Thanks for clear statement of the problem. The point is that if you want to decrease false negatives, you should sufficiently lower the threshold of your decision function. If the false negatives are decreased, as you mentioned, true positives increase but false positives also increase. As a result, recall will increase and precision will decrease.

Thanks for clear statement of the problem. The point is that if you want to decrease false negatives, you should sufficiently lower the threshold of your decision function. If the false negatives are decreased, as you mentioned, true positives increase but false positives can also increase. As a result, recall will increase and precision will decrease.