Timeline for Of all probability matrix $P$ having stationary distribution $\pi$, find the one having smallest diagonal
Current License: CC BY-SA 4.0
11 events
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Jun 18, 2021 at 17:51 | vote | accept | RSMax | ||
Jun 11, 2020 at 23:19 | history | edited | Robert Israel | CC BY-SA 4.0 |
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Jun 11, 2020 at 20:51 | history | edited | Robert Israel | CC BY-SA 4.0 |
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Jun 11, 2020 at 19:49 | comment | added | Robert Israel | Linear programming in Excel. But the algorithm I presented in the EDIT is simpler (though maybe not so easy to program in Excel). Maybe I should make it more explicit. | |
Jun 11, 2020 at 17:14 | comment | added | RSMax | If by linear programming you mean something else than Newton's method, I am very much interested. As I need to ultimately code the result, anything easier and/or more efficient than Newton's method will be received with open arms. Would you mind elaborating a bit? Thank you sir. | |
Jun 11, 2020 at 17:08 | comment | added | RSMax | Whether I minimize the trace or the sum the squares of all elements on the diagonal, my current issue (under the Newton's method approach) is that my "main" unknown is a matrix ($\boldsymbol{P}$, whereas the Lagrange multipliers are scalars, and I'm confused about how to wrap those 3 under a common structure so that I can recursively update it and converge toward a solution. Should I add rows and columns to the unknown $\boldsymbol{P}$ matrix so that it incorporates $\lambda_1$ and $\lambda_2$ on the main diagonal? | |
Jun 11, 2020 at 17:03 | comment | added | RSMax | I totally agree with the EDIT section, this is something I have observed in practice using Excel's solver. What remains encouraging is that through this scheme, $P_{11}$ will nonetheless be less than $\pi_1$, given that $\pi_1 \geq 0.5$, ergo the diagonal has still been minimized in some way. | |
Jun 11, 2020 at 17:03 | history | edited | Robert Israel | CC BY-SA 4.0 |
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Jun 11, 2020 at 15:50 | history | edited | Robert Israel | CC BY-SA 4.0 |
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Jun 11, 2020 at 15:04 | comment | added | RobPratt | Alternatively, an objective to minimize $\sum_i P_{i,i}^2$ yields a (convex) quadratic programming problem. | |
Jun 11, 2020 at 14:41 | history | answered | Robert Israel | CC BY-SA 4.0 |