An optimal strategy to blend two probability estimates

[broccoli7]

Two mariners report to the skipper of a ship that they are distances d1 and d2 from the shore. The skipper knows from historical data that the mariners A & B make errors that are normally distributed and have a standard deviation of s1 and s2. What should the skipper do to arrive at the best estimate of how far the ship is from the shore?

Spoiler http://bayesianthink.blogspot.com/2013/02/the-case-of-two-mariners.html

 

[Stephen Tashi]

What should the skipper do to arrive at the best estimate of how far the ship is from the shore?

He should first define what he means by "best".

 

[jbriggs444]

Two mariners report to the skipper of a ship that they are distances d1 and d2 from the shore. The skipper knows from historical data that the mariners A & B make errors that are normally distributed and have a standard deviation of s1 and s2. What should the skipper do to arrive at the best estimate of how far the ship is from the shore?

Spoiler http://bayesianthink.blogspot.com/2013/02/the-case-of-two-mariners.html

The answer given in the spoiler is sub-optimal.

The skipper can arrive at a [STRIKE]better[/STRIKE] closer estimate by ignoring both mariners whenever both estimates are negative.

 

[Stephen Tashi]

The answer given in the spoiler is sub-optimal.

The skipper can arrive at a [STRIKE]better[/STRIKE] closer estimate by ignoring both mariners whenever both estimates are negative.

If making a negative error implies causing a collision that is certainly the case. But it's not possible to say what is optimal until the captain defines what quantity he is trying to optimize. For example, does he want an estimator with the minimum expected square error or minimum expected absolute error? Or does he want a maximum liklihood estimator etc.

An error of -0.3 is a smaller when squared than an error of +0.5.

 

[jbriggs444]

I may be babbling a bit here, but...

From a Bayesian perspective the skipper has some unspecified distribution in mind for the ship's possible distance from shore. The figures reported by the two mariners are (hopefully independent!) pieces of evidence that may lead him to revise that initial estimate. If the skipper's initial estimate has zero probability for being anywhere on the landward side of the shore line then no evidence will change that estimation.

The problem appears to ask for a single parameter that is related to this distribution and is optimal without having specified the Bayesian prior and without, as you have pointed out, having specified the criteria for optimality.