what's the best way of Inferring chance of heads with a coin of Unknown Bias

Question

hi what would be the best strategy to infer the range of probabilities of getting heads with a coin of Unknown Bias that is variable?

Thanks

EDIT: Problem specification I'm working on a similar problem with a game AI.

I'm working on a AI to play a game that consists of multiple nodes in a graph network. Each node in the network switches between two values, either A or B, based on a probability variable unknown to the player in every round. This probability variable also changes with each subsequent round based on what move the players makes.

Its a two player game, and if at the end of the game there are more node with the value A, player A win and if more nodes with value B then player B wins. Each player has 5 moves that affect the probability by multiples of 5:

move 1 affect the probability variable of each node by 5%, move 2 by 10%, and so on. Each move has a set energy cost, the greater the percentage change of the move, the greater the energy cost. The chosen move affect all the nodes in the graph network. The value switch happens once both players have made their moves

So the optimal strategy for each player would be to increase the probability variable of a node if its has the opponents value and decrease it if it has the players value. Every node in the graph is assigned a random probability at the start of the game, and the probability variable of each node is unknown to both players.

so I assume I would need to infer the probability of each node in a specific round for my AIagent to decide on a move, so what would be the best strategy accomplish this? I assume it would be similar to inferring the probability of a biased coin, but I'm not sure how that would work considering that the probability variable of the nodes changes in each round

edit 2: Players are not made aware of the probability of a node, only their values i.e either A or B. The players also can not target specific nodes, their chosen move affect every node in the network by increasing the probability of switch for nodes with the opponent's value and decreasing probability of switch for nodes with the players value. However the players are aware of the entire switching history of each Node

Answer 1

This is an interesting question since the problem setup has not been completely specified. The usual answer would be to flip the coin $N$ times and then compute a sample mean (the fraction of times the coin landed on heads). This is a uniformly minimum variance and unbiased estimator. If you care about a different metric, not variance, then the sample mean might not be the best estimator.

It is also possible that the number of samples $N$ is up to the us, and could depend on some random process of our choosing. In that case, we should define a cost associated with flipping a coin to choose the best stopping time. The number of samples we draw would likely be related to the variance of the coin flips; if the coin always lands on heads, we need fewer samples than if there is a equal chance of landing on heads or tails. For example, we might want to draw samples until we form a confidence interval around the estimated mean that is small enough for our use case..