# Can you ever have known parameters?

Maybe a bit of a philosophical question - but can you ever truly have known parameters in data? I have a set of data for which the dataset is complete, but the parameters will still be estimates i believe?

• It depends what you mean by complete. If it means that you have the entire population, then your parameter estimate is no longer an estimate. This assumes, of course that the population stays constant. And a word on the careless language here: a parameter has an estimator and a particular value of the estimator is an estimate. – rocinante Mar 17 '14 at 22:38
• So say I want to find the mean and variance of the closing price of the S&P over the last month. I have the entire population, but I don't know the distribution. So surely they are still estimates? – user40124 Mar 18 '14 at 10:46
• If all you want to know is the mean and variance of the index over the last month, why do you think you need the distribution? You have the entire population, so no statistic you calculate is an estimate. Your estimate becomes invalid if you want to say something about the future/past mean/variance/etc/ of the closing price. If you want to make some inference about the long term mean/variance/etc of the closing price and are just using a short term period of two months as a "representative" value, then it is an estimate. – rocinante Mar 19 '14 at 0:00
• How i find the mean (the formula) is dependent on what I believe the distribution to be surely? – user40124 Mar 19 '14 at 16:45
• No, of course not. – rocinante Mar 19 '14 at 18:18

## 3 Answers

There are two circumstances where you can have "known" parameters - parameters that, when your model reports them, you know them to be true:

1. Genuinely complete information, meaning you have a data set covering every single individual (or whatever) in the population, with all the possibly relevant variables known. This is, of course, a somewhat rare circumstances, but in this case you're not estimating a parameter, you're calculating it, which is a somewhat different beast philosophically. For example, if you have a desert island with 100 people on it, and you ask each one their age, the mean age you calculate is the mean, with no uncertainty.
2. You're working on a simulated or otherwise constructed data set where you fixed a parameter in the data to be a particular value, and you're checking to make sure your estimator is returning the correct, known answer.

It depends what you mean by 'known' and what your data set represents.

If you mean known absolutely and with no uncertainty, then -- philosophically -- there is not much in science (hence, data science) that is 'known' in this way.

However, in every data set, there are parameters that can be assumed to be true with respect to the model underpinning the analysis that you're going to carry out.

So here what matters is what your data set represents.

If it contains physical measurements, then philosophically, every measurement is an estimate up to some degree of precision.

If it contains samples of a population, then it is an estimate of some population parameter.

In physical sciences some parameters are often known. For instance, you have fundamental constants such as the ratio of an electron charge to a mass, or the speed of sound. In social sciences there are no fundamental constants, and nothing is usually truly known, that's why there's so much advanced statistics developed in these fields to cover the lack of fundamental understanding of phenomena.

Another example, is the radioactive decay. We know that the counts are from Poisson. All you need to do is to work on the electronics of counters so it doesnt interfere too much. It's a big deal to know the distribution. You only need to estimate the rate of decay. Compare it to social sciences where at best you guess that it's maybe Poisson.

UPDATE: @whuber's comments made me reread the question, and I noticed that you mentioned

the dataset is complete

If this means that you got the population, then this changes my answer. When you have the population, there's no more estimation required. You simply compute the parameter. You estimate when there's only sample from population.

• The constants in physical science are not known: practically by the definition of "physical" and "science," they are measured. In some cases they have been measured to extremely high precision (13 or more significant figures), but that does not alter their ontological status. Moreover, it has been established that historically the confidence limits on estimates of physical parameters tend to be (way) too short, which rather emphasizes the fact of their uncertainty! – whuber Apr 18 '14 at 18:31
• @whuber, let us say I'm measuring electromagnetic pollution. Do you think I start with estimating the speed of light, because it is uncertain or unknown? Trust me, nobody measures speed of light, because it is known, google it. – Aksakal Apr 18 '14 at 18:39
• In that application you hypothesize, the speed of light is not a statistical parameter. One has not set out to estimate it but is adopting it as a model assumption. The Google hits that refer to the speed of light as "exact" are misleading, because they define the speed in terms of units of time and distance. One of those, at least, has an uncertainty. This is explained on the physics site. One top Google hit documents the present uncertainty in this estimate at speed-light.info/measure/speed_of_light_history.htm. – whuber Apr 18 '14 at 19:05
• With a little more Googling I found a reference you might enjoy: Figure 4.1 (p. 58) of Morgan & Henrion's Uncertainty: books.google.com/…. It documents published estimates and confidence intervals for the speed of light from 1870 through 1960. It shows those intervals had less than 40% coverage (and it didn't really improve with more recent experiments). – whuber Apr 18 '14 at 19:14
• @whuber, that's not the point I'm making here. Unless your objective is to measure the speed of light, you consider it known. You also know that it's constant. In social sciences nothing is known to be constant, and nothing has precision of 13 digits. Basically, you can point to anything, and its value is unknown in economics or psychology, you may need to estimate every single parameter in the model, which itself is most likely misspecified. For instance, I never had to measure the speed of light in my years in Physics. – Aksakal Apr 18 '14 at 19:27