1.5 Foundations
There are several fundamental principles discussed in this section. They are :
(a) Misuse of Classical Inference Procedure
(b) Frequentist Perspective
(c) Conditional Perspective
(d) Likelihood Principle
(a) Misuse of Classical Inference Procedure:
Example 7:
Let
In classical inference problem,
,
the rejection rule is
,
as
Assume the true mean and
.
Suppose , then
and we reject .
Intuitively, seems to strongly indicate should be true. However, for a large sample size, even as is very close to 0, the classical inference method still indicates the rejection of . The above result seems to contradict the intuition.
Note: it might be more sensible to test, for example,
.
Example 8:
Let
In classical inference problem,
,
the rejection rule is
,
as
If then and we do not reject . However, as ,
,
we then reject .
(b) Frequentist Perspective:
Example 9:
Let
In classical inference problem,
,
the rejection rule is
,
as By employing the above rejection rule, about 5% of all rejection of the null hypothesis will actually be in error as is true. However, suppose the parameter values and occur equally often in repetitive use of the test. Thus, the chance of being true is 0.5. Therefore, correctly speaking, 5% error rate is only correct for 50% repetitive uses. That is, one can not make useful statement about the actual error rate incurred in repetitive use without knowing for all .
(c) Conditionalt Perspective:
Example 10:
Frequentist viewpoints:
are independent with identical distribution,
.
Then,
can be used to estimate .
In addition,
Thus, a frequentist claims 75% confidence procedure.
Conditional viewpoints:
Given , is 100% certain to estimate correctly, i.e., .
Given , is 50% certain to estimate correctly, i.e., .
Example 11:
X1 / 2 / 3
/ 0.005 / 0.005 / 0.99
/ 0.0051 / 0.9849 / 0.01
/ 1.02 / 196.98 / 0.01
: some index (today) indicating the stock (tomorrow) will not go up or go down
: some index (today) indicating the stock (tomorrow) will go up
: some index (today) indicating the stock (tomorrow) will go down
: the stock (tomorrow) will go down.
: the stock (tomorrow) will go up
Frequentist viewpoints:
To test
,
by the most powerful test with , we reject as since
.
Thus, as , we reject and conclude the stock will go up. This conclusion might not be very convincing since the index does not indicate the rise of the stock.
Conditional viewpoints:
As ,
.
Thus, and are very close to each other. Therefore, based on conditional viewpoints, about 50% chance, the stock will go up tomorrow.
Example 12:
Suppose there are two laboratories, one in Kaohsiung and the other in Taichung. Then, we flip a coin to decide the laboratory we will perform an experiment at:
Head: Kaohsiung ; Tail: Taichung
Assume the coin comes up tail. Then, the laboratory in Taichung should be used.
Question: should we need to perform another experiment in Kaohsiung in order to develop report?
Frequentist viewpoints: we have to call for averaging over all possible data including data obtained in Kaohsiung.
Conditional viewpoints: we don’t need to perform another experiment in Kaohsiung. We can make statistical inference based on the data we have now.
The Weak Conditionality Principle:
Two experiments or can be performed to draw information about . Then, the actual information about should depend only on the experiment that is actually performed.
(d) Likelihood Principle:
Definition:
For observed data x, the function , considered as a
function of , is called the likelihood function.
Likelihood Principle:
All relevant experimental information is contained in the likelihood function for the observed x. Two likelihood functions contain the same information about if they are proportional to each other.
Example 13:
: the probability that a coin comes up head.
Suppose we want to know if the coin is fair, i.e.,
,
with Then, we flip a coin in a series of trials, 9 heads and 3
tails. Let
: the number of heads.
Two likelihood functions can be used. They are:
1. Binomial:
In this example,
2. Negative Binomial:
In this example, we throw a coin until 3 tails come up. Therefore,
and
By likelihood principle, and contain the same information. Thus, intuitively, the same conclusion should be achieved based on the two likelihood functions. However, classical statistical inference would result in bizarre conclusions from frequentist point of view.
1. Binomial:
The reject rule is
is some constant. Thus, in this example,
Thus, we do not reject and conclude the coin is fair.
2. Negative Binomial:
The reject rule is
is some constant. Thus, in this example,
Thus, we reject and conclude the coin is not fair.
Note:
The “robust” Bayesian paradigm which takes into account uncertainity in the prior is fundamentally correct paradigm.
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