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Also posted here: http://lesswrong.com/r/discussion/lw/3ol/is_there_a_way_to_quantify_the_relationship/

Okay, so maybe you could say this.

Suppose you have an index I. I could be a list of items in belief-space (or a person’s map). So I could have these items (believes in evolution, believes in free will, believes that he will get energy from eating food, etc..) Of course, in order to make this argument more rigorous, we must make the beliefs finer.

For now, we can assume the non-existence of a priori knowledge. In other words, facts they may not explicitly know, but would explicitly deduce simply by using the knowledge they already have.

Now, maybe Person1 has a map in j-space with values of (0,0,0.2,0.5,0,1,…), corresponding to the degree of his belief in items in index I. So the first value of 0 corresponds to his total disbelief in evolution, the second corresponds to total disbelief in free will, and so on.

Person2 has a map in k-space with values of (0,0,0.2,0.5,0,0.8, NaN, 0, 1, …), corresponding to the degree of his belief in everything in the world. Now, I include a value of NaN in his map, because the NaN could correspond to an item in index I that he has never encountered. Maybe there’s a way to quantify NaN, which might make it possible for Person1 and Person2 to both have maps in the same n-space (which might make it more possible to compare their mutual information using traditional math methods).

Furthermore, Person1’s map is a function of time, as is Person2’s map. Their maps evolve over time since they learn new information, change their beliefs, and forget information. Person1’s map can expand from j-space to (j+n)th space, as he forms new beliefs on new items. Once you apply a distance metric to their beliefs, you might be able to map them on a grid, to compare their beliefs with each other. A distance metric with a scalar value, for example, would map their beliefs to a 1D axis (this is what political tests often do). A distance metric can also output a vector value (much like what a MBTI personality test could do) to a value in j-space. If you simply took the difference between the two maps, you cold also output a vector value that could be mapped to a space whose dimension is equal to the dimension of the original map (assuming that the two maps have the same dimension, of course).

Anyways, here is my question: Is there a better way to quantify this? Has anyone else thought of this? Of course, we could use a distance metric to compare their distances with respect to each other (of course, a Euclidean metric could be used if they have maps in the same n-space.

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As an alternative question, are there metrics that could compare the distance between a map in j-space with a map in k-space (even if j is not equal to k)? I know that you have p-norms that correspond to some absolute scalar value when you apply the p-norms to a matrix. But this is sort of difference. And could mutual information be considered a metric?

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