# descriptive statistics as generative models

It is common to distinguish between descriptive and inferential statistics.

However, it seems to me that any descriptive statistic can be immediately used to generate realizations of the data. In the simplest (often uncomputable) case this can be done through uniform sampling of all possible data samples which exactly match the value of the statistic. Such a simple approach, while reminiscent of Monte Carlo simulations or maximum-entropy models, actually has no parameters or mechanistic specifications (at least not in an explicitly defined sense).

But it seems that this above approach is still a generative model of the data -- albeit one for which it is hard, or may be impossible, to compute the likelihood. Is this correct? And if yes, does this imply that there is no fundamental distinction between descriptive and inferential statistics?

Here is a simple example. Suppose I have a data sample of $m$ real numbers, such that each number is in the range of $[0, b]$, where $m$ and $b$ represent some known constraints of the system in question.

I also know that the mean of my data sample is $\bar{x}$, but I don't know anything else about the distribution of these numbers. In other words, I can make no assumptions that the numbers are distributed uniformly or in any other way.

I can construct a generative model which samples uniformly all sets $S=\{a_1, a_2, ..., a_m\}$, such that all $a_i \in [0, b]$ and $\frac{1}{m}\sum_{a_i \in S} a_i = \bar{x}$. This model will, by definition, match the mean of my data but may otherwise be a poor representation of its distribution.

Now suppose I gradually acquire additional arbitrary descriptive statistics about my data (e.g. the first quartile, the kurtosis, or any other statistic). I can incorporate these statistics into my generative model in exactly the same way as above (i.e. through uniform sampling of all sets that match these statistics). With more descriptive statistics, my model becomes a more accurate representation of the data.

Here is again my question. It seems that the descriptive statistics (and the initial constraints of the system) allow me to construct an arbitrarily precise generative model of my data without performing any inference. Is this correct? And if yes, does this imply that there is no fundamental distinction between descriptive and inferential statistics?

Any pointers to the relevant literature would also be greatly appreciated.

• What does "uniform sampling of all sets of real numbers which have mean x" mean? – Qiaochu Yuan Aug 3 '17 at 5:11
• @QiaochuYuan, thank you -- it means to draw uniformly from a distribution of all sets $S=\{a_1, a_2, ..., a_m\}$ which satisfy $\frac{1}{m}\sum_{a \in S} a = \bar{x}$ (where all $a \in S$ are real numbers, and $m$ is an arbitrary positive integer). I updated the question above. – Akim Aug 3 '17 at 12:59
• Is $m$ fixed? If not, how is it chosen? Also, what do you mean by uniformly sampling a random real number? – Qiaochu Yuan Aug 3 '17 at 19:25
• @QiaochuYuan, thanks for the questions; I have rephrased to make the problem better defined, without altering my original intent. $m$ can be constant, and the real numbers $a$ can be constrained to lie in some range $[0, a_{\rm max}]$. – Akim Aug 3 '17 at 20:17

You are hinting at some interesting ideas related to 're-sampling'. But I will need to make some minor changes in your example in order to explore them.

(1) Suppose you have $n$ observations $X_1, \dots, X_n$ sampled at random from $\mathsf{Unif}(0,b),$ with sample mean $\bar X.$ If you generate $n$ new random variables $Y_i$ from $\mathsf{Unif}(0, 2\bar X)$ you will get roughly what you want. But $2\bar X$ might turn out to be somewhat below $b$ or somewhat above $b$, so you could re-scale them as $Z_i = (\frac{b}{2\bar X})Y_i \sim \mathsf{Unif}(0,b).$

A brief demonstration in R statistical software illustrates this for $n = 15$ and $b = 10.$

# Original sample and description
n = 15;  b = 10; x = runif(n, 0, b)
a = mean(x);  a
## 4.132791
summary(x)
#   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
# 0.2708  1.3120  4.5370  4.1330  6.2430  9.1780

## New sample y  and adjustment z
y = runif(n, 0, 2*a);  z = (b/(2*a))*y
summary(y)
#  Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
# 1.289   1.754   4.237   4.492   7.492   8.265
summary(z)
#  Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
# 1.559   2.122   5.125   5.434   9.065   9.999

## Test showing z is consistent with UNIF(0, b)
ks.test(z, "punif", 0, b)

One-sample Kolmogorov-Smirnov test

data:  z
D = 0.23621, p-value = 0.3202      # P-value > .05:  Do not reject UNIF(0,b)
alternative hypothesis: two-sided


(2) A related idea is the nonparametric bootstrap. Suppose you don't know the population distribution (except that you know it has a mean $\mu),$ but you have have $n$ observations and their sample mean. Then you can bootstrap your sample (by re-sampling) to get a confidence interval for the population mean $\mu$.

You can read about 'nonparametric bootstrap' on Wikipedia or other Internet sites. Also, you can take a look at my related Answer to another Question.

• Thank you very much your answer! These pointers are very helpful, but my question concerns a more conceptual problem -- namely whether it is possible to construct a generative model without performing any inference (e.g. estimating latent variables). I have expanded my question to try and make this point clearer. – Akim Aug 4 '17 at 15:43
• My (1) seems close to your goal. Definition of inference can be slippery. Some would say that computing $\bar X$ is point estimation of pop $\mu.$ (2) is somewhat related, but admittedly a reach. Also popular and important inferential topic, maybe worth investigating even if not Answer here. // Hope you get interesting answers. – BruceET Aug 4 '17 at 15:53