# Forecasting the number of parties to the TPNW

    This article shares a statistical forecast of the number of parties to the United Nations Treaty on the Prohibition of Nuclear Weapons. It invites related commentary, encouraging contributors to “be bold but not reckless,” subject to the Wikimedia standards of writing from a neutral point of view, citing credible sources, and raising other questions and concerns on the associated '“Discuss”' page.
The accompanying figure displays the history of the number of parties to the United Nations Treaty on the Prohibition of Nuclear Weapons (TPNW) with prediction limits on the number of parties into 2032. The red and orange dashed lines give 98 and 80 percent prediction limits from random permutations of the interarrival times observed between the dates that new states have become parties to the TPNW, rescaled to reflect the uncertainty in the mean of the interarrival times. The blue dashed line is the median of the simulations.
The predictions in this figure should be interpreted skeptically, especially further in the future. They assume that the distribution of interarrival times will not change until it abruptly stops at saturation, i.e., with all 193 members of the UN plus the 2 observer states as parties to the TPNW. With those assumptions, 98 percent of the simulated trajectories ended with saturation sometime between 2028 and 2038.
Of course, the system that has been encouraging new states to join the TPNW will likely change long before then. The range of possibilities depicted in this figure seems reasonable for the next couple of years but will become increasingly untenable as the list of states not (yet) parties looks increasingly like the existing nuclear weapon states and their close allies.
There is also a countervailing trend that is increasing the number of nuclear weapon states. As this is being written, there are currently 9 nuclear weapon states -- 8 new states in the 77 years since the first use of nuclear weapons in 1945, just over one new state each decade (actually 9.7 years). North Korea first tested a nuclear weapon 2006-10-09, not quite 16 years ago as this is being written. That's longer than the 9.7 years average interarrival time between the 8 new nuclear weapon states but not enough longer to seriously suggest that there will NOT be more.
We should remember that North Korea first tested a nuclear weapon just over 4 years after US President George W. Bush decried an "Axis of Evil" on 2002-01-29, consisting of North Korea, Iran, and Iraq, and then proceeded to invade Iraq and seriously threaten Iran. Have the behaviors of the US, Russia and China become substantively less bellicose since George W. Bush decried an "Axis of Evil"?
This is similar to the traditional concern that extrapolation is much riskier than interpolation and is consistent with the observation that "All models are wrong, but some are useful."
The Appendix includes a companion R Markdown vignettes that makes the production of the accompanying figure completely reproducible. Work on this began 2020-08-10, when there were only 44 parties to the TPNW, and the interest was in predicting when the fiftieth would join. On 2020-10-24, Honduras became that fiftieth party. That then fixed the date on which it will enter into force (EIF) as 2021-01-22.

### Statistical theory

The time between acceptances, etc., of the TPNW is an example of a renewal process. The simplest renewal process assumes that this renewal time is exponentially distributed.
The first version of this analysis was completed in 2020 - August - before the fiftieth state became a party to the TPNW: We wanted to forecast when that might happen. After it did, we then wanted to forecast the distribution of the number of parties when the treaty entered into force.
For the current analysis, we summarized the bivariate observations on time between dates upon which at least one new party accepted, approved, ratified or acceded to the treaty and then added random permutations of this interarrival distribution to the date of the simulation to forecast the number of nuclear weapon states on 2021-01-22, the date at which the TPNW entered into force. An alternative to random permutations is bootstrapping (statistics). It would be interesting to try bootstrapping here; we would not expect the results to be noticeably different.
In trying to forecast to saturation, when all 195 UN members and observers were parties to the TPNW, the distribution of the simulated extrapolations seemed too narrow to be credible more than a couple of years in the future. We then prepared a normal probability plot of the logarithms of the interarrival times. They seemed reasonably normal, so we simulated variations in the mean with the appropriate Student's t distribution to produce the figure above.

## Appendix. Companion R Markdown vignette

Statistical details that make the research in article reproducible are provided in an R Markdown vignette to "Simulate numbers of parties to the TPNW":
* Forecasting the number of parties to the TPNW/Simulate number of parties to the TPNW