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I wish people would just give simple explanations for this stuff and not make it more complicated than it needs to be. Something like a visual explanation show

I wish people would just give simple explanations for this stuff and not make it more complicated than it needs to be. Something like a visual explanation showing how a conditional distribution is like a single row or column from a 2D table would be so much easier for people to understand.

mjburgess 4y ago
But also not accurate. Probability interpreted as having to do with sets is just the classical formulation, and really, it's more like measure-theory than "probability" which is a specific application. The bayesian interpretation, which is easily the most general, is happy to take the arguments of the probability function inscrutably as bare propositions.
- nerdponx 4y ago
  It's not not accurate, it's just not complete from the perspective of modern theory. I agree that conditional probability is best demonstrated graphically. The sample space is a big rectangle, and events are represented as polygons or other closed shapes, with area proportional to their probability. Conditional probability elegantly falls out of this representation by treating one of these event-shapes as a new "sub sample space" and computing fractions of areas.
  - mjburgess 4y ago
    Yes, but it rather poisons the well against the bayesian, who says that conditional probability is basic; there are no unconditional probabilities. So this area model at once creates a lot of confusion elsewhere.
    - nerdponx 4y ago
      I don't know. I think you need to start with the "world in a rectangle model" to build intuition. Then you can blow your students' minds with the possibility that the "rectangle" is actually the unknowably massive timeline of the entire universe, from the Big Bang until its heat death, and all of our models and reasoning are conditional on some subset of that gigantic world-rectangle. In my opinion, the power of this philosophical leap is lost without the basic geometric intuition.
- pfisherman 4y ago
  It’s not totally accurate, but it gives a pretty good intuition that an average middle school student could easily grasp. Think about it like PCA. Do you lose some information? Yes. But you can do a lot with only the first few principle components (i.e. basic concepts). Things like events, sets, measures, etc add important details like why you have to divide the selected row / column by its sum to get a true conditional distribution; how many dimensions the array should have; constraints, properties, and relationships of subarrays, etc. But that all becomes a lot easier to understand once you have the basic intuition. And once you get good enough you can start to let go of the grounding in arrays and ground in other concepts and then think about it more abstractly.

mjburgess 4y ago

But also not accurate. Probability interpreted as having to do with sets is just the classical formulation, and really, it's more like measure-theory than "probability" which is a specific application. The bayesian interpretation, which is easily the most general, is happy to take the arguments of the probability function inscrutably as bare propositions.

nerdponx 4y ago
It's not not accurate, it's just not complete from the perspective of modern theory. I agree that conditional probability is best demonstrated graphically. The sample space is a big rectangle, and events are represented as polygons or other closed shapes, with area proportional to their probability. Conditional probability elegantly falls out of this representation by treating one of these event-shapes as a new "sub sample space" and computing fractions of areas.
- mjburgess 4y ago
  Yes, but it rather poisons the well against the bayesian, who says that conditional probability is basic; there are no unconditional probabilities. So this area model at once creates a lot of confusion elsewhere.
  - nerdponx 4y ago
    I don't know. I think you need to start with the "world in a rectangle model" to build intuition. Then you can blow your students' minds with the possibility that the "rectangle" is actually the unknowably massive timeline of the entire universe, from the Big Bang until its heat death, and all of our models and reasoning are conditional on some subset of that gigantic world-rectangle. In my opinion, the power of this philosophical leap is lost without the basic geometric intuition.
pfisherman 4y ago
It’s not totally accurate, but it gives a pretty good intuition that an average middle school student could easily grasp. Think about it like PCA. Do you lose some information? Yes. But you can do a lot with only the first few principle components (i.e. basic concepts). Things like events, sets, measures, etc add important details like why you have to divide the selected row / column by its sum to get a true conditional distribution; how many dimensions the array should have; constraints, properties, and relationships of subarrays, etc. But that all becomes a lot easier to understand once you have the basic intuition. And once you get good enough you can start to let go of the grounding in arrays and ground in other concepts and then think about it more abstractly.