# Forecasting nuclear proliferation

This essay is on Wikiversity to encourage a wide discussion of the issues it raises moderated by the Wikimedia rules that invite contributors to “be bold but not reckless,” contributing revisions written from a neutral point of view, citing credible sources, and raising other questions and concerns on the associated '“Discuss”' page.

This article (i) describes efforts to model the time between the first test of a nuclear weapon by one nation and the next over the 74 years of history since the first such test by the US,[1] (ii) forecasts nuclear proliferation over the next 74 years with statistical error bounds quantifying the uncertainty, and (iii) reviews some of the geopolitical questions raised by this effort. Our modeling effort considers the possibility that the rate of nuclear proliferation may have slowed over time.

In brief, current international policy seems to imply that nuclear proliferation can be ignored. The analysis in this article of the statistical and non-statistical evidence suggests that nuclear proliferation is likely to continue unless (a) a nuclear war destroys everyone's ability to make more such weapons for a long time, or (b) an international movement has far more success than similar previous efforts in providing effective nonviolent recourse for grievances of the poor, weak and disfranchised.

Statistical details are provided in R Markdown vignettes on “Forecasting nuclear proliferation” and "GDPs of nuclear weapon states" in an appendix, below. Those vignettes should allow anyone capable of accessing the free and open-source software R and RStudio to replicate this analysis and modify it in any way they please to check the robustness of the conclusions.

## The data

The “nuclearWeaponStates” dataset[2] in the Ecdat package for R[3] was used for this study. Those data combine information from the “World Nuclear Weapon Stockpile” maintained by Ploughtshares,[4] the Wikipedia article on “List of states with nuclear weapons”, and multiple articles in the Bulletin of the Atomic Scientists. This includes the five states that officially had nuclear weapons when the United Nations Treaty on the Non-Proliferation of Nuclear Weapons (Non-Proliferation Treaty, NPT) entered into force in 1970 (the US, Russia, the UK, France and China) plus four others that first tested nuclear weapons since (India, Israel, Pakistan, and North Korea).

There seems to be a fairly broad consensus on the dates of the first tests of 8 of these 9 nuclear weapon states. Some reports claim that France and Israel had such close collaboration on nuclear weapons development in the late 1950s that the first test of a nuclear weapon by France on 1960-02-13 effectively created two nuclear-weapon states, not one.[5] The current study used the date of the 1979-09-22 Vela Incident for Israel. A 2019 report by Professor Avner Cohen, professor at the Middlebury Institute of International Studies, and the Director of the Education Program and Senior Fellow at the James Martin Center for Nonproliferation Studies, said that, “there is a scientific and historical consensus that [the Vela incident] was a nuclear test and that it had to be Israeli”,[6] conducted probably with South Africa. A robustness analysis could involve simply deleting Israel as a separate nuclear-weapon state.

## Plotting the time between the “first test” by one nuclear-weapon state and the next

Figure 1. Years between new nuclear-weapon states. CN = China, FR = France, GB = UK, IL = Israel, IN = India, KP = North Korea, PK = Pakistan, RU = Russia. NPT = Treaty on the Non-Proliferation of Nuclear Weapons (Non-Proliferation Treaty). INF = Intermediate-Range Nuclear Forces Treaty. The US is not on this plot, because it had no predecessors.

A plot of times between "first tests" by the world's nuclear-weapon states as of 2020-04-29 suggests that the process of nuclear proliferation has slowed; see Figure 1.

This plot also marks the effective dates of both the Treaty on the Non-Proliferation of Nuclear Weapons (Non-Prolireration Treaty, NPT) and the Intermediate-range Nuclear Forces (INF) Treaty (1970-03-05 and 1988-06-01, respectively), because of the suggestion that those treaties may have slowed the rate of nuclear proliferation.

A visual analysis of this plot suggests that nuclear proliferation is still alive and well, and neither the NPT nor the INF treaty impacted nuclear proliferation. The image is pretty bad: There were only 5 nuclear-weapon states when the NPT entered into force in 1970.[7] When US President George W. Bush decried an "Axis of evil" in his State of the Union message, 2002-01-29,[8] there were 8. As this is written 2020-04-21, there are 9.

Toon et al. (2007) noted that in 2003 another 32 had sufficient fissile material to make nuclear weapons if they wished. Moreover, those 32 do NOT include either Turkey nor Saudi Arabia. On 2019-09-04, Turkish President Erdogan said it was unacceptable for nuclear-armed states to forbid Turkey from acquiring its own nuclear weapons.[9]

Similarly, in 2006 Forbes reported that Saudi Arabia has "a secret underground city and dozens of underground silos for" Pakistani nuclear weapons and missiles.[10] In 2018 the Middle East Monitor reported that "Israel 'is selling nuclear information' to Saudi Arabia".[11] This is particularly disturbing, because of the substantial evidence that Saudi Arabia may have been and may still be the primary recruiter and funder of Islamic terrorism.[12]

This analysis suggests that the number of nuclear-weapon states will likely continue to grow until some dramatic break with the past makes further nuclear proliferation either effectively impossible or sufficiently undesirable.

This article first reviews the data and history on this issue. We then discuss modeling these data as a series of annual Poisson observations of the number of states conducting a first test of a nuclear weapon each year (1 in each of 8 years since 1945; 0 in the others).

A relatively simple model for the inhomogeneity visible in Figure 1 is Poisson regression assuming that log(Poisson mean) is linear in the time since the first test of a nuclear weapon by the US on 1945-07-16.[13] This model is plausible to the extent that this trend might represent a growing international awareness of the threat represented by nuclear weapons including a hypothesized increasing reluctance of existing nuclear-weapon states to share their technology. The current process of ratifying the new Treaty on the Prohibition of Nuclear Weapons supports the hypothesis of such a trend, while the lack of universal support for it and the trend visible in Figure 1 clearly indicate that nuclear proliferation is still likely to continue. We use this model to extend the 74 years of history of nuclear proliferation available as this is being written on 2020-04-21 into predicting another 74 years into the future.

## How did the existing nuclear-weapon states develop this capability?

There are, of course, multiple issues in nuclear proliferation: a new nuclear-weapon state requires at least four distinct things to produce a nuclear weapon: motivation, money, knowledge, and material. And many if not all of the existing nuclear-weapon states got foreign help, as outlined below and summarized in the accompanying table.

Disclaimer: Complete answers to each of these questions for every nuclear-weapon state can never be known with certainty. The literature found by the present authors is summarized in the accompanying table with citations to the literature in the following discussion but should not be considered any more authoritative than the sources cited, some of which may not be adequate to support all the details and the generalizations in the accompanying table.

However, this analysis should be sufficient to support the general conclusions of this article.

Country Motivation Money Knowledge Material Foreign Help
Who Why
US Nazi threat self own scientists + immigrants, esp. fr. Germany & Italy in collaboration with the UK and Canada. Congo + self GB (incl. Canada) Nazi threat
USSR (RU) Hiroshima & Nagasaki bombs + western invasions during WW II, after WW I, and before self own scientists + espionage in the US & captured Germans self US (espionage) US scientists wanted to protect USSR
UK (GB) USSR self Manhattan Project Canada ?
France (FR) USSR + Suez Crisis self self self ?
China (CN) 1st Taiwan Strait Crisis 1954–1955, the Korean Conflict, etc. self USSR self RU US threat
India (IN) loss of territory in the China-Himalayan border dispute-1962 self students in UK, US Canadian nuc reactor ?
Israel (IL) hostile neighbors self self + France France + ??? ?
Pakistan (PK) Loss of E. Pakistan in 1971 Saudis + self US, maybe China? self? US USSR in Afghanistan
CN ?
N.Korea (KP) threats fr. US self? US via Pakistan? self? PK +? ?

Table 1. Where did the existing nuclear-weapon states get the motivation, money, knowledge, and material for their nuclear-weapons program?

Figure 2. Gross Domestic Product (GDP) of nuclear-weapon states in billions of 2019 US dollars at Purchasing Power Parity (PPP) before (dashed line), during (thick solid line) and after (thinner solid line) their nuclear-weapons program leading to their first test of a nuclear weapon. (Country codes as with Figure 1.) The dotted line indicates the total cost of the Manhattan Project that developed the very first nuclear weapon from 1942 to the end of 1945.

To help us understand the differences in sizes of the different nuclear-weapon states, Figure 2 plots the evolution of GDP in the different nuclear-weapon states. The following subsections provide analysis with references behind the summaries in Table 1 and Figure 2.

### Motivation

Virtually any country that feels threatened would like to have some counterweight against aggression by a potential enemy.

• The US funded the Manhattan project believing that Nazi Germany likely had a similar project.
• India lost territory to China in the 1962 Sino-Indian War, which reportedly convinced India to abandon a policy of avoiding nuclear weapons.[19]
• Pakistan's nuclear weapons program began in 1972 in response to the loss of East Pakistan (now Bangledesh) in the 1971 Bangladesh Liberation War.[20] On November 29, 2016, Moeed Yusuf claimed that the threat of a nuclear war between India and Pakistan was the most serious foreign policy issue facing then-President-elect Trump.[21] That may have been an overstatement, but the possibilities of a nuclear war between India and Pakistan should not be underestimated. There have been lethal conflicts between India and Pakistan at least as recent as 2019. If that conflict goes nuclear, it could produce a “nuclear autumn” during which a quarter of humanity not directly impacted by the nuclear war would starve to death, according to simulations by leading climatologists.[22]
• Israel has faced potentially hostile neighbors since its declaration of independence in 1948.[23]
• North Korea first tested a nuclear weapon on 2006-10-09,[24] less than five years after having been named as part of an "Axis of evil" by US President George W. Bush on 2002-01-29.[25] Chomsky claimed that the relations between the US and North Korea have followed "a kind of tit-for-tat policy. You make a hostile gesture, and we'll respond with a crazy gesture of our own. You make an accommodating gesture, and we'll reciprocate in some way." He gave several examples including a 1994 agreement that halted North Korean nuclear-weapons development. "When George W. Bush came into office, North Korea had maybe one [untested] nuclear weapon and verifiably wasn't producing any more."[26]

All this suggests that it will be difficult to reduce the threat of nuclear proliferation and nuclear war without somehow changing the nature of international relations so weaker countries have less to fear from the demands of stronger countries.

### Money

It's no accident that most of the world's nuclear-weapon states are large countries with substantial populations and economies. That's not true of Israel with only roughly 9 million people nor North Korea with roughly 26 million people in 2018. France and the UK have only about 67 and 68 million people, but they are also among the world leaders in the size of their economies.

Pakistan is a relatively poor country. It reportedly received financial assistance from Saudi Arabia for its nuclear program.[27]

Another reason for a possible decline in the rate of nuclear proliferation apparent in Figure 1 is the fact that among nuclear-weapon states, those with higher GDPs tended to acquire this capability earlier, as is evident in Figure 2.

### Knowledge

In 1976, John Aristotle Phillips, an "underachieving" undergraduate at Princeton University, "designed a nuclear weapon using publicly available books and papers."[28] Nuclear weapons experts disagreed on whether the design would have worked. Whether Phillips' design would have worked or not, it should be clear that the continuing progress in human understanding of nuclear physics inevitably makes it easier for people interested in making such weapons to acquire the knowledge of how to do so.

Before that, the nuclear age arguably began with the 1896 discovery of radioactivity by the French scientist Henri Becquerel. It was further developed by Pierre and Marie Curie in France, Ernest Rutherford in England, and others, especially in France, England and Germany.[29] In 1933 after Adolf Hitler came to power in Germany, Leo Szilard moved from Germany to England. The next year he patented the idea of a nuclear fission reactor. Other leading nuclear scientists similarly left Germany and Italy for the UK and the US. After World War II began, the famous Manhattan Project became a joint British-American project, which produced the very first test of a nuclear weapon.[30]

After Soviet premier Joseph Stalin learned of the atomic bombings of Hiroshima and Nagasaki, the USSR (now Russia) increased the funding for their nuclear-weapons program. That program was helped by intelligence gathering about the German nuclear weapon project and the American Manhattan Project.[31]

The UK's nuclear-weapons program was built in part on their wartime participation in the Manhattan Project, as noted above.

France was among the leaders in nuclear research until World War II. They still had people with the expertise needed after the 1956 Suez Crisis convinced them they needed to build nuclear bombs, as noted above.[32]

China got some help from the Soviet Union during the initial phases of their nuclear program.[33]

The first country to get nuclear weapons after the Non-Proliferation Treaty was India. Their Atomic Energy Commission was founded in 1948, chaired by Homi J. Bhabha. He had published important research in nuclear physics while a graduate student in England in the 1930s, working with some of the leading nuclear physicists of that day.[34]

Meanwhile, Israel's nuclear weapons program initially included sending students abroad to study under leading physicists like Enrico Fermi at the University of Chicago. It also included extensive collaboration with the French nuclear-weapons program.[35]

Pakistan got "dual use" production technology and complete nuclear-capable delivery systems from both the US and China.[36] Pakistan got secret help from the US in the 1980s in violation of US law to secure Pakistani cooperation with US support for anti-Soviet resistance in Afghanistan.[37] (In 1995 the Wisconsin Center on Nuclear Arms Control reported that Pakistan’s most reliable nuclear delivery platforms were French-made Mirage fighters,[38] though they also had US-made F-16s they could modify to carry those weapons.)

Abdul Qadeer Khan, a leader in Pakistan's nuclear weapons program, has also faced multiple allegations of being one of the world's leading nuclear proliferators in operating a black market in nuclear weapons technology. North Korea, Iran and other countries have allegedly received help from Pakistan for their nuclear weapons programs with at least some of it coming via A. Q. Khan's black market dealings.[39] Some of this technology was reportedly obtained from the US in the 1980s with the complicity of US government officials who wanted Pakistan's help for groups in Afghanistan fighting the Soviets.[40]

Vikram Sood, a former head of India's foreign intelligence agency, said, "America fails the IQ test" in discussing A. Q. Khan's nuclear black market, adding that Pakistan may have given nuclear-weapons technology to al Qaeda "just weeks prior to September 11, 2001."[41] It may not be wise to accept Sood's claim at face value, given the long-standing hostility between India and Pakistan. In April 2002 Milhollin, Founder and then Executive Director of the Wisconsin Project on Nuclear Arms Control, said that Al Qaeda "is interested in getting weapons of mass destruction, [and if it] can organize a 19-person group to fly airliners into buildings, it can smuggle a nuclear weapon across a border."[42] In 2005 Robert Gallucci, a leading researcher and expert on nuclear proliferation who served in high level positions in the Reagan, G. H. W. Bush and Clinton administrations because of this expertise, wrote that there was an unacceptably high probability "that Al Qaeda or one of its affiliates will detonate a nuclear weapon in a US city ... . The loss of life will be measured ... in the hundreds of thousands. ... Consider the more likely scenarios ... . An Al Qaeda cell ... purchases 50 or so kilograms of highly enriched uranium. Today, the sellers might be Pakistan or Russia; tomorrow they might be North Korea or Iran. ... Another scenario ... involves the acquisition ... of a completed nuclear weapon."[43]

And the US is helping Saudi Arabia obtain nuclear power, in spite of (a) the evidence that the Saudi government including members of the Saudi royal family were involved at least as early as 1999 in preparations for the suicide mass murders of September 11, 2001,[44] and (b) their on-going support for Al Qaeda in Yemen, reported as recently as 2018.[45]

### Material

Reportedly the most difficult part of making nuclear weapons today is obtaining sufficient fissile material. Toon et al. (2007) said, "Thirteen countries operate plutonium and/or uranium enrichment facilities, including Iran", but Iran did not have sufficient fissile material in 2003 to make a nuclear weapon. Another 20 were estimated to have had sufficient stockpiles of fissile material acquired elsewhere to make nuclear weapons. They concluded that 32 (being 13 minus 1 plus 20) additional countries have sufficient fissile material to make nuclear weapons if they want.[46]

Toon et al. (2007) also said, "In 1992 the International Atomic Energy Agency safeguarded less than 1% of the world’s HEU [Highly Enriched Uranium] and only about 35% of the world inventory of Pu [Plutonium] ... . Today [in 2007] a similarly small fraction is safeguarded."

HEU is obtained by separating 235U, which is only 0.72 percent of naturally occurring uranium.[47] Weapons-grade uranium has at least 85 percent 235U.[48] Thus, at least 0.85/0.0072 = 118 kg of naturally occurring uranium are required to obtain 1 kg that is weapons-grade. Toon et al. (2007) estimated that 25 kg of HEU would be used on average for each 235U-based nuclear weapon. Plutonium, by contrast, is a byproduct of energy production in standard 238U nuclear reactors.

Much of the uranium for the very first test of a nuclear weapon by the US came from the Congo,[49] but domestic sources provided most of the uranium for later US nuclear-weapons production.[50] The Soviet Union (USSR, now Russia) also seems to have had adequate domestic sources for its nuclear-weapons program, especially including Kazakhstan, which was part of the USSR until 1990; Kazakhstan has historically been the third largest source of uranium worldwide after Canada and the US.[49] The UK presumably got most of its uranium from Canada.

The French nuclear-weapons program seems to have been built primarily on plutonium.[51] This required them to first build standard 238U nuclear reactors to make the plutonium. Then they didn't need nearly as much uranium to sustain their program.

China has reportedly had sufficient domestic reserves of uranium to support its own needs,[49] even exporting some to the USSR in the 1950s in exchange for other assistance with their nuclear defense program.[52]

India's nuclear weapons program seems to have been entirely (or almost entirely) based on plutonium.[53]

Israel seems not to have had sufficient uranium deposits to meet its own needs. Instead, they purchased some from France until France ended their nuclear-weapons collaboration with Israel in the 1960s. To minimize the amount of uranium needed, nearly all Israeli nuclear weapons seem to be plutonium bombs.[54]

It's not clear where Pakistan got most of its uranium: Its reserves in 2015 were estimated at zero, and its historical production to that point was relatively low.[49] By comparison with the first seven nuclear-weapon states, it's not clear where Pakistan might have gotten enough uranium to produce 83 plutonium bombs and 44 uranium bombs, as estimated by Toon et al. (2007, Table 2, p. 1976.) As previously noted, the US helped the Pakistani nuclear-weapons program in the 1980s and accused China of providing similar assistance, a charge that China has repeatedly and vigorously denied. China has provided civilian nuclear reactors, which could help produce plutonium but not 235U.[55]

According to the Federation of American Scientists, "North Korea maintains uranium mines with an estimated four million tons of exploitable high-quality uranium ore ... that ... contains approximately 0.8% extractable uranium."[56] If that's accurate, processing all that would produce 4,000,000 times 0.008 = 32,000 tons of pure natural uranium, which should be enough to produce the weapons they have today.

### Conclusions regarding motivation, money, knowledge, and material

1. There seems to be no shortage of motivations for other countries to acquire nuclear weapons. The leaders of the Soviet Union had personal memories of being invaded not only by Germany during World War II but also by the US and others after World War I. The UK had reason to fear the Soviets in their occupation of Eastern Europe. The French decided after Suez they couldn't trust the US to defend them. China had been forced to yield to nuclear threats before starting their nuclear program, as did India, Pakistan and North Korea. Israel has fought multiple wars since their independence in 1948.

2. The knowledge and material required to make such weapons in a relatively short order are also fairly widely available, even without the documented willingness of current nuclear powers to secretly help other countries acquire such weapons in some cases.[57]

3. Unless there is some fundamental change in the structure of international relations, it seems unwise to assume that there will not be more nuclear-weapon states in the future, with the time to the next "first test" of a nuclear weapon following a probability distribution consistent with the previous times between "first tests" of nuclear weapons by the current nuclear-weapon states.

## Distribution of the time between Poisson “first tests”

Possibly the simplest model for something like the time between "first tests" in an application like this is to assume they come from one exponential distribution with 8 observed times between the 9 current nuclear-weapon states plus one censored observation of the time between the most recent one and a presumed next one. This simple theory tells us that the maximum likelihood estimate of the mean time between such "first tests" is the total time from the US "Trinity" test to the present, 74.8 years, divided by the number of new nuclear-weapon states, 8, not counting the first, which had no predecessors. Conclusion: Mean time between "first tests" = 9.3 years.[58]

However, Figure 1 suggests that the time between "first tests" of succeeding nuclear-weapon states is increasing. The decreasing hazard suggested by this figure requires mathematics that are not as easy as the censored data estimation as just described.

Figure 3. Semilog plot of the years between new nuclear-weapon states. (Country codes as with Figure 1.)

To understand the current data better, we redo Figure 1 with a log scale on the y axis in Figure 3.

Figures 1 and 3 seem consistent with the following:

• If the mean time between "first tests" is increasing over time, as suggested by Figures 1 and 3, then the distribution cannot be exponential, because that requires a constant hazard rate.[59]
• Even though nuclear proliferation has been slowing since 1950, it seems not to have slowed fast enough to support the assumption that nuclear proliferation can be ignored, which seems to be implied by current international policy. It could accelerate in the future if more states began to perceive greater threats from other nations.
• Fortunately we can simplify this modeling problem by using the famous duality between exponential time between events and a Poisson distribution for numbers of events in specific intervals of time. By modeling Poisson counts of "first tests" each year, we can use techniques for Poisson regression for models suggested by Figure 3. The simplest such model might consider log(Poisson mean numbers of "first tests" each year) to be linear in the time since the first test of a nuclear weapon (code-named "Trinity").[60]
• The image in Figure 3 suggests the time between “first tests” by new nuclear-weapon states may be increasing, but not necessarily liearly. Easily tested alternatives to linearity could be second, third and fourth powers of the "timeSinceTrinity".[61]

We used Poisson regression to model this as a series of the number of events each year.[62]

## Parameter estimation

For modeling and parameter estimation, we model the number of “first tests” of a new nuclear-weapon state each year (1 in 8 years, 0 in the remaining 66 years between 1945 and 2019) with log(Poisson mean number of “first tests” each year) as polynomials in “timeSinceTrinity” = the time in years since the Trinity test by the US, 1945-07-16. The standard p-value for the Wald test of the linear model was 0.21 -- not statistically significant.

George Box famously said that, "All models are wrong, but some are useful.".[63]

Burnham and Anderson (1998) and others claim that better predictions can generally be obtained using Bayesian Model Averaging (BMA).[64] In this case, we have two models: log(Poisson mean) being constant or linear in “timeSinceTrinity”. It is standard in the BMA literature to assume a priori an approximate uniform distribution over all models considered with a penalty for estimating each additional parameter to correct for the tendency of the models to overfit the data. With these standard assumptions, this comparison of these two models estimated a 21 percent posterior posterior probability for the model linear in “timeSinceTrinity”, leaving 79 percent probability for the model with a constant Poisson mean.

Figure 4. BMA constant-linear and quartic fits to time between new nuclear-weapon states. (Country codes as with Figure 1.)

We also experimented with fitting up to quartic models in “timeSinceTrinity”.[65] These prediction lines were added to Figure 3 to produce Figure 4.

Comparing predictions between the constant-linear and constant-quartic mixtures might help us understand better the limits of what we can learn from the available data. A visual analysis of the right (quartic mixture) panel in Figure 4 makes one wonder if the quartic, cubic and quadratic fits are really almost as good as the linear, as suggested by minor differences in the posterior probabilities estimated by the algorithm used.

However, the forecasts of nuclear proliferation will be dominated by the constant component of the BMA mixture. Its posterior probability is 79 percent for the constant-linear mixture and 48.59 percent for the quartic mixture. That means that the median line and all the lower quantiles of all simulated futures based on these models would be dominated by that constant term.

Moreover, the quartic, cubic and quintic lines in the right (quartic mixture) panel of Figure 4 do not look nearly as plausible, at least to the present author, as the constant and linear lines.[66] That, in turn, suggests that the constant linear mixture may be more plausible than the quartic mixture.

We then used Monte Carlo simulations with 5,000 random samples to compute central 60 and 80 percent confidence limits for the mean plus 80 percent prediction, and (0.8, 0.8) tolerance limits for future nuclear proliferation, as discussed in the next three sections of this article.[67]

## Confidence limits

Figure 5. Confidence limits for the mean time between “first tests,” past and future per the constant-linear mixture. (Country codes as with Figure 1.)

To get confidence limits, we simulated 5,000 Poisson mean numbers of "first tests" by new nuclear-weapon states for each of the 74 years used in the two BMA fits and another 74 years beyond. These simulations were later used to compute confidence limits for the model estimates of the Poisson mean and prediction and tolerance limits for the actual number of nuclear-weapon states.[67] First, however, we inverted the simulated Poisson means to get simulated exponential times, then summarized them to get simulated mean, median, and 60 and 80 percent confidence limits of the mean time to the next new nuclear weapon state. We then added those simulation summary statistics from the constant-linear model in Figure 3 to produce Figure 5.

The fairly flat shape of the median and lower 10 and 20 percent lines in Figure 5 seem consistent with a model that is a sum of a mixture of log-normal distributions with the dominant component having a posterior probability of either 79 or 48.59 percent and a constant mean, as noted in Figure 4. The substantial curvature of the solid line forecast looks hopeful, with a mean of simulated means for the constant-linear mixture being almost 200 years between successive "first tests" by new nuclear-weapon states by the end of the forecasted period, 2093.

The fact that the mean of the simulations exceeds the upper confidence limit for 2093 seems odd but can be explained by noting that this is a mixture of log-normal distributions, and the mean of a log-normal can exceed any quantile of its distribution if the standard deviation is sufficiently large.[68]

Figure 6: Confidence limits for the mean time between “first tests,” past and future per the quartic mixture. (Country codes as with Figure 1.)

Note further that the distribution for each year in Figure 5 is a mixture of log-normal distributions, which means that their reciprocals, the mean numbers of "first tests" each year, will also be a mixture of log-normals with the same standard deviations on the log scale. This standard deviation is larger the farther we extrapolate into the future.

The increase over time in the mean time between "first tests" in Figures 5 and 6 suggests a desirable decrease in the rate of nuclear proliferation.

However, we are more concerned with the shorter times between "first tests", and they seem all too probable, as we shall see when we simulate and compute their cumulative sums. To do that, we append these simulated predictions to a plot of the evolution of the number of nuclear-weapon states through the historical period.[69]

Figure 7: Number of nuclear-weapon states, past and predicted mean; BMA constant-linear model on the left; quartic fit on the right.

These numbers are plotted in Figure 7 for both BMA models considered. The slope of the median lines are steeper than the recent history, but the statistical evidence does not support the naive interpretation of a slowing in nuclear proliferation that one might get from considering only the most recent data.

Comparing the forecasts between the constant-linear and quartic BMA mixtures shows that the higher order quartic mixture widens the confidence limits, making the 20th percentile essentially flat with almost no additional nuclear proliferation, while the mean quickly escapes the upper limit. That sharply rising mean suggests that less than 10 percent of the simulations predict nuclear arms races that involve many nation states and many more non-state armed groups. These outcomes are not likely, but the probabilities of such outcomes seem too large to be dismissed without further consideration, especially when gambling with the future of civilization. (Replications of these simulations with different sets of random numbers confirmed the stability of the images in Figure 7.)

Ignoring the simulations of uncontrolled nuclear arms races, the median lines in Figure 7 predict between 16.3 and 14.5 at the end of the current simulated period, 2093, adding either 7.3 and 5.5 (for the constant-linear and quartic mixtures, respectively) to the current 9 nuclear-weapon states. Those median numbers are a little less than double the number of nuclear-weapon states today.

We extend this analysis by adding prediction intervals to these plots.

## Prediction limits

Figure 8: Number of nuclear-weapon states, past and predicted; BMA constant-linear model on the left; quartic fit on the right.

The simplest bounds on the future are prediction intervals, which combine the statistical uncertainty in the estimates of mean numbers of nuclear-weapon states with the random variability in the outcomes. We simulated 80 percent equal-tailed prediction limits and added them to Figure 7 to produce Figure 8.

For both Bayesian mixture models, the most likely scenarios, especially the median line and the space between the 60 percent confidence limits, predict a continuation of nuclear proliferation. It's difficult to imagine how that could continue without also substantively increasing the risk of nuclear war and therefore also of the extinction of civilization.

Figure 9: Probabilities of the time to the next 5 new nuclear-weapon states using the constant-linear and quartic BMA models (left and right panels, respectively).

We can also summarize the simulations to estimate the probabilities of having 1, 2, 3, 4, and 5 new nuclear weapon states for each year in the prediction period between 2020 and 2093 in Figure 9. This is another way of evaluating the sensibility of pretending there will be no further nuclear proliferation: Not likely.

Ninety-four percent of the simulations per the constant-linear model had at least one more nuclear-weapon state by 2093 and a 40 percent chance of at least 1 by 2025. The quartic model predicts a 73 percent chance of at least one more nuclear-weapon state by 2093 and a 29 percent chance of at least one by 2024.

The conclusions from both models include the following:

The current structure of international relations
seems to threaten the extinction of civilization.

To better quantify the uncertainty in modeling, we can also construct tolerance intervals for the time to the next new nuclear-weapon state.

## Tolerance limits

Figure 10: Number of nuclear-weapon states with prediction and tolerance limits; BMA constant-linear model on the left; quartic fit on the right.

We want to add statistical tolerance limits to Figure 8 in addition to the prediction limits. To do this, we add Poisson simulations to the 80 percent confidence limits in Figure 7 rather than adding Poisson simulations to all the individual simulations summarized in Figure 7 to produce Figure 8. The results appear in Figure 10.

The upper limit lines in Figure 10 are higher than those in Figure 8. It gives us a bit more humility regarding the value of current knowledge. However, the difference is not enough to substantively alter our conclusions, namely that nuclear proliferation is likely until something makes it impossible for anyone to make more nuclear weapons for a very long time.

## Discussion

A growing number of leading figures have said that as long as the world maintains large nuclear arsenals, it is only a matter of time before there is a nuclear war. Concerns like this have been expressed by two former US Secretaries of Defense (Robert McNamara[70] and William Perry, two former US Secretaries of State Henry Kissinger and George Schultz, former US Senator Sam Nunn[71] and others with, for example, the Nuclear Threat Initiative. Daniel Ellsberg has said that a nuclear war will most likely generate a nuclear winter that lasts several years during which 98 percent of humanity will starve to death if they do not die of something else sooner.[72]

Banerjee and Duflo, two of the three who won the 2019 Nobel Memorial Prize in Economics, have noted that neither economic nor political stability are assured for any country, including the United States, China and India. In particular, they predict that economic growth will almost certainly slow substantially in the latter two, leaving many poor people in desperate economic straits.[73] Internal problems in the US, China, India or any other nuclear-weapon state could push political leaders to pursue increasingly risky foreign adventures, like Argentina did in 1982,[74] possibly leading to a war that could produce nuclear Armageddon.[75]

The evidence compiled in the present work only seems to increase the urgency of limiting the threat of nuclear war and nuclear proliferation in particular.

In the 20 years following the first test of a nuclear weapon on 1945-07-16 by the US, four more nations acquired such weapons. In the 50 years since the Non-Proliferation Treaty took effect in 1970, another four acquired them.[76] Our analysis of the available data considering only the dates of these first tests suggests that nuclear proliferation may have been slowing throughout this period. However, that apparent trend was not statistically significant in the model we fit.

Bayesian Model Averages (BMA) is known to generally produce better predictions than single model fits. Accordingly, we've estimated confidence, prediction, and tolerance limits for the number of new nuclear-weapon states 74 years into the future based on two BMA models with mixtures of either a constant with a linear model or a constant with terms up to quartic in the time since the very first test of a nuclear weapon.

We can expect that some non-nuclear nations and terrorist groups would eagerly pursue nuclear weapons if such seemed feasible unless some unprecedented change in international law provided them with effective nonviolent recourse to perceived threats.

Moreover, these weapons will likely become more available with the passage of time unless (a) a nuclear war destroys everyone's ability to make more such weapons for a long time, or (b) there is a major change in the structure of international relations that has far more success than similar previous efforts in limiting the ability of new nations and non-state actors to acquire nuclear weapons.

## Monitoring nuclear proliferation

Organizations like the Wisconsin Project on Nuclear Arms Control, the Federation of American Scientists, the Stockholm International Peace Research Institute, and other similar organizations seem to have made substantive contributions to the apparent reduction in the rate of nuclear proliferation visible in most of the plots included in this article.

In 2017 the Nuclear Verification Capabilities Independent Task Force of the Federation of American Scientists published seven recommendations for improving the process of nuclear monitoring and verification:[77]

1. A network of 4-5 independent Centers of Nonproliferation Authentication
2. The P5+1 and Iran should publicize important implementation steps.
3. Periodic public updates on monitoring & U.S. support to the IAEA.
4. The P5+1 and others should encourage Iranian openness.
5. A trusted body of outside experts should be created to monitor the Iranian nuclear agreement.
6. NGOs concerned with nonproliferation should aggressively protect both the information and the physical safety of its sources.
7. Funders of nonproliferation NGOs should strengthen cyber security.

These seven recommendations seem likely to contribute to the trend towards a reduction in the rate of nuclear proliferation visible in many of the figures included in this article.

However, the Federation of American Scientists was founded in 1946, and only one of the current nuclear-weapon states had such weapons before they were founded. When Stockholm International Peace Research Institute was founded in 1966, there were five nuclear-weapon states. When the Wisconsin Project on Nuclear Arms Control was founded in 1986, there were seven. Two more nations have joined the list of nuclear-weapon states since this Wisconsin Project was founded. Even if these seven recommendations are fully implemented, it seems unlikely that those actions by themselves will end nuclear proliferation. We can hope that they will contribute slowing the rate of nuclear proliferation already implicitly considered in the model fit and forecasts discussed above. Sadly the recent actions by the US and Russia in embarking on major "modernization" programs seem to be cause for concern.

## The Trump administration and nuclear weapons

Several actions of the Trump administration have raised concern about a new arms race, escalating bellecosity of the US in international relations and even possibly accelerating the threat of further nuclear proliferation.

• Terminated the 1987-88 Intermediate-Range Nuclear Forces Treaty alleging both Russian non-compliance and concerns about the continuing growth of China's missile forces. This was announced 20 October 2018 and completed on 1 February 2019.
• On May 21, 2020, President Trump announced that the United States would withdraw from the 2002 Treaty on Open Skies, alleging Russian violations.[78]
• The Trump administration has expressed a desire to build smaller more "usable" nukes. The use of such weapons by themselves seem less likely to produce a nuclear winter or autumn but could increase the chances of a full scale nuclear war using the larger weapons that would more likely produce nuclear Armageddon.

## Conclusions

It seems likely that nuclear proliferation will continue until an international movement has far more success than similar previous efforts in ending it. The seven recommendations of the Wisconsin Project on Nuclear Arms Control mentioned above may or may not slow nuclear proliferation enough to prevent nuclear Armageddon destroying civilization, dramatically shorten the lives of nearly all humans on earth.

Might it be possible to energize existing organizations concerned about nuclear proliferation to the point that they have unprecedented success in achieving nearly complete nuclear disarmament and in strengthening international law so the poor, weak and disfranchised have effective nonviolent means for pursuing a redress of grievances?

## Appendix. Companion R Markdown vignettes

Statistical details that make the research in article reproducible are provided in two R Markdown vignettes on "Forecasting nuclear proliferation" and "GDPs of nuclear-weapon states":

## References

• Kenneth P. Burnham; David R. Anderson (1998), Model Selection and Multimodel Inference: A practical information-theoretic approach, Wikidata Q62670082.

## Notes

1. This is being written on 2020-04-26. For the purposes of the present analysis, this is considered to be 74 years since the first test of a nuclear weapon on 1945-07-16.
2. Spencer Graves (20 March 2020), "nuclearWeaponStates: Nations with nuclear weapons", Ecdat: Data Sets for Econometrics, Wikidata Q88894684
3. Yves Croissant; Spencer Graves (20 March 2020), Ecdat: Data Sets for Econometrics, Wikidata Q56452356
4. World nuclear weapon stockpile, Ploughshares Fund, Wikidata Q63197617
5. Warner D. Farr (September 1999), The Third Temple's Holy of Holies: Israel's Nuclear Weapons, Air War College, Wikidata Q88922617
6. Ofer Aderet (22 September 2019), "U.S. Covered Up an Israeli Nuclear Test in 1979, Foreign Policy Says", Haaretz (in English and Hebrew), Wikidata Q88921529
7. Smlouva o nešíření jaderných zbraní (in English, Russian, French, and Spanish), Wikidata Q91335914. See also reaty on the Non-Proliferation of Nuclear Weapons.
8. Ece Toksabay (4 September 2019), "Erdogan says it's unacceptable that Turkey can't have nuclear weapons", Reuters, Wikidata Q91338524; Tom O'Connor (4 September 2019), "Turkey has U.S. nuclear weapons, Now it says it should be allowed to have some of its own", Newsweek, ISSN 0028-9604CS1 maint: multiple names: authors list (link), Wikidata Q91342138 .
9. "AFX News Limited: "Saudia Arabia working on secret nuclear program with Pakistan help - report"", Forbes (in English, Russian, Georgian, Polish, Korean, Portuguese, Japanese, Chinese, and Estonian), 28 March 2006, ISSN 0015-6914, Wikidata Q91342270; see also w:Nuclear program of Saudi Arabia.
10. "Israel 'is selling nuclear information' to Saudi Arabia", Middle East Monitor, 31 May 2018, Wikidata Q91343477; see also w:Nuclear program of Saudi Arabia.
11. Medea Benjamin (2016), Kingdom of the Unjust: Behind the US Saudi connection, OR Books, Wikidata Q55616039; see also Winning the War on Terror.
12. A vignette on “Forecasting nuclear proliferation” describes fitting such models to the available data in a way that allows anyone able to run the free and open-source software R (programming language) to reproduce the analysis outlined in this article and experiment with alternatives: Spencer Graves; Douglas A. Samuelson (2020), "Forecasting nuclear proliferation" (PDF), Ecfun: Functions for Ecdat, Wikidata Q89780728.
13. David S. Fogelsong (1995), America's Secret War against Bolshevism: U.S. Intervention in the Russian Civil War, 1917-1920, University of North Carolina Press, Wikidata Q91370284. That doesn't count French invasion of Russia|numerous other invasions that are a sordid part of Russian history, which educated Russians throughout history would likely remember, even if their invaders may not.
14. The UK and France would have had many reasons to fear the intentions of the USSR during the early period of the Cold War: The first test of a nuclear weapon by the USSR came just over three months after the end of the 1948-49 Berlin Blockade. Other aspects of Soviet repression in countries they occupied in Eastern Europe contributed to the failed Hungarian Revolution of 1956.
15. David Fromkin (28 October 2006), "Stuck in the Canal", The New York Times, ISSN 0362-4331, Wikidata Q91382112. See also w:France and weapons of mass destruction.
16. w:China and weapons of mass destruction#Nuclear weapons; see also w:First Taiwan Strait Crisis, Morton Halperin (December 1966), "The 1958 Taiwan Straits Crisis: A documentary history" (PDF), RAND Corporation Research Memoranda (RM-4900-ISA), Wikidata Q63874609, and w:Daniel Ellsberg.
17. Charles Pierson (8 September 2017), "The Atomic Bomb and the First Korean War", CounterPunch, ISSN 1086-2323, Wikidata Q63874136. See also w:Daniel Ellsberg.
18. Bruce Riedel (28 June 2012), "JFK's Overshadowed Crisis", The National Interest, ISSN 0884-9382, Wikidata Q91391545. See also India and weapons of mass destruction. India and China have continued to have conflicts. See, for example, the Wikipedia articles on w:China-India relations and the w:2017 China-India border standoff.
19. Pakistan and weapons of mass destruction. India-Pakistan relations have been marked by frequent conflict since the two nations were born with the dissolution of the British Raj in 1947. This history might help people understand the need that Pakistani leaders may have felt and still feel for nuclear parity with India, beyond the loss of half their population and 15 percent of their land area in the 1971 Bangladesh Liberation War.
20. Moeed Yusuf (26 November 2016), "An India-Pakistan Crisis: Should we care?", War on the Rocks, Wikidata Q91271615.
21. Toon et al. (2007)
22. Arab-Israeli conflict. Threats perceived by Israel continue, including the 2018 Gaza border protests that have continued at least into 2020. One might therefore reasonably understand why Israel might feel a need for nuclear weapons and why others might believe that the 1979-09-22 Vela incident was an Israeli nuclear test.
23. Comprehensive Nuclear-Test-Ban Treaty: Background and Current Developments, 1 September 2016, Wikidata Q59596578; The US Congressional Research Service in 2016 reported, “The Comprehensive Nuclear-Test-Ban Treaty Organization (CTBTO) PrepCom's international monitoring system detected data indicating that North Korea had conducted a nuclear test on January 6, 2016. ... On October 9, 2006, North Korea declared that it had conducted an underground nuclear test.” For the present purposes, we use the October date declared by North Korea, not the January date reported by CTBTO. See also 2006 North Korean nuclear test.
24. Noam Chomsky (May 2016), Who Rules the World?, Picador, OL 20029949W, Wikidata Q86247233, pp. 131-134. Chomsky includes in this game of tit-for-tat the total destruction of North Korean infrastructure during the Korean War in the early 1950s, including huge dams that controlled the nation's water supply, destroying their crops, and raising the spectre of mass starvation. Gabriel Kolko (1968), "Report on the destruction of dikes: Holland 1944-45 and Korea 1953", Against the crime of silence: Proceedings of the Russell International War Crimes Tribunal, Wikidata Q91455702 noted that German General Syss-Inquart ordered similar destruction of dikes in Holland in 1945, which condemned many Dutch civilians to death by starvation. For that crime Syss-Inquart became one of only 24 of the people convicted at the Nurenberg war crimes trial to have been sentencted to death. Chomsky noted that this is "not in our memory bank, but it's in theirs."
25. Bruce Riedel (28 January 2008), Saudi Arabia: Nervously Watching Pakistan, Brookings Institution, Wikidata Q84288832.
26. "Student Plans Complete Nuclear Bomb", Spokane Daily Chronicle, 9 October 1976, Wikidata Q91459264. See also w:John Aristotle Phillips.
27. Espionage and the Manhattan Project (1940-1945), Office of Scientific and Technical Information, Wikidata Q91461780. See also Soviet atomic bomb project.
29. w:Nuclear weapons and Israel. See also "Israel’s Nuclear Weapon Capability: An Overview", July 1, 1996, by the Wisconsin Project on Nuclear Arms Control.
30. For Chinese help to Pakistan, see Matthew Godsey; Valerie Lincy (2019), "Gradual Signs of Change: Proliferation to and from China over Four Decades" (PDF), Strategic Trade Review, 5 (8): 3–21, ISSN 2406-5269, Wikidata Q95917195.
31. William Burr (4 May 2012), New Documents Spotlight Reagan-era Tensions over Pakistani Nuclear Program, Woodrow Wilson International Center for Scholars, Wikidata Q91463994. William Burr (22 November 2013), Pakistan's Illegal Nuclear Procurement Exposed in 1987: Arrest of Arshed Pervez Sparked Reagan Administration Debate over Sanctions, National Security Archive, Wikidata Q91464530. See also w:Pakistan and weapons of mass destruction.
32. Pakistan: American, Chinese or French Planes Would Deliver its Bomb, Wisconsin Project on Nuclear Arms Control, 1 October 1995, Wikidata Q95919096.
33. A summary of this appears in "Pakistan Nuclear Milestones, 1955-2009" by the Wisconsin Project on Nuclear Arms Control. See also citations on this in the Wikipedia article on w:Abdul Qadeer Khan.
34. E.g., Lyndsey Layton (7 July 2007), Whistle-Blower's Fight For Pension Drags On, The Washington Post, Wikidata Q88306915, and Richard Barlow.
35. Vikram Sood (17 January 2008), "America fails the IQ test", Hindustan Times, ISSN 0972-0243, Wikidata Q88310866
36. Gary Milhollin (15 April 2002), Use of Export Controls to Stop Proliferation, Wisconsin Project on Nuclear Arms Control, Wikidata Q95987528.
37. Gallucci's estimate of the probability of a nuclear attack by a terrorist group has declined substantially since 2005. Back then, he wrote that a terrorist attack with a nuclear weapon in the next five to ten years "is more likely than not". In a private communication on June 4, 2020, he wrote, "I was wrong in my estimate [that such an attack was more likely than not], and glad that I was. I don't understand AQ to be the threat now that it was fifteen years ago, but my concern continues that it is principally the unavailability of fissile material that prevents a terrorist from constructing an improvised nuclear device." The quote from 2005 is available in Robert Gallucci (2005), "Averting Nuclear Catastrophe: Contemplating Extreme Responses to U.S. Vulnerability", Harvard International Review, 26 (4): 84, 83, ISSN 0739-1854, Wikidata Q96062427. Essentially this same quote appears in a longer article by the same name: R. L. Gallucci (1 September 2006), "Averting Nuclear Catastrophe: Contemplating Extreme Responses to U.S. Vulnerability", Annals of the American Academy of Political and Social Science, 607 (1): 51–58, doi:10.1177/0002716206290457, ISSN 0002-7162, Wikidata Q29395474.
38. Bob Graham; Porter Goss; Richard Shelby; Nancy Pelosi (29 January 2003), Joint Inquiry into Intelligence Community Activities before and after the Terrorist Attacks of September 11, 2001 (PDF), United States House Permanent Select Committee on Intelligence, Wikidata Q1702537. See also The 28 pages, which were redacted from the official report published 2003-01-29 and declassified in July 2016 by then-President Obama.
39. Maggie Michael; Trish Wilson; Lee Keath (6 August 2018), "AP Investigation: US allies, al-Qaida battle rebels in Yemen", Associated Press, Wikidata Q61890713.
40. pp. 1975, 1977. The 32 countries they identified included 12 of the 13 that "operate plutonium and/or uranium enrichment facilities", excepting Iran as noted. The other 20 countries acquired stockpiles elsewhere. In addition to the 32 with sufficient fissile material to make a nuclear weapon, Egypt, Iraq and the former Yugoslavia were listed as having abandoned a nuclear-weapons program.
41. See the section on “Highly enriched uranium (HEU)” in the Wikipedia article on w:Enriched uranium.
42. France and weapons of mass destruction. See also Table 2 in Toon et al. (2007), which claims that in 2003, France had enough fissile material for roughly 24,000 plutonium bombs and 1,350 235U bombs.
43. w:India and weapons of mass destruction; see also Toon et al. (2007) and w:List of countries by uranium reserves.
44. Toon et al. (2007).
45. Steven Aftergood; Hans M. Kristensen (16 November 2006), DPRK: Nuclear Weapons Program, Federation of American Scientists, Wikidata Q91520731. See also w:North Korea and weapons of mass destruction.
46. In addition to the 32 currently non-nuclear-weapon states with "sufficient fissile material to make nuclear weapons if they wished", per Toon et al. (2007), the inspector general of the US Department of Energy concluded in 2009 (in its most recent public accounting) that enough highly enriched uranium was missing from US inventories to make at least five nuclear bombs comparable to those that destroyed substantial portions of Hiroshima and Nagasaki in 1945. The issue of missing fissile material is likely much larger than what was reported missing from US inventories, because substantially more weapons-grade material may be missing in other countries, especially Russia, as noted by Patrick Malone; R. Jeffrey Smith (16 July 2018), Plutonium is missing, but the government says nothing, Center for Public Integrity, Wikidata Q91521732.
47. For precursors to the current study that involve censored estimation of time to a nuclear war, see Time to extinction of civilization and Time to nuclear Armageddon.
48. For the exponential distribution, ${\displaystyle h(t)=(-d/dt\log S(t))=\lambda }$ , writing the exponential survival function as ${\displaystyle S(t)=\exp(-\lambda t)}$ .
49. Richard Rhodes (1986), The Making of the Atomic Bomb, Simon & Schuster, OL 2617750W, Wikidata Q7749726. See also w:Trinity (nuclear test).
50. One might also consider a model with the log(Poisson mean) behaving like a "Wiener process" (also called a "Brownian motion"). This stochastic formulation would mean that the variance of the increments in log(hazard) between "first tests" is proportional to the elapsed time. See Wiener Process, Wolfram Research, Wikidata Q91547149 and w:Wiener process. The “bssm” package for R should provide a reasonable framework for modeling this; see Jouni Helske, bssm: Bayesian Inference of Non-Linear and Non-Gaussian State Space, Wikidata Q91626942. However, this author's efforts to use this package for this purpose have so far produced unsatisfactory results. More time understanding the software might produce better results but not necessarily enough better to justify the effort that might be required.
51. We could have used one observation each month, week, or day. Such a change might give us slightly better answers while possibly increasing the compute time more than it's worth.
52. George E. P. Box; Norman Richard Draper (1987), Empirical Model-Building and Response Surfaces, Wiley, Wikidata Q91658340.
53. See also Adrian Raftery (1995), "Bayesian model selection in social research", Sociological Methodology: 111–196, ISSN 0081-1750, Wikidata Q91670340 and Gerda Claeskens; Nils Lid Hjort (2008), Model selection and model averaging, Cambridge University Press, Wikidata Q62568358.
54. Adrian Raftery; Jennifer A. Hoeting; Chris Volinsky; Ian Painter; Ka Yee Yeung, BMA: Bayesian Model Averaging, Wikidata Q91674106. The algorithm used for this retained only the intercept and the coefficient of the highest power in each order. Models like ${\displaystyle b_{0}+b_{1}x+b_{2}x^{2}}$  with ${\displaystyle b_{1}\neq 0}$  were considered but had a posterior probability so low they were not retained in the final mixture of models. The quartic mixture retained only ${\displaystyle b_{0}}$  (constant), ${\displaystyle b_{0}+b_{1}x}$  (linear), ${\displaystyle b_{0}+b_{2}x^{2}}$  (quadratic), ${\displaystyle b_{0}+b_{3}x^{3}}$  (cubic), and ${\displaystyle b_{0}+b_{4}x^{4}}$  (quartic) with posterior probabilities 49.59, 13.24, 13.21, 12.66, and 12.31 percents, respectively.
55. Recall that the estimation methodology here is Poisson regression, not ordinary least squares.
56. Confidence intervals" bound the predicted mean number of nuclear-weapon states for each future year considered. Central 80 percent “prediction intervals" are limits that include the central 80 percent of distribution of the number of nuclear-weapon states. They add the uncertainty in the modeled Poisson process to the uncertainty of estimating the mean of that process for each future year considered. We will also compute (0.8, 0.8) "tolerance intervals"; ${\displaystyle (p,1-\alpha )}$  tolerance intervals have a probability of ${\displaystyle (1-\alpha )}$  of containing a proportion of at least ${\displaystyle p}$  of all future observations.
57. This follows, because quantile ${\displaystyle q}$  of a log-normal is ${\displaystyle \exp[\mu +\sigma \Phi ^{-1}(q)]}$  and the mean is ${\displaystyle \exp[\mu +\sigma ^{2}/2]}$ , so the mean exceeds quantile ${\displaystyle q}$  whenever ${\displaystyle \sigma \Phi ^{-1}(q)<\sigma ^{2}/2}$ , i.e., when ${\displaystyle \Phi ^{-1}(q)<\sigma /2}$ .
58. In these simulations, we assume a zero probability of a nuclear power giving up their nuclear weapons, even though South Africa reportedly discontinued their nuclear weapons program in 1989, prior to its first universal elections in 1994. We could potentially add South Africa to our dataset of nuclear weapon states with the same date as Israel, then model the distribution of the time to when a nuclear-weapon state gives up its nuclear weapons using an exponential distribution. For that, we have one observed time and eight such times that are censored. Standard theory in that case says that the maximum likelihood estimate of the mean time to relinquishing nuclear weapons assuming an exponential distribution is the sum of all the times, censored or observed, divided by the number of times observed, not including the censored times in the denominator. For purposes of illustration, we will assume that South Africa dismantled its nuclear weapons 1989-12-31, though a report of an inspection by the International Atomic Energy Agency dated 1994-08-19 said they had dismantled six nuclear weapons and were still working to dismantle one more. Based on this, the mean lifetime of a nuclear-weapon state can be estimated at 493 years. We could potentially add this to the current modeling effort, but it would not likely change the answers enough to justify the additional effort.
59. Robert McNamara; James G. Blight (2003), Wilson's ghost: reducing the risk of conflict, killing, and catastrophe in the 21st century, PublicAffairs, Wikidata Q64736611.
60. George P. Shultz; William Perry; Sam Nunn (10 April 2019), "The Threat of Nuclear War Is Still With Us", The Wall Street Journal, ISSN 0099-9660, Wikidata Q92101045.
61. Daniel Ellsberg; Amy Goodman; Juan González (6 December 2017), Daniel Ellsberg Reveals He was a Nuclear War Planner, Warns of Nuclear Winter & Global Starvation, Democracy Now!, Wikidata Q64226035.
62. Abhijit Banerjee; Esther Duflo (2019), Good Economics for Hard Times, PublicAffairs, Wikidata Q85764011. Various journalists and academic researchers have expressed concern about increases in ethic violence in various countries and whether electoral transitions of power will continue, even in the US. See, e.g., Brian Klaas (5 September 2019), "Everyone knows the 2020 election will be divisive. But will it also be violent?", The Washington Post, ISSN 0190-8286, Wikidata Q92101761.
63. The risks of a nuclear war producing major global climate problems have been documented in a series of simulations published in refereed academic journals, each more detailed and more disconcerting than the previous. All assume that many firestorms will be produced, because (a) the areas targeted will likely be much more susceptible to firestorms than the underground or isolated sites used to test nuclear weapons, and (b) many of the weapons used will have yields substantially greater than those employed in Hiroshima and Nagasaki. For a discussion of that literature, see Time to extinction of civilization and Time to nuclear Armageddon.
64. This uses a commonly accepted list of existing nuclear-weapon states and when they each first tested a nuclear weapon. The sources used for the data are in the help file for the “nuclearWeaponStates” dataset in the “Ecdat” package for R. See Yves Croissant; Spencer Graves (20 March 2020), Ecdat: Data Sets for Econometrics, Wikidata Q56452356.
65. Nuclear Verification Capabilities Independent Task Force of the Federation of American Scientists (September 2017), Nuclear Monitoring and Verification in the Digital Age: Seven Recommendations for Improving the Process (PDF), Federation of American Scientists, Wikidata Q97136193
66. Riechman, Deb (May 21, 2020). "US says it's pulling out of Open Skies surveillance treaty". Associated Press. Archived from the original on May 21, 2020. Retrieved May 21, 2020.