TF ErrorAna StatInference

Statistical Inference

Frequentist -vs- Bayesian Inference

When it comes to testing a hypothesis, there are two dominant philosophies known as a Frequentist or a Bayesian perspective.

The dominant discussion for this class will be from the Frequentist perspective.

frequentist statistical inference

Statistical inference is made using a null-hypothesis test; that is, ones that answer the question Assuming that the null hypothesis is true, what is the probability of observing a value for the test statistic that is at least as extreme as the value that was actually observed?

The relative frequency of occurrence of an event, in a number of repetitions of the experiment, is a measure of the probability of that event. Thus, if nt is the total number of trials and nx is the number of trials where the event x occurred, the probability P(x) of the event occurring will be approximated by the relative frequency as follows:

[math]P(x) \approx \frac{n_x}{n_t}.[/math]

Bayesian inference.

Statistical inference is made by using evidence or observations to update or to newly infer the probability that a hypothesis may be true. The name "Bayesian" comes from the frequent use of Bayes' theorem in the inference process.

Bayes' theorem relates the conditional probability|conditional and marginal probability|marginal probabilities of events A and B, where B has a non-vanishing probability:

[math]P(A|B) = \frac{P(B | A)\, P(A)}{P(B)}\,\! [/math].

Each term in Bayes' theorem has a conventional name:

• P(A) is the prior probability or marginal probability of A. It is "prior" in the sense that it does not take into account any information about B.
• P(B) is the prior or marginal probability of B, and acts as a normalizing constant.
• P(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.
• P(B|A) is the conditional probability of B given A.

Bayes' theorem in this form gives a mathematical representation of how the conditional probabability of event A given B is related to the converse conditional probabablity of B given A.

Example

Suppose there is a school having 60% boys and 40% girls as students.

The female students wear trousers or skirts in equal numbers; the boys all wear trousers.

An observer sees a (random) student from a distance; all the observer can see is that this student is wearing trousers.

What is the probability this student is a girl?

The correct answer can be computed using Bayes' theorem.

[math] P(A) \equiv[/math] probability that the student observed is a girl = 0.4

[math]P(B) \equiv[/math] probability that the student observed is wearing trousers = 60+20/100 = 0.8

[math]P(B|A) \equiv[/math] probability the student is wearing trousers given that the student is a girl

[math]P(A|B) \equiv[/math] probability the student is a girl given that the student is wearing trousers

[math]P(B|A) =0.5[/math]

[math]P(A|B) = \frac{P(B|A) P(A)}{P(B)} = \frac{0.5 \times 0.4}{0.8} = 0.25.[/math]

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