The document defines and discusses the likelihood function. The key points are: 1) The likelihood function is a function of the parameters of a statistical model that represents the probability of the observed data given the parameters. 2) It is not a probability density function itself, but rather represents the probability or density of the observed outcomes given the parameter values. 3) The likelihood function is used to generate maximum likelihood estimators and in Bayesian inference. It represents the likelihood of the parameter values given the observed data.

1. Definition 2
Likelihood Function

2. Likelihood function
• The Likekihood function of a random
variables X1,X2,...,Xn is defined to be the joint
density of the n random variables, say
fx1,...,xn(x1,...,xn; Θ) , which is considered to be a
function of Θ. In particular, if X1,...,Xn is a
random sample from the density f (x; Θ) , then
the likelihood function is f (x1; Θ)f (x2; Θ).....f
(xn; Θ) . ///

3. • In statistics, the likelihood function (often
simply the likelihood) is a function of the
parameters of a statistical model that plays a
key role in statistical inference.
• but, in statistical usage, a clear technical
distinction is made: the probability of some
observed outcomes given a set of parameter
values is referred to as the likelihood of the set
of parameter values given the observed
outcomes.

4. 1.the likelihood function is a function.
2.the likelihood function is not a probability
density function.
3.if the data are iid then the likelihood is
___________ iid case only.
4.the likelihood is only defined up to constant of
propotionality.
5.the likelihood function is used (i) to generate
estimators (the maximum likelihood
estimator) and (ii) as a key ingredient in
Bayesian inference.

5. • Mathematically, writing X for the set of
observed data and Θ for the set of parameter
values, the expression P(X | Θ), the probability
of X given Θ, can be interpreted as the
expression L( Θ| X) , the likelihood of Θ given
X. The interpretation of L( Θ | X) as a function
of Θ is especially obvious when X is fixed and
Θ is allowed to vary.

6. • Generally, L(Θ | X) is permitted to be any positive
multiple of P(X | Θ). More precisely then, a
likelihood function is any representative from an
equivalence class of functions,
• where the constant of proportionality α > 0 is not
permitted to depend upon Θ. In particular, the
numerical value L(Θ | X) alone is immaterial; all
that matters are likelihood ratios, such as those of
the form that are invariant with respect to the
constant of proportionality α.

7. Notation
• To remind ourselves to think of the likelihood
function as a function of Θ, we shall use the
notation L( Θ;x1,...,xn) or L(.;x1,...,xn) for the
function. ///

8. The Likelihood Principle
• An informal summary of the likelihood principle
may be that inferences from data to hypotheses
should depend on how likely the actual data are
under competing hypotheses, not on how likely
imaginary data would have been under a single
"null" hypothesis or any other properties of merely
possible data.
• Bayesian inferences depend only on the probabilities
assigned due to the observed data, not due to other
data that might have been observed.

9. • A more precise interpretation may be that
inference procedures which make inferences
about simple hypotheses should not be
justified by appealing to probabilities assigned
to observations that have not occurred.
• The usual interpretation is that any two
probability models with the same likelihood
function yield the same inference for θ.

10. Difference of Probability and Likelihood
• 1.“Probability” and “likelihood” can be both
used to express a prediction and odds of
occurrences.
• 2.“Probability” refers to a “chance” while
likelihood refers to a “possibility.”
• 3.A probability follows clear parameters and
computations while a likelihood is based
merely on observed factors.

11. End of the slide
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12. Definition
• Let Xn=(x1,..,xn) have joint density
p(xn; Θ)=P(x1,...,xn; Θ) where Θ ∈Θ. The
likelihood function L :Θ [0, ∞) is defined by
L( Θ) = L( Θ;xn)
where xn is fixed and Θ varies in Θ.