# An optimal strategy to blend two probability estimates

• broccoli7
In summary, the skipper must first define what he means by "best" and then take into account the historical data and the reports from the two mariners to arrive at a better estimate of the ship's distance from the shore. This will depend on the skipper's initial estimate and the criteria for optimality that he chooses.
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

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.

## 1. What is an optimal strategy for blending two probability estimates?

An optimal strategy for blending two probability estimates involves combining two or more independent estimates of the same probability into a single, more accurate estimate. This is typically done by taking a weighted average of the estimates, where the weights reflect the relative trustworthiness or accuracy of each estimate.

## 2. Why is blending probability estimates important?

Blending probability estimates is important because it can help improve the accuracy and reliability of predictions and decision-making. By combining multiple estimates, we can reduce the impact of individual errors or biases and arrive at a more robust and accurate estimate.

## 3. How do you determine the weights for blending probability estimates?

The weights for blending probability estimates can be determined in various ways, depending on the specific context and data available. In some cases, the weights may be based on the reliability or accuracy of each estimate, or they may be assigned based on the expertise or track record of the sources providing the estimates. Machine learning techniques can also be used to automatically determine the optimal weights.

## 4. Can blending probability estimates be used for all types of data?

Blending probability estimates can be used for any type of data where multiple independent estimates of the same probability are available. This can include numerical data such as stock prices or weather forecasts, as well as qualitative data such as expert opinions or survey responses.

## 5. Are there any limitations to blending probability estimates?

While blending probability estimates can be a powerful tool for improving accuracy, it is not a perfect solution and may not always be appropriate. It requires a certain level of data and expertise to determine accurate weights, and the quality of the resulting blended estimate is highly dependent on the quality of the individual estimates being combined. Additionally, blending probability estimates may not be suitable for critical or high-stakes decisions where a single, precise estimate is needed.

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