Celebrities tend to age differently from normal humans (e.g. Keanu Reeves) and whenever I’m talking with my girlfriend we sometimes get into a discussion on how old is a celebrity. So logically we often play the game: “Guess that celebrity age”. She is far better guesser than me. Putting concrete numbers, we can say that we both make estimations errors normally distributed (hehe. NO! but its a game), and my error standard deviation is 7.3 years, and her’s is 3.3 years. The main question becomes “can we combine these guesses to arrive at a better guess together?”. The answer is yes, and we just need some algebra and calculus (as with everything in life).
Formal representation
To formalize this problem say that two people come up with two guesses, \(A\) and \(B\) and \(\sigma_1^2\) and \(\sigma_2^2\) variances respectively. To generate a better guess \(Y\) we can combine it using a \(p\) probability of blending first guess, and \((1-p)\) of blending the second as such:
\[Y = p * A + (1-p) * B\]Say that our guesses are independent, we can express the variance as such:
\[Var(Y) = Var(A) + Var(B)\]expanding it to:
\[Var(Y) = p^2 * \sigma_1^2 + (1-p)^2 * \sigma_2^2\]Notice: linearity of variance tells us to give square factor to the “multiplier” whenever we are summing up variances.
If we calculate the derivative of the variance with respect to \(p\) we get:
\[Var(Y)^` = 2p * \sigma_1^2 - 2(1-p) * \sigma_2^2 = 0\]If we calculate the second derivative of variance with respect to \(p\) we see its \(2 (\sigma_1^2 + \sigma_2^2)\) which is always positive, meaning we can find the minimum \(p\) by solving the equation above.
and rearranging for \(p\):
\[2p * \sigma_1^2 - (2 - 2p) * \sigma_2^2 = 0\] \[2p\sigma_1^2 - 2\sigma_2^2 + 2p\sigma_2^2 = 0\] \[2p\sigma_1^2 + 2p\sigma_2^2 = 2\sigma_2^2\] \[p(\sigma_1^2 + \sigma_2^2) = \sigma_2^2\] \[p = \frac{\sigma_2^2}{\sigma_1^2 + \sigma_2^2}\]And our original formula becomes:
\[Y = (\frac{\sigma_2^2}{\sigma_1^2 + \sigma_2^2}) * A + (1 - \frac{\sigma_2^2}{\sigma_1^2 + \sigma_2^2}) * B\]Using this formula we can create a better guess given two guesses and their variances.
Example
Let’s take the example of my girlfriend and me again, and we are trying to guess the age of Keanu Reeves.
\(A = 50\) with \(\sigma_1^2 = 7.3\) years
\(B = 60\) with \(\sigma_1^2 = 3.3\) years
\[Y = (\frac{3.3^2}{3.3^2 + 7.3^2}) * 50 + (1 - \frac{3.3^2}{3.3^2 + 7.3^2}) * 60\] \[Y = (0.169) * 50 + (1 - 0.169) * 60\] \[Y = 58.31\]According to wikipedia, Keanu Reeves is 58 years old exactly.