Comparing two hypotheses with uncertainties

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Discussion Overview

The discussion revolves around comparing two hypotheses, each represented by a Gaussian distribution with associated uncertainties. Participants explore methods to assess the probability of one hypothesis being correct given the other, particularly in the context of measuring gravitational acceleration and its significance when uncertainties are considered.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Johannes initiates the discussion by seeking a method to compare two Gaussian hypotheses while accounting for their uncertainties, specifically mentioning integration for probability calculation.
  • Johannes provides a specific example of measuring gravitational acceleration, noting a significant discrepancy between his measurement and the known value.
  • One participant suggests checking instruments as a response to the measurement discrepancy, implying potential measurement error.
  • Another participant emphasizes that the large difference between the measured and known values suggests either a fault in the measurement or an incorrect known value, highlighting the improbability of such a deviation occurring by chance.
  • A later reply questions the approach of considering uncertainty from only one value, advocating for a method that incorporates uncertainties from both measurements in the comparison.
  • One participant mentions using the square root of the sum of the squares to combine uncertainties, assuming they are independent, but notes that this does not qualitatively change the outcome.

Areas of Agreement / Disagreement

Participants express differing views on how to approach the comparison of uncertainties, with some focusing on measurement reliability and others on mathematical methods for incorporating uncertainties. The discussion remains unresolved regarding the best approach to include both uncertainties in the comparison.

Contextual Notes

Participants have not reached a consensus on the appropriate method for comparing the two hypotheses while accounting for uncertainties, and there are unresolved assumptions regarding the independence of uncertainties.

jlicht
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Hi there,

I'm trying to compare two hypotheses, each with a mean and an error (in this case Gaussian as well), to tell the probability of one being correct given the other.
Currently, the only thing I know how to do is calculate the Gaussian probability of the second hypothesis being correct according to the error of the first by integration. However, this doesn't take the error of the second distribution into account.

What would be the correct way of determining whether one (let's say Gaussian) hypothesis is correct given another, taking into account both uncertainties?

Cheers,
Johannes

EDIT:
As an application example, let us say I measured the gravitational acceleration to be 15.0 ± 0.3, while the known value was 9.8 ± 0.1. What would the significance of my newly measured result be, given the uncertainty on both numbers?
 
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As an application example, let us say I measured the gravitational acceleration to be 15.0 ± 0.3, while the known value was 9.8 ± 0.1. What would the significance of my newly measured result be, given the uncertainty on both numbers?

If I was doing this measurement, I would check my instruments.
 
mathman said:
If I was doing this measurement, I would check my instruments.

That is obviously not the point; replace g by something weird and unknown called x in the above, then.
 
If you have a quantity with known value ~ 9.8 and you measured it as ~ 15.0, it seems that either the known value was incorrect or the measurement was faulty. They are too far apart (> ~ 15σ) to be purely by chance.
 
mathman said:
If you have a quantity with known value ~ 9.8 and you measured it as ~ 15.0, it seems that either the known value was incorrect or the measurement was faulty. They are too far apart (> ~ 15σ) to be purely by chance.

Now here's my point: you're computing this based on the uncertainty on one of the values. But this completely ignores the uncertainty on the other value. Isn't there a way to include both in a comparison?
 
It is the square root of the sum of the squares, assuming the uncertainties are independent. The net = .316, so qualitatively there is no change.
 

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