SIGMA ONE

In their discussion of optimizing search strategies for the simultaneous discovery and characterization of variable stars using predetermined but highly non-uniform sampling, Madore & Freedman (2005) investigated the small-sample limits. Their emphasis was on situations where only a handful of observations (e.g., 2–20) might be available from which it would be necessary to determine periods, measure mean magnitudes, and derive full amplitudes. In this paper, we discuss how it is possible to use a single observation to derive confidence intervals (CIs) on the mean (i.e., limits on variability, or estimates of the population variance, or its full amplitude, etc.).

Shortly after the original manuscript was completed, it was pointed out that similar arguments have been made much earlier by Gott (1993). Gott was interested in the estimation of ages and lifetimes of events based on single instances; and he presented CIs for the longevity of an object given a single observation of its present age. He assumed a flat prior (of finite duration), and took the stance that there cannot be anything special about any given time that a random observation is made of an object that has a finite lifetime. We return to a contextual discussion of this example, and a generalization of it to modeling, at the end of this paper.

What information can be derived about the parent population from a single, isolated (non-negative) observation? Here, we are clearly working at the observational information limit, and so a number of plausible assumptions ("priors") will of necessity enter the calculations. A tacit assumption is that the observation in question is drawn from an underlying population having some dispersion¹ that we are trying to estimate. We also assume that this is a physical object and that all values (observed or considered in the calculation) are therefore non-negative.

Lacking any information to the contrary, we take the single observed data point y₁ to be the best available estimate of the true mean, 〈Y〉, thus 〈Y〉 = y₁. It follows directly from the non-negativity assumption that the maximum possible semi-amplitude of Y is A/2 = (〈Y〉 − 0.0) = y₁. Invoking a symmetric prior would give a full amplitude A = 2 × y₁. Having given the flavor of this argument in the above few lines, we now more closely investigate a variety of priors, but first we introduce a new tool, the "g-factor".

We ask the following question: if random samples, y_i, are drawn from the parent distribution and considered one at a time, what is the auxiliary distribution of multipliers, "g-factors," that would convert the observation itself into its own "error," (_i), where _i is defined as being the absolute value of the distance of the data point from the true population mean (i.e., _i ≡ ∣〈Y〉 − y_i∣)? By this definition, g_i × y_i = _i and so g_i = ∣〈Y〉/y_i − 1.0∣. With the g-factor distribution in hand, for any given prior distribution, it is then possible to calculate specific multipliers (g(50.0), g(68.5), g(90.0), etc., by integrating over the g-factor distribution function) that will guarantee that set fractions (50.0%, 68.5%, 90.0%, etc.) of the data set will fall within the range y₁ ± g(50.0) × y₁, y₁ ± g(68.5) × y₁, etc., where y₁ is again our estimate for 〈Y〉.

We now consider five realizations of four plausible underlying population distribution functions—a Poisson distribution, a uniform distribution, an exponential distribution, and two Gaussians.

3.1. Poisson Distribution

3.2. Uniform Parent Population

We now consider a uniform distribution spanning the non-negative interval [0, 2] (Figure 1, upper panel). The distribution of g-factors is expected to be highly skewed: points near zero will require very large factors to make their error bars overlap with the true mean; however, all points greater than the mean (that is, fully half of the ensemble for a uniform prior) will have calculated error bars that overlap the mean for multiplicative g-factors that are all less than 0.5.

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We first solved for the g-factor distribution function by simulation. We successively drew 100,000 observations from a uniform distribution, calculated the factor that needed to be applied to that number such that y₁ ± g_u × y₁ just overlaps the true mean 〈Y_u〉. In Figure 1, the results of that computer simulation are shown, where the distribution function of multiplicative g-factors is found in the lower panel, and the parent distribution of individual observations going into the simulation is given in the upper panel. The lower panel also shows where the 50%, 68.5%, and 90% CIs are found for the g-factor distribution corresponding to this uniform prior.

If the underlying population, from which a single data point y₁ is drawn, is itself uniformly populated and non-negative, then y₁ ± 0.41 × y₁ will contain the mean of the parent population 50% of the time. Other selected CIs are given in Table 1.

Table 1. CIs and g-Factors for Selected Priors

Confidence	Uniform	Exponential	Gaussian (0.50)	Gaussian (0.25)
Interval	g-Factor	g-Factor	g-Factor	g-Factor
50.0%	±0.414	±0.67	±0.31	±0.16
68.5%	±0.563	±1.60	±0.45	±0.24
90.0%	±4.000	±8.5	±1.78	±0.48
95.0%	±9.000	±19.0	±4.0	±0.70
98.5%	±32.333	±65.0	±14.0	±1.2

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After the computer simulations were completed, it made possible for us to derive a closed-form analytic solution for multiplicative factors, g_u as a function of the CI associated with this case of a uniform prior. We give these solutions below; they are based on a simple mapping of the uniform distribution into its g-factors (using the absolute values of the g-factors accounts for the curious shape of the distribution in the lower panel of Figure 1). The CIs are then found by integrating the normalized functional form of that mapping up to the required value of CI:

$$\begin{eqnarray*} g_u &=& (\sqrt{1.0 + 4(\mbox{CI})^2} - 1.0)/(2{\mbox{CI}}), \ \ \ \ [\mbox{CI} \le 2/3]\\ g_u &=& (1.0 - 2{\mbox{CI}})/(2(\mbox{CI} - 1.0)), \ \ \ \ [\mbox{CI} \ge 2/3]. \end{eqnarray*}$$

Exemplary values for common CIs for the uniform prior are given in the second column of Table 1. For the uniform prior, they were calculated from the analytic solution and are confirmed by the simulations; all other solutions for other priors were derived from the computer simulations.

3.3. Exponential Parent Population

Another plausible parent population is the exponential distribution. Here, we investigate the g-factors corresponding to that distribution function. We have chosen to simulate an exponential distribution with mean 〈Y_e〉 = 1.0. The results of the inversion are shown in Figure 2 and the third column of Table 1. The contrast between this and the uniform prior is clear. Because significantly more samples will come with values close to zero, their contribution to the distribution of multiplicative factors will increase. This forces the individual g-factors for specific CIs to higher values as compared to the uniform prior. Inspection of Table 1 confirms this quantitatively for all CIs listed.

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3.4. Two Gaussian Distributions

Since the uniform and the exponential priors are characterized by a single parameter, the mean, they are far more easily constrained by the observation(s). But, for completeness and for illustrative purposes, we consider here two Gaussian distributions each with a mean of unity, but having differing dispersions. We make no claim that these two parameters can be constrained by a single observation. We simply want to illustrate quantitatively that the uniform prior at least and the exponential prior in its extreme both encompass the results for these Gaussian priors as well. That is, by adopting the g-factors for a uniform prior, one will also encompass the CIs derived for a variety of Gaussian distributions that have the same (unit) mean and certainly any dispersion less than 0.5.

The results for these additional priors are given graphically in Figures 3 and 4, and in Columns 5 and 6 of Table 1.

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Our interest in this formalism started with a desire to characterize luminosity variations of astronomical objects (in the first instance through their first two moments, chosen to be their means and amplitudes) having a highly restricted number of observations. Here we have taken that thought experiment to its limit of a singular observation. For an underlying distribution of finite extent (or duration), the dispersion and the full amplitude are equivalent, differing only by a constant scale factor in any given case. Accordingly, our derivation of the variance of a population based on a single observation is conceptually equivalent to Gott's (1993) derivation of future longevity (that is, Gott's longevity is a one-directional semi-amplitude) for an object based on a "single" observation of its present age.

When dealing with physical observations, certain assumptions are tacitly taken for granted. The obvious assumptions are that those quantities are non-negative, and that the underlying population from which they are drawn does not have an infinite range (in time, mass, energy, size, etc.) However, those same assumptions carry additionally useful (prior) quantitative information that can be used to constrain limits on observations as they are obtained. This paper had the intent of making those assumptions explicit and then formalizing the mathematical consequences expressed as CIs on the underlying population as derived from one observation.

While it was our hope that this intentionally short contribution might stimulate others into finding applications not obvious to the author, the referee requested that examples be given of how this formalism might be applied to astronomy. In keeping with the spirit of this paper, we offer a modest example from the past, and predict that there will be future examples; all of this is based on a sample of N = 1.

Laplace developed his "rule of succession" when confronted with a question as to the mathematical (not the physical) probability that the Sun will rise tomorrow given its past (statistical) performance. After observing N events, Laplace derived that the probability of the next occurrence was (N + 1)/(N + 2). This would suggest that on the first day (N = 0) the probability of the Sun rising was actually 50:50. On the second day (N = 1), the probability would have gone up to 66%, and so on. Gott (1993) asked a similar question, not just about the next occurrence of something that has a past persistence, but about the sum of all future occurrences. How long will a thing last, given that we know how old it is now? Gott was reformulating Laplace's question to be, if we have a single measurement (N = 1) of the age of something, what can one say about its total lifetime (its full amplitude) or rather its future longevity (a semi-amplitude). Of course, any such prediction is best described in terms of probabilities, and so rather than predicting a firm lifetime based on a precise age, Gott predicted a forward-looking probability distribution (expressed as a variance) based on a backward-looking age.

One could argue that Gott actually required two observations: one of the time at which something began and another of the time at which the prediction was being made. This may be seen as quibbling but it is, in fact, equivalent to our physical prior, stated above, that none of the quantities to which this method applies can drop below zero or become infinite in amplitude (mass, luminosity, time, etc.).

Grounded with post-dictions on the longevity of the Soviet Union and the Berlin Wall, Gott went on to predict the longevity of a variety of things astronomically big and small: from the expected demise of Nature magazine itself (somewhere between 3.15 and 4800 years) to the probability that we will end as a civilization (in 5100–8 million years with 95% probability), or colonize the Galaxy (the odds are against it). Each of these predictions were based on a single observation, N = 1 and a uniform prior.

It is quite clear that the uniform, non-negative prior distribution of data points discussed above is at one extreme (of simplicity, or of ignorance.) However, this extreme is also rather inclusive. If the true range of the underlying distribution is smaller, more centrally peaked, or more skewed toward the upper bound of the distribution function, than a uniform distribution, then their CIs will also be smaller than those calculated for a uniform prior; under those conditions, the uniform prior is likely to provide a conservative upper bound on the uncertainty.

4.1. Is This Just Another Name for Modeling?

I thank Wendy Freedman for pointing out both the prior nature and the a posteriori relevance of Gott's (1993) Nature study. Discussions with David Hogg were both stimulating and illuminating, as always. And finally, I thank both the editor-in-chief, Jay Gallagher, for his patience in dealing with this paper as it slowly evolved, and the anonymous referee, who carefully read the paper and gave many helpful suggestions.