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]

Method of Maximum Likelihood

The principle of maximum likelihood is the cornerstone of Frequentist based hypothesis testing and may be written as

The best estimate for the mean and standard deviation of the parent population is obtained when the observed set of values are the most likely to occur;ie: the probability of the observing is a maximum.

Least Squares Fit

Applying the Method of Maximum Likelihood

Our object is to find the best straight line fit for an expected linear relationship between dependent variate [math](y)[/math] and independent variate [math](x)[/math].

If we let [math]y_0(x)[/math] represent the "true" linear relationship between independent variate [math]x[/math] and dependent variate [math]y[/math] such that

[math]y_o(x) = A + B x[/math]

Then the Probability of observing the value y_i with a standard deviation \sigma_i is given by

[math]P_i = \frac{1}{\sigma \sqrt{2 pi}} e^{- \frac{1}{2} \left ( \frac{y_i - y_0(x_i)}{\sigma_i}\right)^2}[/math]

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