Parametric Statistical Models

At the end of this lecture, you will be able to do the following:

• Construct a statistical model .
• Give several examples of statistical models involving commonly used distributions (e.g. Gaussian, Poisson)
• Distinguish between parametric and non-parametric statistical models.
• Determine whether or not a parameter in a statistical model is identified.
• What are the main notions of statistical inferences?

The goals of statistics : Estimation, confidence intervals, and hypothesis testing

In this unit, we introduce a mathematical formalization of statistical modeling to make a principled sense of the Trinity of statistical inference.

Statistical modelling

An example of a statistical model.

Statistical model

statistical model : definition

Types of Statistical Models

Parametric, nonparametric and semiparametric models

Examples of parametric and nonparametric models

Mixtures of Gaussians

Gaussian Mixtures

→ two subpopulations within one big population

→ mixture of two gaussians

Give a bigger weight to the first distribution

• Unimodal → only one maximum
• multimodal → several local maximum

Another representation of Gaussian Mixtures

→ ex. 3 같은 경우는 여자와 남자의 예시를 생각해보면, 대략 반반으로 나뉘는 것을 알 수 있고, 따라서 웨이트는 두 개가 같은 1/2 로 고정된다.

Identifiability

Injection , 단사함수 = 일대일함수

references.

https://learning.edx.org/course/course-v1:MITx+18.6501x+2T2022/block-v1:MITx+18.6501x+2T2022+type@sequential+block@u02s02_parainference/block-v1:MITx+18.6501x+2T2022+type@vertical+block@u02s02_parainference-tab1

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