We consider the fundamental problem of communicating an estimate of a real number $x\in[0,1]$ using a single bit. A sender that knows $x$ chooses a value $X\in\set{0,1}$ to transmit. In turn, a receiver estimates $x$ based on the value of $X$. We consider both the biased and unbiased estimation problems and aim to minimize the cost. For the biased case, the cost is the worst-case (over the choice of $x$) expected squared error, which coincides with the variance if the algorithm is required to be unbiased. We first overview common biased and unbiased estimation approaches and prove their optimality when no shared randomness is allowed. We then show how a small amount of shared randomness, which can be as low as a single bit, reduces the cost in both cases. Specifically, we derive lower bounds on the cost attainable by any algorithm with unrestricted use of shared randomness and propose near-optimal solutions that use a small number of shared random bits. Finally, we discuss open problems and future directions.
翻译:我们考虑的是使用一个位数来通报实际数字的估计数[0,1]美元[0,1]美元这一根本问题。知道美元的一个发送者选择一个值(X\in\inset{0,1}美元)来传输。反过来,一个接收者根据美元值来估算美元x美元。我们既考虑偏向和不公正的估计问题,又力求将成本降到最低。对于有偏向的情况,成本是预期的方差的最坏情况(而不是选择美元),如果算法需要公正的话,成本与差异相吻合。我们首先审视共同的偏向和不偏向的估计方法,并在不允许共享随机性时证明它们的最佳性。然后我们展示如何通过少量的共享随机性来降低两种情况的成本。具体地说,我们通过不加限制地使用共享随机性的方法,从任何算法所能实现的成本中得出较低的界限,并提出使用少量共享随机数的近最佳解决办法。最后,我们讨论开放的问题和今后的方向。