Reality of Bible Codes Strongly Confirmed by Extensive Additional Research |
By Richard E. Sherman, FCAS, MAAA, and Nathan Jacobi, Ph.D.
In the article Non-Random Equidistant Letter Sequence Extensions in the Hebrew Bible published in the May/June 2010 digest, we summarized the results of 4˝ years of BCD's most recent research. The specific findings from this research were published in BCD articles from November/December 2006 through March/April 2010. A rigorous analysis of these findings conclusively supported the rejection of the null hypothesis that Bible codes can be explained by random causes. The phenomenon cannot be explained by chance. This conclusion was based on a statistically significant population of 1,064 observations of initial search term ELSs that were examined for the possible existence of extensions in good Hebrew. A total of 475 extended ELSs were discovered in the course of this investigation.
In this article, we summarize the results of 3˝ years of previous BCD research (from July 2001 through December 2004). These findings also conclusively confirm that the null hypothesis that Bible codes can be explained by random causes should be rejected. A total of 366 extended ELSs were discovered during this earlier investigation. Our record-keeping then did not include documentation of how many original search terms were examined for extensions.
Taken all together, then, BCD's research during the eight years of these two investigations resulted in the compilation of 841 extended ELSs. The gap between the indicated discovery rates of ELS extensions explainable by chance (18.7% to 25.6%) and the discovery rates exhibited from the Hebrew Bible (36.7% to 37.2%) is far too large to be attributable to chance. Consequently, the reality of the phenomenon of Bible codes is conclusively demonstrated by eight years of our extensive investigations.
The discovery rate due to chance is based on three findings:
The first extension rate is based on BCD's most recent 4˝ years of research. For 55.357% of these initial ELSs, no extension was found on either side. This translates into a discovery rate of 25.6%, as detailed later in this article.
The second and third indicated discovery rates were those exhibited during an experiment conducted by BCD comparing ELS extension finding in a section of the book of Ezekiel and a corresponding portion of Hebrew text consisting of a translation of Leo Tolstoy's novel, War and Peace published as Non-Random Equidistant Letter Sequence Extensions in Ezekiel.
The ELS extension discovery rates indicated by BCD's most recent investigation and from its previous investigations are displayed in the following table.
The difference between these two sets of discovery rates (18.7% to 25.6% versus 37%) is so large that it cannot be explained by random fluctuations, given the very large number of observations (over 1,850) on which it is based.
Extended Bible codes are found by first locating a single word ELS with a given skip and then by searching for extensions in good Hebrew on both sides, continuing at the same skip. The longer an extended ELS is, the more improbable it is. A Markov chain model was developed for estimating the probability of chance occurrence of extended ELSs of a variety of different lengths. Monte Carlo simulations of the extension discovery process were also used to determine the probability of different outcomes, by chance, around the expected outcome.
As disclosed in the May/June 2010 BCD issue, over the past 4˝ years, 1,064 search terms were examined to determine if extensions in good Hebrew existed around them. For 55.2% of the initial search terms, no extension was found. This indicated an extension discovery rate of 25.6%. Based on a Markov chain model, this discovery rate implies that 102.2 extended ELSs should consist of three or more extensions. In actuality, 315 such extended ELSs were discovered during both of BCD's extended investigations. Using computer simulation with 100,000 trials, the number of extended ELSs with three or more extensions varied from 68 to 147, as illustrated in Graph 1 below.
Comparison of Number of Extended ELSs with Three or More Extensions
Expected by Chance (Left Histogram) and Actual Number Found (Far Right)
The vertical axis displays the number of trials, out of 100,000, where the given number of findings occurred. For example, the largest number of trials (19,929) coincides with 102 extended ELSs found. The number "102" represents all trials where there were 100 to 104 extended ELSs with three or more extensions. Similarly, the number "317" on the far right represents the actual result of 315 findings, as 317 is the middle number in the range of 315 to 319 findings. That the actual result is radically greater than the entire range of what is possible by chance (i.e., 68 to 147) means that the null hypothesis should be completely rejected.
Table 1 summarizes the results shown in Graph 1 and also displays comparable results for different numbers of extensions. In every case other than the number of extended ELSs with two extensions, the null hypothesis that the actual finding can be attributed to chance should be decisively rejected.
Comparison of Expected and Actual Number of Findings
If the statistical significance standard is at the 0.001 level, or 1 in 1,000, then a result that is 3.09 or more standard deviations out should result in rejection of the null hypothesis. The outcome cannot be explained by chance variation. If the standard is at the one in one million level, then a result that is 4.75 or more standard deviations out should result in rejection of the null hypothesis. At the one in one billion level, 6.00 standard deviations is the boundary value. At the one in one trillion level, 7.34 standard deviations is the boundary value. As the above table shows, standard deviations from expected ranging from 21.5 to 30.4 are indicated. Such large standard deviations indicate rejection of the null hypothesis at a level completely beyond the realm of possibility.
For an introduction to the phenomenon of Bible codes, the reader is referred to the Introduction section of the scientific paper. Additional sections worth reviewing in that paper are headed, "Two Very Different Types of Bible Code Findings", "Background", "Phase One Findings" and "Phase Two Findings".
The Earlier Investigation
During the period from July 2001 to December 2004, a Hebrew expert (Nathan Jacobi, Ph.D.), examined over 800 ELSs for the presence of extensions in good Hebrew, either immediately before or after the given ELS. The initial search terms were selected on the basis of likely relevance to current events or to religious topics (Appendix C). Generally, two types of searches were performed. In the first type, for each search term, ELSs with the 10 shortest skips in the Hebrew Bible were examined for possible extensions. In the second type, ELSs with the 10 shortest skips where at least one letter of the ELS appeared in a selected passage were examined. In each case, the passage was selected on the basis of the relevance of the literal text to the topic under investigation. The beginning and ending of the selected text were not modified to "optimize" the number of extended ELSs found.
For 55.357% of these initial ELSs, no extension was found on either side. This translates into a discovery rate of 25.6%. Suppose d is the probability of discovering an extension around an existing ELS, either before or after it. For no extension to be found on either side, no extension must appear twice (both before and after the initial ELS). So, (1.0 – d)2 = 55.357%, (1.0 – d) = 74.402% and d = 25.6%. And we have the formula,
(1) d = 1.0 – (square root of the %-age of searches where no extension was found).
If an extension is found, one new opportunity to find yet another extension is created. That opportunity will consist of the new letters that are now next to the extension that had just been discovered. There is no new opportunity on the other side of the ELS where an extension wasn't found, since that opportunity has already been counted.
In Appendix A, a Markov chain model was used to derive the formula for the total number of final ELSs consisting of k extensions expected to be found by searching around n initial ELSs, given a discovery rate of d:
(2) n ( k + 1 ) dk ( 1 – d )2.
Formula (2) was used to determine the expected number of lengthy ELSs displayed in Table 2 for several length categories from both BCD investigations.
Comparison of Actual with Expected Number of Extended
ELSs by Length Category (Assuming a 25.6% Discovery Rate)
Combining the Current and Previous Investigations
The presence of extended ELSs with only one extension was surprisingly rare, compared to expectations assuming randomness (271 actual versus 534 expected). Correspondingly, for extended ELSs with three or more extensions, the presence of lengthy extensions far exceeded what would be expected by chance. In other words, there was a clear tendency for there to not be any extension around the original term, or for there to just be one extension, or for the number of extension(s) to be noticeably longer than expected by chance.
If we combine the results in the above table for three or more extensions, we see that although 102.16 ELSs were expected by chance, 315 were actually discovered. The greatest number of ELSs with three or more extensions produced from 100,000 trials of the Markov chain model simulation was 147. It is evident that the null hypothesis that the lengthy extension findings are due to chance should be rejected.
Corresponding results for compilations for each given number of extensions, or more than the given number, are as follows:
Comparison of Actual with Expected Number of Extended ELSs by
Given Length (or Longer) Category (Assuming a 25.7% Discovery Rate)
The consistent, dramatic excesses of the actual number of extended ELSs for each category of number of extensions (more than one) are conclusive evidence that the Hebrew Bible was intentionally encoded with extended ELSs.
Tables 2 and 3 also indicate that the terseness of Hebrew and the absence of vowels can result in the "discovery" of some longer extended ELSs, simply due to chance.
By working with the information in Table 3 above, we can construct Table 4.
Number of Lengthy Extended ELS's
and Indicated Discovery Rates
For example, Table 4 shows that while there were 841 total ELSs with at least one extension, there were 570 with at least two extensions. That means that for 67.8% of the ELSs where one or more extensions had been found, at least one more extension was found. The discovery rate between the 1+ and 2+ categories was calculated by taking 1.0 minus the square root of the probability of more extensions (67.8%). Or, working backwards, if the discovery rate is 43.2%, then the rate of not discovering an extension is 56.8% and the rate of not discovering an extension on either side of the original term would be 56.8% times 56.8%, or 32.3%. And 67.8% is 100% minus 32.3%.
In Table 5, we have a side-by-side comparison of the discovery rates indicated by the most recent 4˝ years of investigation and the earlier 3˝ years of investigation. We see that the indications are very consistent.
Comparison of Number of Lengthy Extended ELS's
Recent and Earlier BCD Investigations
Appendices B and D provide a listing of all ELSs for which extensions were found. These extended ELSs are presented in descending order of length, first in terms of the number of actual extensions found, and second in descending order of the total number of letters in the extension(s).
Key findings from BCD's two lengthy investigations are:
- The null hypothesis that the actual number of long ELSs discovered is due to chance must be rejected. The frequency of long extended ELSs far exceeds that explainable by chance, clearly supporting the claim that some real Bible codes do exist. The chi square p-value indicated from Table 2 is smaller than 1.0 E -310 for codes with one to 10 extensions.
- Longer ELSs can be "discovered," even in ordinary texts, with a fair degree of frequency. This affirms the claim of code skeptics that "ELS codes" can be found in any book.
- The extension model provides a definitive benchmark for testing purported claims of the discovery of real Bible codes. It provides the expected number of extended ELSs that will consist of any given number of extensions. The mere discovery of a few lengthy extended ELSs does not provide credible evidence of the potential reality of Bible codes.
- If one multiplies the number of initial ELSs investigated (1,884) by the ratio of actual to expected ELSs in one of the longer categories, estimates can be obtained of the expected number of initial ELSs to have been investigated from the Hebrew Bible to find as many long ELSs as were actually discovered. For example, that expected number of extended ELSs with seven or more extensions was 0.84, as compared with the 28 extended ELSs that were discovered.
- Chances are significant that some of the shorter extended ELSs could be due to chance. For that reason, the content of extended ELSs should not be viewed as a reliable basis for making any predictions or for supporting any political or religious views that might be expressed in such extended ELSs.
- One hundred thousand trials were run of a simulation of the total number of ELSs with three or more extensions, given 1,884 initial ELSs and a discovery rate of 25.6%. The number of trials with various numbers of ELSs with 3+ extensions is shown in Table 6. The exceptionally large gap between the simulated results (which range between 68 and 147) and the actual number of 315 ELSs illustrates how infinitesimal is the probability of chance occurrence of findings as extensive as those described in this paper.
Number of Trials with Given Numbers of ELSs with 3+ Extensions
- While the results of this analysis are sensitive to key assumptions such as the discovery rate from a non-encoded text, the gap between the results expected by chance from a non-encoded text and the actual findings is so great that it would not be bridged by any reasonable variations in these key assumptions and procedures. Furthermore, we have tried to give the benefit of the doubt on the side of underestimating the number of extensions present in many extended ELSs.
- Even though the probability of chance occurrence of a collection of extended ELSs as extensive as those presented in this paper is extremely remote, there are compelling reasons to believe that the chance presence of this collection of extended ELSs is a virtual impossibility. The discovery rate of 25.6% underlying our calculations is that for finding an extension in good Hebrew. However, a high percentage of the extended ELSs consist of either relevant or plausible content. The proportion of these extended ELSs that are mysterious is relatively low, and the proportion of nonsense content is very small. While categorization of extended ELSs according to the quality of their content is inherently a subjective exercise, the reader is invited to examine the English translations of the extended ELSs disclosed in Appendices B and D to form their own opinion on this matter. Further, for each extended ELS, a page number on the www.biblecodedigest.com website is provided. These articles elaborate on the possible relevance and plausibility of the content of each of these extended ELSs. It is often additional observations such as these that make it clear that chance is not a rational explanation for the presence of so many plausible extended ELSs.
- The above considerations make the findings of this analysis conclusive in rejecting the null hypothesis that the number and length of the extended ELSs found is attributable to chance.
Similar findings for ELSs with 5+ extensions are shown in Table 7 below. Again, there is a very large gap between the simulated results (ranging between 0 and 25) and the actual number of 101 ELSs with five or more extensions.
Number of Trials with Given Numbers of ELSs with 5+ Extensions
The number of trials with various numbers of ELSs with 7+ extensions is shown in Table 8 below. Again, there is a very large gap between the simulated results (ranging from 0 to 7) and the actual number of 28 ELSs with seven or more extensions.
Number of Trials with Given Numbers of ELSs with 7+ Extensions
The number of trials with various numbers of ELSs with 9+ extensions is shown in Table 9 below. There is a very large gap between the simulated results (0 to 3) and the actual number of eight ELSs with nine or more extensions.
Number of Trials with Given Numbers of ELSs with 9+ Extensions
In the five appendices to this paper, the following topics are covered:
- A Markov Chain ELS Extension Model
- Comparison of the Number of Lengthy ELSs Found with Those Expected by Chance—For Extended ELSs with Three or More Extensions
- Summary of Searches for Extensions
- ELSs with One or Two Extensions
- Commentary on the Content of Extended ELSs
Appendices B, C and D focus on disclosing many of the most important details of the research completed.
About the Researchers
Richard E. Sherman is a Fellow of the Casualty Actuarial Society and a Member of the American Academy of Actuaries. He received a B.A. and M.A. in Mathematics from the University of California at San Diego, and passed three Ph.D. qualifying exams. He has 37 years of experience as a consulting actuary in serving numerous Fortune 500 corporations, major public entities, law firms and insurance companies in applying probability, statistics and econometric forecasting to risk management problems. He was a Principal with PricewaterhouseCoopers, the world's largest accounting and consulting firm, for seven years. He has authored five professional papers in actuarial journals and over 70 articles in trade publications. He directs the biblecodedigest.com website.
Nathan Jacobi, Ph.D., was educated in Israel from age six through his receipt of a doctorate (1945-1969), gaining a very thorough knowledge of both Biblical and contemporary Hebrew. In graduate school, he served as an interpreter (from English to Hebrew) of courses taught at the Weizman Institute of Science. He lectured at the Tel-Aviv University in Hebrew. He has over 30 years of experience in research, development and scientific computing in applied physics, aerospace and geophysics. He has taught numerous classes on Hebrew in recent years (1998-2003 and 2009 to present). In 2006, after living in the United States for many years, Nathan and his wife have moved back to Israel to be near their two children. In 2008, BCD conducted a two-part interview with Jacobi to discuss his life and his background in Hebrew.
Richard E. ("Ed") Sherman, FCAS, MAAA
President, Richard E. Sherman & Associates, Inc.
1257 Siskiyou Blvd. (183)
Ashland, OR 97520
(541) 821-4238 (cell)
(541) 488-7759 (fax)
Nathan Jacobi, Ph.D.
Kochav Hashahar 202
D.N. East Benjamin 90641
(212) 202-3247 (from USA)
Continue to Appendix A
Bombshell examines two massive, recently discovered clusters of codes in the Hebrew Old Testament. To read more about Bombshell, click here, or click below to order from Amazon today!