| --- |
| title: "House Oversight: Estate Documents (Nov 12) (HOUSE_OVERSIGHT_013609)" |
| source: "House Oversight: Estate Documents (Nov 12)" |
| sourceUrl: "https://www.justice.gov/epstein" |
| date: "2026-01-01" |
| category: "House Oversight" |
| eftaNumber: "HOUSE_OVERSIGHT_013609" |
| ocrPages: 1 |
| ocrChars: 2204 |
| ocrElapsed: 0.0 |
| parseTier: "external-legacy" |
| engine: "engine undisclosed (ep-nov-12.greg.technology mirror)" |
| externalSource: "greg-ep-nov-12" |
| externalLicense: "not granted" |
| externalCredit: "ep-nov-12.greg.technology" |
| externalUrl: "https://ep-nov-12.greg.technology" |
| --- |
| |
| probabilities, so that the sum of the decimal fraction parts of all the boxes in each |
| horizontal row add up to 100%, or as a real number, 1.00. Recall that in the |
| example we've been using, the binary expansion of the natural number 729, the |
| transition incidence matrix is M, = |
| and its Markov matrix is top row, 1/4, 3/4 |
| 0.75 |
| and bottom row 3/5, 2/5, i.e. Mt.p = |
| Matrix multiplication of Mt,p by itself |
| 0.60 0.40 |
| repeatedly is equivalent to tracking the temporal evolution of the transition matrix's |
| probabilities until the resulting matrices move toward, converge onto, a steady state; |
| each self matrix multiplication step represents what results from the passage of one |
| unit of time. The convergence to equilibrium values is continuous and gradual. |
| When the steady state is reached, both rows become identical. For this example, |
| Mi.p X Mi.p Or Mi,p? = 0.5125 0.4875. |
| 0.3900 0.6100 ÷ Mip * = 0.4527 0.5472 |
| 0.4377 0.5622: M4,08 = |
| 0.4445 0.5554 |
| 0.4443 0.5556' |
| Mt,p |
| 16 = 0.4444 0.5555 |
| which for the first four decimal places remain the same for |
| 0.4444 0.5555 |
| additional times of self multiplication. Note the convergence of the top and bottom |
| rows to the same asymptotic values. Books discussing the multiplicative and other |
| behavior of these nonnegative matrices are numerous and frequently appear in |
| matrix algebra texts under the rubric of the Frobenius-Perron theorems. |
| Using the entropy formalism of Claude Shannon as developed previously, Hm |
| is computed as the sum across either of the identical rows of each probability times |
| its logarithm, pix log(pipzxlog(p)) remembering from above that we are working in |
| base 2 logarithms and to change the minus sign (resulting from taking the |
| logarithms of decimal fractions) to plus: Hm (Mr.p) = 4444 × 10g(4444) + 5555 x |
| 1og(.5555) = .9911 The nonuniformity of the box occupancy probabilities is reflected |
| in the difference between the topological (maximal estimate) and metric (minimal |
| estimate) entropies and is therefore quantifiable and computable: HT - Hm # 0 = 1.00 |
| - 0.9911 = 0. 0089. If the maximal and minimal estimates of the entropy were equal |
| and all the probabilities boxes in each row asymptotically contained the same |
| 109 |
| HOUSE_OVERSIGHT_013609 |
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|
