it is possible that a 25/25/50 split is appropriate. In order to make a judgment, we’d have to know more about what exactly A1 and A2 are, and why they can be considered a “split” of A.
Consider, for example, the case of a coin being flipped—we don’t know if the coin is fair or not. Let us say A is that the coin comes up “heads” and B is that the coin comes up “tails”. The split, then, could reflect a second flip, after the first flip is decided, if and only if it is heads; A1 might be “heads-heads”, A2 might be “heads-tails”. Then a 25/25/50 split makes sense; A1 and A2 are not independent.
If, on the other hand, we have discovered that it isn’t a coin at all, but a three-sided die, two faces of which have the same symbol, and one face of which has another symbol; we label the faces with the similar symbol A1 and A2, and the face with the sameedit: other symbol B. We still don’t know whether or not the die is fair—maybe it is weighed—but the position of maximal ignorance is 33/33/33, because even if it -is- weighted, we don’t know which face it is weighted in favor of; A1 and A2 are independent.
So—what are A1 and A2, and how independent are they? We have equations that can work this out with sample data, and your prior probability should reflect your expectation of their independence. If you insist on maximal ignorance about independence—then you can assume independence. Most things are independent; it is only the way the problem is constructed that leads us to confusion here, because it seems to suggest that they are not independent (consider that we can simply rename the set of conclusions to “A, B, and C”—all the names you have utilized are merely labels, after all, and in effect, what you have actually done is to introduce C, with an implication that A and C should maybe be considered partially dependent variables). If you insist on maximal ignorance about that, as well, then you can, I suppose, assume 50% independence, which would be something like splitting the difference between the die and the coin. And there’s an argument to be made there, in that you have, in fact, implied that they should maybe be considered partially dependent variables—but this comes down to trying to interpret what you have said, rather than trying to understand the nature of probability itself.
“If you then say we can split A into A1 and A2, you have added information to the problem. Like the Monty Hall problem, information can change the odds in unexpected ways!”—It’s not clear which is the baseline.
The point there is that there is no contradiction because the informational content is different. “Which is the baseline” is up to the person writing the problem to answer. You’ve asserted that the baseline is A vs B; then you’ve added information that A is actually A1 and A2.
The issue here is entirely semantic ambiguity.
Observe what happens when we remove the semantic ambiguity:
You’ve been observing a looping computer program for a while, and have determined that it shows three videos. The first video portrays a coin showing tails. The second video portrays two coins; the left coin shows heads, the right coin shows tails. The third video also portrays two coins; the left coin shows heads, the right coin shows heads.
You haven’t been paying attention to the frequency, but now, having determined there are three videos you can see, you want to figure out how frequently each video shows up. What are your prior odds for each video?
33/33/33 seems reasonable. I’ve specified that you’re watching videos; the event is which video you are watching, not the events that unfold within the video.
Now, consider an alternative framing: You are watching somebody as they repeat a series of events. You have determined the events unfold in three distinct ways; all three begin the same way, with a coin being flipped. If the coin shows heads, it is flipped again. If the coin shows tails, it is not. What are your prior odds for each sequence of events?
25/25/50 seems reasonable.
Now, consider yet another framing: You are shown something on a looping computer screen. You have determined the visuals unfold in three distinct ways; all three begin the same way, with a coin being flipped. If the coin shows heads, it is flipped again. If the coin shows tails, it is not. What are your prior odds here?
Both 25/25/50 and 33/33/33 are reasonable. Why? Because it is unclear whether or not you are watching a simulation of coin flips, or something like prerecorded videos; it is unclear whether or not you should treat the events within what you are watching as events, or whether you should treat the visuals themselves you are watching as the event.
Because it is unclear, I’d lean towards treating the visuals you are watching as the event—that is, assume independence. However, it would be perfectly fair to treat the coin tosses as events also. Or you could split the difference. Prior probabilities are just your best guess given the information you have available—and given that I don’t have access to all the information you have available, both options are fair.
Now, the semantic ambiguity you have introduced, in the context of this, is like this:
You’re told you are going to watch a computer program run, and what you see will begin with a coin being flipped, showing heads or tails. What are your probabilities that it will show heads or tails?
Okay, 50⁄50. Now, if you see the coin shows heads, you will see that it is flipped again; we now have three possibilities, HT, HH, and TT. What are your probabilities for each event?
Notice: You didn’t specify enough to know what the relevant events we’re assigning probabilities to even are! We’re in the third scenario; we don’t know if it’s a video, in which case the relevant event is “Which video we are watching”, or if it is a simulation, in which case the relevant event is “The outcome of each coin toss.” Either answer works, or you can split the difference, because at this point a large part of the probability-space is devoted, not to the events unfolding, but towards the ambiguity in what events we’re even evaluating.
If you have two options, A and B, 50% odds is maximal ignorance; you aren’t saying they have equivalent odds of being true, you’re saying you have no information by which to make an inference which is true.
If you then say we can split A into A1 and A2, you have added information to the problem. Like the Monty Hall problem, information can change the odds in unexpected ways!
There’s no contradiction here—you have more information than when you originally assigned odds of 50⁄50. And the information you have added should, in real situations, inform how to distribute the odds. If A1 and A2 are sufficiently distinct (independent), it is possible that a 33/33/33 split is appropriate; if they aren’t, it is possible that a 25/25/50 split is appropriate. In order to make a judgment, we’d have to know more about what exactly A1 and A2 are, and why they can be considered a “split” of A.
Consider, for example, the case of a coin being flipped—we don’t know if the coin is fair or not. Let us say A is that the coin comes up “heads” and B is that the coin comes up “tails”. The split, then, could reflect a second flip, after the first flip is decided, if and only if it is heads; A1 might be “heads-heads”, A2 might be “heads-tails”. Then a 25/25/50 split makes sense; A1 and A2 are not independent.
If, on the other hand, we have discovered that it isn’t a coin at all, but a three-sided die, two faces of which have the same symbol, and one face of which has another symbol; we label the faces with the similar symbol A1 and A2, and the face with the
sameedit: other symbol B. We still don’t know whether or not the die is fair—maybe it is weighed—but the position of maximal ignorance is 33/33/33, because even if it -is- weighted, we don’t know which face it is weighted in favor of; A1 and A2 are independent.So—what are A1 and A2, and how independent are they? We have equations that can work this out with sample data, and your prior probability should reflect your expectation of their independence. If you insist on maximal ignorance about independence—then you can assume independence. Most things are independent; it is only the way the problem is constructed that leads us to confusion here, because it seems to suggest that they are not independent (consider that we can simply rename the set of conclusions to “A, B, and C”—all the names you have utilized are merely labels, after all, and in effect, what you have actually done is to introduce C, with an implication that A and C should maybe be considered partially dependent variables). If you insist on maximal ignorance about that, as well, then you can, I suppose, assume 50% independence, which would be something like splitting the difference between the die and the coin. And there’s an argument to be made there, in that you have, in fact, implied that they should maybe be considered partially dependent variables—but this comes down to trying to interpret what you have said, rather than trying to understand the nature of probability itself.
“If you then say we can split A into A1 and A2, you have added information to the problem. Like the Monty Hall problem, information can change the odds in unexpected ways!”—It’s not clear which is the baseline.
The point there is that there is no contradiction because the informational content is different. “Which is the baseline” is up to the person writing the problem to answer. You’ve asserted that the baseline is A vs B; then you’ve added information that A is actually A1 and A2.
The issue here is entirely semantic ambiguity.
Observe what happens when we remove the semantic ambiguity:
You’ve been observing a looping computer program for a while, and have determined that it shows three videos. The first video portrays a coin showing tails. The second video portrays two coins; the left coin shows heads, the right coin shows tails. The third video also portrays two coins; the left coin shows heads, the right coin shows heads.
You haven’t been paying attention to the frequency, but now, having determined there are three videos you can see, you want to figure out how frequently each video shows up. What are your prior odds for each video?
33/33/33 seems reasonable. I’ve specified that you’re watching videos; the event is which video you are watching, not the events that unfold within the video.
Now, consider an alternative framing: You are watching somebody as they repeat a series of events. You have determined the events unfold in three distinct ways; all three begin the same way, with a coin being flipped. If the coin shows heads, it is flipped again. If the coin shows tails, it is not. What are your prior odds for each sequence of events?
25/25/50 seems reasonable.
Now, consider yet another framing: You are shown something on a looping computer screen. You have determined the visuals unfold in three distinct ways; all three begin the same way, with a coin being flipped. If the coin shows heads, it is flipped again. If the coin shows tails, it is not. What are your prior odds here?
Both 25/25/50 and 33/33/33 are reasonable. Why? Because it is unclear whether or not you are watching a simulation of coin flips, or something like prerecorded videos; it is unclear whether or not you should treat the events within what you are watching as events, or whether you should treat the visuals themselves you are watching as the event.
Because it is unclear, I’d lean towards treating the visuals you are watching as the event—that is, assume independence. However, it would be perfectly fair to treat the coin tosses as events also. Or you could split the difference. Prior probabilities are just your best guess given the information you have available—and given that I don’t have access to all the information you have available, both options are fair.
Now, the semantic ambiguity you have introduced, in the context of this, is like this:
You’re told you are going to watch a computer program run, and what you see will begin with a coin being flipped, showing heads or tails. What are your probabilities that it will show heads or tails?
Okay, 50⁄50. Now, if you see the coin shows heads, you will see that it is flipped again; we now have three possibilities, HT, HH, and TT. What are your probabilities for each event?
Notice: You didn’t specify enough to know what the relevant events we’re assigning probabilities to even are! We’re in the third scenario; we don’t know if it’s a video, in which case the relevant event is “Which video we are watching”, or if it is a simulation, in which case the relevant event is “The outcome of each coin toss.” Either answer works, or you can split the difference, because at this point a large part of the probability-space is devoted, not to the events unfolding, but towards the ambiguity in what events we’re even evaluating.