Frequentist vs Bayesian reasoning, the essence

2016 May 5
by Daniel Lakeland

A jeweler reaches into his desk drawer and pulls out 3 photographs of diamonds on velvet backgrounds, he is very proud of the workmanship in cutting these expensive diamonds. "What do you think of that?" He says to his two friends, both of whom are statisticians.

Bayesian: Well, I know something about photography and light and diamonds, so I know that if I were to take the digital photos and apply a Gaussian blur to them then in each of the photographs the center of the diamond would be one of the pixel values that had at least brightness B. (Plausibility of the location is measured by brightness)

Frequentist: I see that the diamonds are at x1,y1 and x2,y2 and x3,y3 so if I assume all of the photographs in your desk drawer are of diamonds on velvet backgrounds, I can reject the hypothesis that the diamonds are on average in the center of those other photographs. (Repeated sampling would almost never produce 3 photos in a row with the diamond this far off center)

Jeweler: Yes, yes, but which one should I give to my wife for our anniversary?



3 Responses leave one →
  1. May 6, 2016

    "Jeweler: Yes, yes, but which one should I give to my wife for our anniversary?"

    That must surely depend on his prior, no?

  2. Rahul permalink
    May 9, 2016

    Yes, see but he had an explicit prior?

    What I want to see is a Jeweler analogy where you demonstrate the "Bayesian-ness" of his ways in spite of his not referring to or using any explicit prior.

    I want to contrast a Bayesian Jeweler that uses no explicit prior with his frequentist counterpart.

    • Daniel Lakeland
      May 9, 2016

      Rahul, that may be stretching a punchline a little too far. But this Bayesian who says he can detect the diamond in the image by looking at a local average of brightness could have a strong prior over the threshold for brightness, or he could say that he knows nothing about what the threshold is, and hence would have a flat prior on the threshold function and could get the threshold via a maximum likelihood calculation (with flat prior). Either way, choosing to make a connection between brightness and location of that type is a Bayesian idea, there's no frequency distribution that justifies the choice of a gaussian blur and threshold. Even if there were just one image and no others from which we could imagine sampling, the scientific facts on which the idea is based holds.

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