Think about your recent wine tasting retreat. While you were relaxing and refreshing, you were also tasting/testing wine samples and guessing the quality of the product. You were knowing the unknown using samples and the process of induction. In other words, you were inferring about the true quality of the product based on the taste of the wine sample. A method of generalization from observations.
Think about your recent visit to the theater after being impressed by the trailer. You must have judged how good the movie will be, based on the sample they showed you. If you are like me, you would have told yourself that this is the last time you are watching a movie based on the trailer. But you keep going back. Here again, you are inferring about the film based on the sample teaser. Knowing the unknown via inference.
Do those robocalls for election survey or ‘get out the vote’ campaign bother you. Be nice to them. They are taking a sample survey to estimate/guess the election results. They are trying to know the unknown. Inferring the true outcome from the sample data.
Abraham Wald in 1942 used the sample of fighter planes that returned with bullet holes to judge/infer where to put the armor plates. It is where the holes aren’t there. The truth is that the planes with bullet holes in the wings returned, while the others did not.
Statistical Inference
Deriving general conclusion from what we observe (sample data) is central to increasing our knowledge about the world. Statistical inference helps us with this.
Imagine we wish to know about incomes or IQ levels of adult population in the US. The ideal thing to do is to test every adult for IQ and record their income levels. In other words, get the data for the entire population. But this is not practical. We can, however, take a sample that represents the population, know the IQ and incomes of this sample. We can make a general statement about the population from the sample. The statement could be “the average IQ of adults in the US is 100.” We get this based on a large enough representative sample. The average IQ from the sample is an estimate of the true IQ for the population. Since we are basing it on the sample, the estimate will be sensitive to how much data (sample size) we have.
Imagine we wish to know about the climate of a region we want to relocate to, i.e., what is the temperature distribution and how much does it rain or how many hurricanes does it get per year. Proper measurements of rainfall and temperature are a recent phenomenon. Maybe the last 80 years with good quality. We don’t have an observational record of what happened before that. But, we can take the data we have (sample of climate observations) and estimate the temperature distribution and the hurricane counts and make a statement about the overall climate in this place so you can relocate. The generalization can be “the average temperature of this place is 60F with four hurricanes per year.” Don’t complain after you move that the area is getting more hurricanes. The sample may not have represented the climate history. Our inference is based on what we had. We can always update it.
Come 2020, the media and polling companies will predict the outcome of the presidential election. They will make voter calls or take online surveys to infer which side the election will swing. Ask for sample breakdowns and survey questions. You will know what biases are built into it.
In the most recent lessons on the generalized extreme value distribution, we assumed that the data on extreme wind speed (block maxima) would most likely follow an extreme value distribution since maximum values converge to an extreme value distribution in the limit. So the data we observed is a sample that represents the true GEV functional form
![G(z) = e^{-[1+\xi(\frac{z-\mu}{\sigma})]^{-1/\xi}_{+}}](https://i0.wp.com/www.dataanalysisclassroom.com/wp-content/ql-cache/quicklatex.com-e5749a92028de2816cd08caaa14721b5_l3.png?resize=175%2C27&ssl=1 "Rendered by QuickLaTeX.com")
This compact GEV function is the population. A true form of all extreme wind values in this place. The data we observed for a few years is a sample that represents this function (population). We can use the sample to draw conclusions (inference) about this GEV function.
The true location, scale and shape parameters for this GEV function are 
and 
. Based on the data we observed, an estimate (a good guess) of this location, scale and shape parameters is 47.44, 5.03 and 0. We are drawing inference about the true function from the sample data we have at hand.
Hence, the objective of inference is to use a ‘sample’ to compute a number which represents a good guess for the true value of the parameter. The true population parameter is unknown, and what we estimate from the sample will enable us to obtain the closest answer in some sense.
The general rule, or a formula which is used to get the estimate, is called an estimator.
For example, if you compute the mean (
) and variance (
) of the data, they are good guesses (estimates or estimators) of the mean (
) and variance (
) of the population.
A different sample will yield a different estimate for the parameter. So we have to think of the estimate as a range of values, an interval or a probability distribution instead of a single value. The truth may be in this interval if we have good representative samples.
Over the next few lessons, we will learn some methods of estimation from data, how reasonable the estimates are, or the criteria for being good, their range or intervals, and many more concepts.
In summary, statistical inference is about understanding the entire population from samples and sample statistics.
The sample has to be representative of the population for good inference.
In other words, the sample distribution must be similar to the population distribution. Extreme winds data would be a good representation of an extreme value function as population, not normal distribution.
Joe and his girlfriend broke up. They inferred their future based on the sample dates. Is it a good inference?
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