On way to prove your judgement sound is to have a computer program roll it out a few thousand times. The drawback is that it won't give you any better understanding of the position.
A better way of learning is to challenge your opponent to play it as a proposition. In a proposition you agree to play the game to a finish an agreed number of games, starting from the disputed position.
The format goes like this: If Player A thinks a position is a take and Player B believes it is a drop, they will agree to play out the position say 50 times. Player A plays the side receiving the cube on 2 and also receives a payment of one point per game from B. The reason for this payment - or spot as it is known - is interesting: if the position is really a drop then A should lose, on average, more than one point per game by taking. So B, if he is right, will show a profit. If the position is really a take then A will lose on average less than one per point per game. With the spot thrown in, A will show a profit.
A second type is when two players cannot agree on the correct play in a position. The position is then played out say 30 times with one player making his move, and 30 times with the other player making his. The difference between the two sets of results is the end payment.
There is a third type of proposition to cover the case where one player believes a position is a double and another believes that you should not double.
The players involved in all these propositions learn a great deal about the positions and the strategies involved. For example, in the original game it is possible you were gammoned because you gave up your anchor in your opponent's board too soon. After a few times of playing it as a proposition you learn that your original strategy was not correct and you adjust it accordingly. Once you have played out a position 50 or 60 times you learn a great deal about it.
Remember that a large element of good backgammon is to store and use reference positions when assessing any particular position. Playing propositions strengthens a player's ability. They are also immensely satisfying when you are proved right.