# Bootstrap Methods for the Empirical Study of Decision-Making and Information Flows in Social Systems

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## Abstract

**:**

## 1. Introduction

## 2. Estimating Entropy

- Uncertainty principle. When all k entries of $\overrightarrow{p}$ are equal, $H(\overrightarrow{p})$ should be a monotonic, increasing function of k.
- Consistency under coarse-graining. $H({p}_{1},{p}_{2},{p}_{3})$ is equal to $H({p}_{1},{p}_{2}+{p}_{3})+({p}_{2}+{p}_{3})H({p}_{2},{p}_{3})$.

- 1′.
- Uncertainty principle. When all k entries of $\overrightarrow{n}$ are equal, $\widehat{H}(\overrightarrow{n})$ should be a monotonic, increasing function of k.
- 2′.
- Consistency under coarse-graining. $\widehat{H}({n}_{1},{n}_{2},{n}_{3})$ is equal to $\widehat{H}({n}_{1},{n}_{2}+{n}_{3})+\frac{{n}_{2}+{n}_{3}}{n}\widehat{H}({n}_{2},{n}_{3})$, where n is the total number of observations.
- 3′.
- Asymptotic convergence. As n goes to infinity, $\widehat{H}(\overrightarrow{n})\to H(\overrightarrow{p})$.

## 3. Distances between Distributions

#### 3.1. Kullback–Leibler Divergence

#### 3.2. Jensen–Shannon Divergence

#### 3.3. The Bhattacharyya Bound

**Figure 2.**Prediction error curves and the existence of multiple classes. Solid curve: the Bhattacharyya bound for prediction of trial outcome for the period 1800 to 1820. Triangle symbols and solid line: actual prediction error, when drawing samples (words) from all trials within a class (guilty or not-guilty). As expected, the curve lies strictly below the Bhattacharyya bound. Diamond symbols and dashed line: actual prediction error, when drawing samples from a single trial. The prediction error actually rises (more samples lead to a less accurate prediction), suggesting that the underlying model (trials sample from one of two distributions) is incorrect. We restrict the set of trials here to those with at least one hundred (semantically-associated) words, so as to make the resampling process more accurate.

#### 3.4. Summary

## 4. Correlation, Dependency and Mutual Information

#### 4.1. Mutual Information

**Figure 3.**Predictability of the Kandahar and Helmand time streams. Top: a dramatic asymmetry on short timescales provides strong suggestion of anticipatory, and potentially causal, effects transmitted from Kandahar to Helmand province on rapid (less than two-week) timescales. Bottom: the consistent, opposite asymmetry is seen in the reverse process. A rise in the predictability of Kandahar by Helmand on longer (one-month) timescales, mirrored in the top panel, suggests potentially longer-term seasonal or constraint-based information common to both systems.

#### 4.2. The Data Processing Inequality

## 5. The Bootstrap Estimators In Practice

#### 5.1. The Bayesian Prior Hierarchy

- Draw a random integer, ${k}^{\prime}$, between k and ${k}^{2}$ inclusive.
- Draw a distribution ${p}^{\prime}$ with ${k}^{\prime}$ bins from ${D}_{\mathrm{NSB}}$. Then ${p}^{\prime}$ is approximately uniform in entropy over ${k}^{\prime}$ bins.
- Randomly partition the ${k}^{\prime}$ bins into k bins.
- Coarse grain the distribution ${p}^{\prime}$, given the partition of Step 3, to get p.

`ranksb`and

`rancom`algorithms from [31].

#### 5.2. Coarse-Graining Consistency

**Table 1.**RMS Violations of coarse-graining consistency for entropy (Condition 2) for the Wolpert & Wolf (WW), Nemenman, Shafee & Bialek (NSB), and the bootstrap. The bootstrap estimator leads to a factor of ten or more improvement in coarse-graining consistency; as the amount of data increases, the bootstrap approaches full consistency faster. The average entropy of the three-state distributions is approximately 1.2 bits. These results are for the ${D}^{\prime}$ prior of Section 5.1.

Sampling | WW (RMS bits) | NSB (RMS bits) | Bootstrap (RMS bits) |
---|---|---|---|

1× | 0.2333 | 0.1376 | 0.0046 |

2× | 0.1697 | 0.0865 | 0.0031 |

4× | 0.1165 | 0.0463 | 0.0014 |

8× | 0.0720 | 0.0239 | 0.0008 |

16× | 0.0423 | 0.0121 | 0.0005 |

**Table 2.**RMS Violations of coarse-graining consistency for mutual information (Equation (27)) for the Wolpert & Wolf (WW) estimator, Nemenman, Shafee & Bialek (NSB) for Mutual Information, and the bootstrap. The bootstrap estimator again leads to a factor of ten or more improvement in coarse-graining consistency; as the amount of data increases, the bootstrap approaches full consistency faster. The average mutual information of the $2\times 3$ distribution is approximately 0.25 bits.

Sampling | WW (RMS bits) | NSB-MI (RMS bits) | Bootstrap (RMS bits) |
---|---|---|---|

1× | 0.0316 | 0.0387 | 0.0026 |

2× | 0.0196 | 0.0323 | 0.0012 |

4× | 0.0119 | 0.0208 | 0.0006 |

8× | 0.0069 | 0.0124 | 0.0004 |

16× | 0.0039 | 0.0066 | 0.0003 |

#### 5.3. Bias Correction and the Reliability of Error Estimates

**Figure 4.**Left panel: bias for the 16-state entropy estimation case, under prior ${D}^{\prime}$. Dotted line: naive estimator; Dashed line, *-symbol: Wolpert and Wolf estimator; Dashed line, □-symbol: NSB estimator. Solid line: Bootstrap estimator. Right panel: one-sigma (solid line) and two-sigma (dashed line) error bar reliability; as the sampling factor increases, both rapidly approach their asymptotic values (thin horizontal lines). Average entropy for this prior is 2.4 bits.

**Figure 5.**Left panel: bias for the estimation of mutual information on a $4\times 4$ joint probability distribution, under prior ${D}^{\prime}$. Dotted line: naive estimator; Dashed line, *-symbol: Wolpert and Wolf estimator; Dashed line, □-symbol: NSB estimator. Solid line: Bootstrap estimator. Right panel: one-sigma (solid line) and two-sigma (dashed line) error bar reliability; as the sampling factor increases, both rapidly approach their asymptotic values (thin horizontal lines). Average mutual information under this prior is 0.55 bits.

**Figure 6.**Left panel: bias for the estimation of mutual information on a $16\times 16$ joint probability distribution, under prior ${D}^{\prime}$. Dotted line: naive estimator; Dashed line, *-symbol: Wolpert and Wolf estimator; Dashed line, □-symbol: NSB estimator. Solid line: Bootstrap estimator. Right panel: one-sigma (solid line) and two-sigma (dashed line) error bar reliability; as the sampling factor increases, both rapidly approach their asymptotic values (thin horizontal lines). Average mutual information for this prior is 1.33 bits.

**Table 3.**RMS errors for estimation of the entropy information of a 16-state system, for the Wolpert & Wolf (WW) estimator, Nemenman, Shafee & Bialek (NSB) for Mutual Information, and the bootstrap. The bootstrap estimator has rms errors comparable to the NSB.

Sampling | WW (RMS bits) | NSB (RMS bits) | Bootstrap (RMS bits) |
---|---|---|---|

1× | 1.0442 | 0.3305 | 0.3605 |

2× | 0.8302 | 0.2223 | 0.2304 |

4× | 0.5990 | 0.1537 | 0.1564 |

8× | 0.3854 | 0.1039 | 0.1051 |

16× | 0.2341 | 0.0730 | 0.0734 |

**Table 4.**RMS errors for estimation of the mutual information of a $4\times 4$ joint probability, for the Wolpert & Wolf (WW) estimator, Nemenman, Shafee & Bialek (NSB) for Mutual Information, and the bootstrap. The bootstrap estimator has rms errors comparable to the NSB.

Sampling | WW (RMS bits) | NSB (RMS bits) | Bootstrap (RMS bits) |
---|---|---|---|

1× | 0.3106 | 0.1924 | 0.2783 |

2× | 0.2494 | 0.1446 | 0.1722 |

4× | 0.1851 | 0.1041 | 0.1132 |

8× | 0.1277 | 0.0727 | 0.0761 |

16× | 0.0845 | 0.0523 | 0.0534 |

**Table 5.**RMS errors for estimation of the mutual information of a $16\times 16$ joint probability, for the Wolpert & Wolf (WW) estimator, Nemenman, Shafee & Bialek (NSB) for Mutual Information, and the bootstrap. The bootstrap estimator has rms errors comparable to the NSB.

Sampling | WW (RMS bits) | NSB (RMS bits) | Bootstrap (RMS bits) |
---|---|---|---|

1× | 0.6920 | 0.1028 | 0.1417 |

2× | 0.5267 | 0.0685 | 0.0642 |

4× | 0.3696 | 0.0417 | 0.0373 |

8× | 0.2409 | 0.0260 | 0.0251 |

16× | 0.1487 | 0.0175 | 0.0173 |

#### 5.4. Summary

## 6. Conclusions

## Acknowledgements

## Conflict of Interest

## Appendix

## A. The Trial of John Long, as Reported on 18 September 1820

## B. SIGACT for IED Explosion, 3 April 2005, Kandahar Province

**Title**

**Text**

**Additional Data**

`{:reportkey=>"DCEAC77F-A84D-45F3-88B3-33B9B5A95B20", :type=>:explosivehazard, :category=>:iedexplosion, :trackingnumber=>:2007-033 -005423-0737, :region=>:rcsouth, :attackon=>:enemy, :complexattack=>false, :reportingunit=>:other, :unitname=>:other, :typeofunit=>:noneselected, :friendlywia=>0, :friendlykia=>0, :hostnationwia=>0, :hostnationkia=>1, :civilianwia=>0, :civiliankia=>1, :enemywia=>0, :enemykia=>0, :enemydetained=>0, :mgrs=>"42RTV5110332180", :latitude=>30.99694061, :longitude=>66.39333344, :originatorgroup=>:unknown, :updatedbygroup=>:unknown, :ccir=>:"", :sigact=>:"", :affiliation=>:enemy, :dcolor=>:red, :classification=>:secret, :start=>2005-04-03 03:15:00 UTC, :province=>:kandahar, :district=>:spinboldak, :nearestgeocode=>"AF241131834", :nearestname=>"Spin Boldak"}`

**Source and Post-processing**

## C. Additional Characterizations

**Figure A1.**Estimator bias as a function of the entropy of the underlying distribution for the naive (dotted line), NSB (dashed line) and bootstrap (solid line) estimators. Distributions are over sixteen categories, drawn from ${D}^{\prime}$, and binned in 0.25 bit increments; the bias is for estimates made with sixteen samples (i.e., $1\times $ oversampling). Ranges shown are one-sigma error bars for the bias in the bin. As can be seen, all estimators tend to overestimate small entropies, and underestimate large entropies, with the cross-over point (and overall bias) depending on the method. As in the main text, Section 5.3, we neglect cases where the empirical distribution has entropy zero; this is one source of the positive bias at the lowest entropy bins.

**Figure A2.**The distribution of entropies for distributions sampled from the ${D}_{1}$ (dotted line), ${D}_{\mathrm{NSB}}$ (dashed line) and ${D}^{\prime}$ (solid line) priors. The ${D}_{1}$ prior produces distributions very strongly skewed towards high entropies, while the ${D}_{\mathrm{NSB}}$ distribution is nearly flat for entropies larger than one bit.

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**MDPI and ACS Style**

DeDeo, S.; Hawkins, R.X.D.; Klingenstein, S.; Hitchcock, T.
Bootstrap Methods for the Empirical Study of Decision-Making and Information Flows in Social Systems. *Entropy* **2013**, *15*, 2246-2276.
https://doi.org/10.3390/e15062246

**AMA Style**

DeDeo S, Hawkins RXD, Klingenstein S, Hitchcock T.
Bootstrap Methods for the Empirical Study of Decision-Making and Information Flows in Social Systems. *Entropy*. 2013; 15(6):2246-2276.
https://doi.org/10.3390/e15062246

**Chicago/Turabian Style**

DeDeo, Simon, Robert X. D. Hawkins, Sara Klingenstein, and Tim Hitchcock.
2013. "Bootstrap Methods for the Empirical Study of Decision-Making and Information Flows in Social Systems" *Entropy* 15, no. 6: 2246-2276.
https://doi.org/10.3390/e15062246