Predictions using unknown probability distributions

[HunterD2]

My question requires a little bit of background so I will start with that. In a game that focuses on the Pacific Theater of War during WWII, you have two sides: the Allies and the Japanese. It is a turn-based game where each player gives their orders for the day and engagements (when opposing units meet) play out when both players end their turns. During this phase where the orders are actually played out, you get these "in the moment" reports that are heavily influenced by fog of war (i.e. the information is not 100% accurate). Plane engagements will be the focus of this question. During air combat, each player has a different view of what is going on which is determined by said fog of war. The problem becomes more complicated as each player's fog of war is not equal as the allied player has less severe fog of war due to better signals intelligence. Essentially, when air combat is completed, the game calculates the actual number of planes shot down and then presumably runs that number through a different random number generator for each player. For example, you may have a case where 5 allied planes and 3 Japanese planes were actually shot down but the "in the moment" report for the Allies say 6 allied and 6 Japanese planes were shot down while the Japanese "in the moment" reports 2 Japanese and 9 Allied losses. It becomes even more complicated as each side predicts its own losses more accurately than the enemies losses. Now, my question is, if you had a large number of "in the moment" reports from both sides but did not know the probability distribution of any of the probabilities involved can you conceivably predict the number of actual enemy aircraft shot down? My answer was no as you wouldn't know your margin of error for predicting how many planes you shot down nor would you know the margin of error your opponent has at reporting their own losses. You also would not have access to the actual number of planes shot down in these reports to deduce the margin of errors. The only certain number you would know is the total number of aircraft that engaged. I realize that this is a somewhat complicated description so if you need me to clear up any part of my description just let me know.

 

[Stephen Tashi]

Data by itself, accurate or otherwise, does not mathematically imply any particular probability distributions or allow probability distributions to be inferred. The scenario for inferring probability distributions is to assume that some mathematical model or family of models is producing the data and then use the data to infer the details of the probability distrbutions based on such assumptions.

Presumably, you have a mathematical model (in the form of certain algorithms) for generating both the actual and inaccurate data. If a player is allowed to know the algorithms you use, he might be able to make inferences about the actual situation given only inaccurate data. Whether inferences are possible depends on the particular details of what you are doing.

 

[FactChecker]

Nonparametric statistics may apply. It's not clear to me that you ever see any true numbers, so you may be limited to probabilities for the reports, not the actuals.