#statistical inference

• reference

There are 2 main approaches to statistical inference, frequentist and Bayesian, differing in their interpretation of uncertainty.

<aside> 💡 Frequentist: The parameter is not a random variable (fixed parameter).

Bayesian: The parameter is a random variable.

</aside>

— simple explanation video —

http://youtube.com/watch?v=GEFxFVESQXc&t=128s

1. Frequentist

Frequentist statistics are a type of statistical inference that tests whether an event (such as your hypothesis) occurs. Frequentists treat probabilities as equivalent to “frequencies” – the number of times something happens.

Frequentist statistics view probability as depending on the outcome of the experiment if you repeat it infinitely. So, if you flip a coin infinite times and get heads half the time, then the probability of getting heads (or tails by inference) is 50%.

In frequentist reasoning, the parameters (e.g., the mean) that you try to estimate from your population are assumed to be fixed. You assume there’s a single true parameter that you’ll try to estimate, and the parameters are not modeled as a probability distribution.

Then, the random data points the experiment obtains are used to estimate the population parameter. Using this estimate, you can make probability computations regarding the data.

For example, if the height of a certain population is the parameter you try to estimate, you may hypothesize that the average height is 170 cm. This 170 cm is then taken as the true value. Then, you obtain data points (e.g., the heights of 75 different people in the population). You then use these data points to perform statistical tests.

• Last time we derived the Frequentist result that when f=40%, if you repeatedly conduct surveys of N=200 people, then 95% of those surveys will produce values for F that are between 33% and 48%, which was our definition of F being "close" to f. The Frequentist answers a subtly different question: if we knew f(which we don't), what would the survey result F be likely to be? So I drew it with a backward pointing arrow instead.
• Frequentist는 “만약 우리가 f값을 안다면(실제로는 모르지만), 설문 결과로 얻게 될 F는 얼마가 될 것인가?”에 집중한다. 즉, 200명을 대상으로 '반복적으로' 설문조사를 시행한다면, 그 결과값 F는 95%의 확률로 33%와 48% 사이에 위치하게 될 것이라고 주장한다.

2. Bayesian

The Bayesian worldview is based on Bayes’ Theorem. This is a mathematical formula that considers the probability of an event based on prior knowledge of conditions related to that event. This formula tells you how your current belief should be updated with data to get your new beliefs.

$$ P(A|B)\;=\;\frac{P(A) \cdot P(B|A)}{P(B)} $$

In a Bayesian setting, A corresponds to the parameters and B to the data.