4 ms·

Stochastic geometric series

10 points by efavdb 11y ago

thanatropism 11y ago
This is nice. So, follow-up question. In economics, there's "input-output models" which amount to the summation of adjacency matrices for inter-sectoral demand linkages such that S = I + A + A^2 + A^3+... = (I-A)^(-1) We know this to be true because matrix A is markovian, so it's a bounded operator, so we have a Neumann series. What happens if we have probability distributions for the elements of A? This is the context where it arises: we define a vector of x interindustrial demands that must satisfy itself through production coefficients A (the weights in the graph of cross-sector dependencies), such that x = Ax but then we give this balanced system an additive shock ("consumer demands") b so that x' = x+e = Ax+b but we still want to satisfy Ax' = x'. So: A(x+e) = Ax+b => A*e = b => e = (I-A)^(-1) b This is the shock to the interindustrial demands provoked by b. We have to know that (I-A) is invertible, which by the Perron-Frobenius theorem is true if A happens to be markovian. This matters because the interindustrial linkage graph A is very noisy (it's hammered into place at the national accounts offices from disperse information), and much basic demand-shock appraisal (in a multi-sector context at least) still uses this basic ("Leontieff") model; and more sophisticated models apply variants of this too; but it's very hard to get anything said about confidence intervals -- other than by means of simulation. So -- maybe it's an interesting, applied problem to work on!
- efavdb 11y ago
  Awesome idea!
  - thanatropism 11y ago
    Can I maybe email you if this is (in your estimation) a conversation worth continuing? I have a feeling it gets fast lost in an aging HN thread. At any rate: probably the most important problem to be posed by such a "random input-output theory" is stochastic divergence. Random matrix theory, of which I can make little sense, does present some knowledge about the distribution of eigenvalues given some stochastic structure, which numerically gives us probability estimates to whether we're breaking Perron-Frobenius and posing an explosive system. But the relevant problem is in reverse: what is the set of acceptable stochastic conditions such that we can guarantee that S = I+A+Aˆ2+... = some finite value?
    - efavdb 11y ago
      Hi, I'm very happy to talk some more about this. My email is jslandy@gmail.com (will delete this post after you email).

10 points by efavdb 11y ago

This is nice. So, follow-up question. In economics, there's "input-output models" which amount to the summation of adjacency matrices for inter-sectoral demand linkages such that S = I + A + A^2 + A^3+... = (I-A)^(-1) We know this to be true because matrix A is markovian, so it's a bounded operator, so we have a Neumann series. What happens if we have probability distributions for the elements of A? This is the context where it arises: we define a vector of x interindustrial demands that must satisfy itself through production coefficients A (the weights in the graph of cross-sector dependencies), such that x = Ax but then we give this balanced system an additive shock ("consumer demands") b so that x' = x+e = Ax+b but we still want to satisfy Ax' = x'. So: A(x+e) = Ax+b => A*e = b => e = (I-A)^(-1) b This is the shock to the interindustrial demands provoked by b. We have to know that (I-A) is invertible, which by the Perron-Frobenius theorem is true if A happens to be markovian. This matters because the interindustrial linkage graph A is very noisy (it's hammered into place at the national accounts offices from disperse information), and much basic demand-shock appraisal (in a multi-sector context at least) still uses this basic ("Leontieff") model; and more sophisticated models apply variants of this too; but it's very hard to get anything said about confidence intervals -- other than by means of simulation. So -- maybe it's an interesting, applied problem to work on!

efavdb 11y ago
Awesome idea!
- thanatropism 11y ago
  Can I maybe email you if this is (in your estimation) a conversation worth continuing? I have a feeling it gets fast lost in an aging HN thread. At any rate: probably the most important problem to be posed by such a "random input-output theory" is stochastic divergence. Random matrix theory, of which I can make little sense, does present some knowledge about the distribution of eigenvalues given some stochastic structure, which numerically gives us probability estimates to whether we're breaking Perron-Frobenius and posing an explosive system. But the relevant problem is in reverse: what is the set of acceptable stochastic conditions such that we can guarantee that S = I+A+Aˆ2+... = some finite value?
  - efavdb 11y ago
    Hi, I'm very happy to talk some more about this. My email is jslandy@gmail.com (will delete this post after you email).