*Disclaimer: I think what I've said is sufficiently obvious and basic that I really doubt that it's original, but I can't easily find any other source that lays out the points I made here. If you are aware of such a source, please let me know in the comments here and I'll credit it. I'd also be happy to be pointed to any relevant literature. It's also possible that I'm overlooking some obvious rejoinders that render my claims wrong or irrelevant; if so, I appreciate criticism on that front.*

The logarithmic timeline is a timeline where time is presented on a logarithmic scale. Note that this differs from the idea of plotting logarithms of quantities with respect to time (a common practice when understanding exponential growth). In those plots, the vertical axis (the dependent variable plotted as a function of time) is plotted logarithmically. With the logarithmic timeline, the time axis itself is plotted logarithmically. If we're plotting quantities as a function of time, then using a logarithmic timeline has an effect that's in many ways the *opposite* of the effect of using a logarithmic scale for the quantity being plotted.

Wikipedia has a page on the logarithmic timeline (see also this detailed logarithmic timeline of the universe and this timeline of the far future), but I haven't seen the topic discussed much in the context of forecasting precision and accuracy, so I thought I'd do a post on it (I'll list some relevant literature I found at the end of the post).

**TL;DR**

Here's an overview of the sections of the post:

- What the logarithmic timeline means for understanding forecasts, and how it differs from the linear timeline.
- Crudely, the logarithmic timeline is useful because uncertainties accumulate over time, with the amount of uncertainty accumulated being roughly proportional to how far out we are in the future.
- Mathematically, The logarithmic timeline is suitable for processes whose time evolution is functionally described in terms of the product of time with a parameter whose precise value we are uncertain about.
- The logarithmic timeline can also be important for the asymptotic analysis of more general functional forms, if the dominant term behaves in the manner described in #3.
- I don't know if the logarithmic timeline is correctly calibrated for comparing the
*value*of particular levels of forecasting precision and accuracy. - The logarithmic timeline is related to hyperbolic discounting.
- If using the logarithmic timeline, point estimates for how far out in time something will happen should be averaged using geometric means rather than arithmetic means. Similar averaging would need to be done for interval estimates or probability distribution estimates for the time variable.
- I don't know if empirical evidence bears out the intuition that forecast accuracy should be time-independent if we use the logarithmic timeline.

**#1: What the logarithmic timeline means for understanding forecasts**

First off, we are using the origin point for the logarithmic timeline as the present. There are other logarithmic timelines that are better suited for other purposes. Using the origin of the universe is better suited for physics. But when it comes to understanding forecasts based on our best knowledge of what has transpired so far, the present is the natural origin.

Let's first understand the implicit assumption embedded in the use of a *linear* timeline for understanding forecasts. With a linear timeline, a statement of the form "technological milestone *x* will happen in year 2017" has equivalent *prima facie* precision as a statement of the form "technological milestone *y* will happen in year 2057" despite the fact that the year 2017 is (as of the time of this writing) just 3 years in the future and the year 2057 is 43 years in the future. But a little reflection shows that this doesn't jibe with intuition: making predictions to single years 43 years in advance is more impressive than making predictions to single years a mere 3 years in advance. Similarly, saying that a particular technological innovation will happen between 2031 and 2035 involves making a more precise statement than saying that a particular technological innovation will happen between 2015 and 2019.

We want a timeline where the equivalent in the far future of a near-future year is an interval comprising more than one year. But there are many such choices of monotone functions. I believe that the logarithmic one is best. In other words, I'm advocating for a situation where you find "between 5 and 10 years from now" as precise as "between 14 and 28 years from now", i.e., it is the quotient of the endpoint to the startpoint (the multiplicative distance) that matters rather than the difference between them (the additive distance).

But why use the logarithm rather than some other monotone transformation? I proffer some reasons below.

**#2: A crude explanation for the logarithmic timeline**

If you're mathematically sophisticated, skip ahead straight to the math.

Here's a crude explanation. Suppose you're trying to estimate the time in which the cost per base pair of DNA sequencing drops to 1/8 of its current level. You have an estimate that it takes between 4 and 11 years to halve. So the natural think to do is say "to get to 1/8, it has to go through three halvings. In the best case, that's 3 times 4 equals 12 years. In the worst case, that's 3 times 11 equals 33 years. So it will happen between 12 and 33 years from now."

Note that the length of the interval for getting to 1/8 is 33 - 12 = 21, three times the length of the interval for getting to half (11 - 4 = 7). But the ratio of the upper to the lower endpoint is the same in both cases (namely 11/4).

*None of the numbers above are significant; I chose them for the benefit of people who prefer worked numerical examples before, or instead of, delving into mathematical formalism.*

Note also that while this particular example had an exponential process, we don't need the process to be exponential *per se* for the broad dynamics here to apply. We do need some mathematical conditions, but they aren't tied to the process being exponential (in fact, exponential versus linear isn't a robust distinction for this context because either can be turned to the other via a monotone transformation). I turn to the mathematical formalism next.

**#3: The math: logarithmic timeline is natural for a fairly general functional form of evolution with time**

Consider a quantity *y* whose variation with time *t* (with *t = 0* marking the current time) is given by the general functional form:

*y = f(kt)*

where *f *is a monotone increasing function, and *k* is a parameter that we have some uncertainty about. Let's say we know that *a < k < b* for some known positive constants *a* and *b*. We now need to answer a question of the form "at what time will *y* reach a specific value *y*_{1}?"

Since *f* is monotone increasing, it is invertible, so solving for *t* we obtain:

*t = f*^{ -1}(*y*_{1})/*k*

There's uncertainty about the value of *k*. So *t* ranges between the possibilities *f*^{ -1}(*y*_{1})/*b* and *f*^{ -1}(*y*_{1})/*a*. In particular, if we divide the endpoint of the interval by the starting point, we get *b/a*, a quantity independent of the value of *y*_{1}. Thus, the use of the logarithmic timeline is a robust choice.

What sort of functional forms match the above description? Many. For instance:

- A linear functional form
*y = kt + c*where*k*is a positive constant and*c*is a constant. Note that even though there are two parameters here, the value of*c*is determined by evaluating at*t = 0*knowing the present value, and is not a source of uncertainty. - An exponential functional form
*y = Ce*where^{kt}*C*and*k*are positive constants. Note that even though there are two parameters here, the value of*c*is determined by evaluating at*t = 0*knowing the present value, and is not a source of uncertainty. - A quadratic functional form
*y = (kt)*^{2}+*c*where*k*is a positive constant and*c*is a constant. Note that even though there are two parameters here, the value of*c*is determined by evaluating at*t = 0*knowing the present value, and is not a source of uncertainty.

Of course, not every functional form is of this type. For instance, consider the functional form *y = t ^{k}*. Here, the parameter is in the exponent and does not interact multiplicatively with time. Therefore, the logarithmic timeline does not work.

**#4: Asymptotic significance of the logarithmic timeline**

A functional form may involve a *sum* of multiple functions, each involving a different parameter. It does not precisely fit the framework above. However, for sufficiently large *t*, one piece of the functional form dominates, and if that piece has the form described above, everything works well. For instance, consider a functional form with two parameters:

*y = e ^{kt} + mt + c*

Both *k* and *m *are parameters with known ranges (*c *is determined from them and the value at 0). For sufficiently large *t*, however, this looks close enough to *y = e ^{kt }*that we can use that as an approximation and find that the logarithmic timeline works well enough. Thus, the logarithmic timeline could be asymptotically significant.

**#5: Does the logarithmic timeline correctly measure the benefits of a particular level of forecasting precision?**

We've given above a reason why the logarithmic timeline correctly measures precision from the perspective of forecasting *ability*. But what about the perspective of the *value* of forecasting? Does knowing that something will happen between 5 years and 10 years from now deliver the same amount of value as knowing that something will happen between 14 years and 28 years from now? Unfortunately, I don't have a clear way of thinking about this question, but I can think of plausible intuitions supporting the logarithmic timeline choice: the farther out in the future we are talking, the less valuable it is to know exact dates, and ratios just happen to capture that lower level of value correctly.

**#6: Relation with hyperbolic discounting**

Gunnar_Zarncke points out in a comment that the logarithmic timeline is related to hyperbolic discounting, a particular form of discounting the future that bears close empirical relation with how people view the future. Hyperbolic discounting gives differential weight 1/*t* to a time instant *t *in the future. This relates with the logarithmic timeline because *d*(ln *t*)*/dt = *1/*t*. This could potentially be used to provide a rational basis for hyperbolic discounting, vindicating the rationality of human intuition.

A follow-up comment by Gunnar_Zarncke links to an earlier LessWrong comment of his that in turn links to research showing that people's subjective perception of time fits the logarithmic timeline model.

**#7: Point estimates and geometric means**

Another implication of the logarithmic timeline is that if we have a collection of different point estimates for points in time when a specific milestone will be attained, the appropriate method of averaging is the *geometric mean* rather than the arithmetic mean. The geometric mean is the averaging notion that corresponds to taking the arithmetic mean on the logarithmic scale.

For instance, if three people are asked for a project estimate, and they give the numbers of 2 years, 8 years, and 32 years, then the geometric mean estimate is the cube root of 2 X 8 X 32. This turns out to be 8. The arithmetic mean estimate is the (2 + 8 + 32)/3 = 14.

Note that, thanks to the AM-GM inequality, the geometric mean is never larger than the arithmetic mean, and they're equal only when all the quantities being averaged are equal to each other to begin with. This suggests that, if people tend to be optimistic about how quickly things will happen when they use arithmetic means, they'll appear even more optimistic when using geometric means. On the other hand, the logarithmic timeline might also result in the optimism not seeming so bad.

Similar geometric averaging would need to be done for interval estimates or probability distribution estimates for the time variable.

**#8: Empirically, is forecast accuracy time-independent once we switch to the logarithmic timeline?**

I consider this the most important question. Namely, as an empirical matter, are people about as good at figuring out whether something will happen between 5 and 10 years from now as they are at figuring out whether something will happen between 14 and 28 years from now?

I do believe that empirical evidence confirms what intuition knows: on the linear timeline, forecast accuracy decays. Thus, for instance, when people are asked for the precise year when something happens, estimates for things that will happen farther out in the future are later. When people are asked to estimate GDP per capita values, estimates far out in the future are worse than near-term estimates. But how much worse are the long term forecasts? Is the worsening in keeping with the logarithmic timeline story?

Note that if the general functional form I described above correctly describes a process, then the logarithmic timeline story is validated theoretically, but the empirical question is still open.

Most research I'm aware of just looks at estimates within specified intervals, such as "how much will GDP growth rate be in a give year?" I suspect an analysis of the data from these experiments might allow us to judge the hypothesis of constant accuracy on the logarithmic timeline, but I don't think just looking at their abstracts would settle the hypothesis. But I'd welcome suggestions on possible tests based on already existing data.

Note also that if existing research uses arithmetic means to aggregate estimates for "how far out in the future" something will happen, we'll have to get back to the source data and use geometric means instead.

*There may be research on the subject of evaluating forecast accuracy using a logarithmic timeline (most research on the logarithmic timeline relates to the history of the universe and evolution, rather than the future of humanity or technology). I haven't been able to locate it, and I'd love if people in the comments point me to it.*

**Potentially relevant literature**: I skimmed the paper *Forecasting the growth of complexity and change* by Theodore Modis, *Technology Forecasting and Social Change*, Vol. 69, 2002 (377-404), available online (gated) here. I haven't been able to locate an ungated version. The paper uses a logarithmic timeline for the *past*, taking the present as the origin. A quick skim did not lead me to believe it overlapped with the points I made here. Incidentally, Modis has been critical of Ray Kurzweil's singularity forecast.

See also the discussion at the end of #6 (hyperbolic discounting) linking to the paper On the perception of time by F. Thomas Bruss and Ludger Ruschendorf.

**Addendum**: To clarify the relation between logarithmic timeline, logarithmic scales, linear functions, power functions, and exponential functions, the table below gives, in its cells, the type of function we'd end up graphing:

Growth rate of quantity with respect to time | Ordinary scale | Logarithmic timeline | Logarithmic scale for quantity, ordinary timeline | Logarithmic scale for both |
---|---|---|---|---|

Linear | Linear | Exponential | Logarithmic | Linear with slope 1 |

Power function | Power function | Exponential | Logarithmic | Linear |

Exponential | Exponential | Double exponential | Linear | Exponential |

This suggests an interesting relationship to Hyperbolic discounting (which has been observed in humans and animals): Hyperbolic discounting stops being a fallacy but becomes the norm if you live in a world of logarithmic time. And your post suggests that with regard to predicting the future logaritmic time is the more natural choice, so in that regard we live in a world with logarithmic time.

Great point, thanks! I added a section to the post called "Relation with hyperbolic discounting" linking to your comment and making some further remarks.

Then you may also be interested on time being subjectively perceived on a logorithmic scale - and there is a mathematical model for that. See my comment in the open thread here:

http://lesswrong.com/lw/k13/open_thread_april_8_april_14_2014/at55

Thanks, added a mention in the hyperbolic discounting section and also in the potentially relevant literature section. I'll take a closer look later.