An optimal strategy to blend two probability estimates

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Discussion Overview

The discussion revolves around how a ship's skipper can best estimate the distance from the shore based on reports from two mariners, each with their own error distributions. The focus includes theoretical considerations of estimation strategies, particularly from a Bayesian perspective, and the implications of different definitions of "best" in this context.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants suggest that the skipper should first define what "best" means in terms of estimation criteria.
  • One participant argues that ignoring both mariners' estimates when both are negative could lead to a better estimate, particularly in the context of avoiding collisions.
  • Another participant emphasizes the importance of specifying the Bayesian prior and the criteria for optimality before determining a single optimal parameter.
  • There is a discussion about different types of estimators, such as those minimizing expected square error versus expected absolute error.
  • Concerns are raised about the independence of the mariners' reports and how that affects the skipper's initial estimate.

Areas of Agreement / Disagreement

Participants generally agree that defining the criteria for "best" is crucial, but there is no consensus on what the optimal strategy should be or how to approach the problem without further specifications.

Contextual Notes

Limitations include the unspecified Bayesian prior and the lack of clarity on the criteria for optimality, which affect the discussion of estimation strategies.

broccoli7
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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
 
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broccoli7 said:
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".
 
broccoli7 said:
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.
 
jbriggs444 said:
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.
 
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.
 

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