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A note on Likelihood and Likelihood Based Inference
Dr. James Kurian
Associate Professor, Department of Statistics, Maharaja’s College, Ernakulam, Kerala, India
DOI:
https://doi.org/10.51583/IJLTEMAS.2026.150600082
Received: 21 June 2026; Accepted: 26 June 2026; Published: 07 July 2026
ABSTRACT
Statistical inference aims to draw conclusions about unknown parameters using observed data. Among the
various inferential frameworks, likelihood-based inference occupies a central position because it provides a
direct measure of the support that observed data give to competing parameter values. This paper presents a
concise review of the likelihood function and its role in statistical inference. The concept of likelihood is
introduced for both discrete and continuous probability models, emphasizing its interpretation as a measure of
relative evidence rather than a probability distribution for the parameter. Several illustrative examples, including
binomial, normal, and exponential models, demonstrate the construction and interpretation of likelihood
functions. The paper discusses important properties of likelihood, including invariance under multiplication by
positive constants and the combination of independent likelihoods through multiplication. Connections between
likelihood and Bayesian inference are examined, showing how posterior information can be viewed as the
product of prior and current likelihood information. Fundamental principles of likelihood-based inference,
including the Likelihood Principle and the Stopping Rule Principle, are reviewed with classical examples. The
role of maximum likelihood estimation as a practical method for parameter estimation is also highlighted. The
discussion illustrates how the likelihood function serves as a compact and informative summary of statistical
evidence, providing a unified framework for estimation and inference.
Key words: Likelihood Function; Likelihood-Based Inference; Likelihood Principle; Maximum Likelihood
Estimation; Bayesian Inference; Stopping Rule Principle; Statistical Evidence; Parameter Estimation.
INTRODUCTION
Statistical inference is concerned with drawing conclusions about unknown population characteristics on the
basis of observed data. Over the past century, several inferential paradigms have been developed, among which
likelihood-based inference has emerged as one of the most influential and widely applicable frameworks. The
likelihood approach provides a direct mechanism for quantifying the support that observed data offer to
competing values of an unknown parameter and forms the theoretical foundation for many modern statistical
methods (Fisher, 1922; Cox, 2006). The concept of likelihood was introduced and systematically developed by
Fisher (1922), who distinguished likelihood from probability and emphasized its role in parameter estimation.
While probability describes the chance of observing future data under a specified model, likelihood measures
the relative plausibility of parameter values given the observed data. This distinction has profound implications
for statistical reasoning and has led to the development of a coherent inferential framework based solely on the
information contained in the observed sample (Edwards, 1992). Furthermore, many desirable large-sample
properties of estimators, including consistency, asymptotic normality, and efficiency, arise naturally within the
likelihood framework (Casella & Berger, 2002; Lehmann & Casella, 1998). Likelihood also plays a central role
in Bayesian inference. Bayes' theorem combines prior information with the likelihood derived from observed
data to produce the posterior distribution of the parameter. From this perspective, the likelihood function acts as
the mechanism through which new information is incorporated into prior beliefs, highlighting the close
relationship between Bayesian and likelihood-based approaches (Royall, 1997; Cox, 2006).
The objective of this paper is therefore to provide a concise exposition of the likelihood function and its role in
statistical inference. Through a series of illustrative examples, the paper discusses the construction and
interpretation of likelihood functions, the combination of evidence from independent sources, the relationship
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between likelihood and Bayesian inference, and the implications of the Likelihood Principle and Stopping Rule
Principle. The discussion highlights how the likelihood function for an observed data serves as a compact and
informative summary of statistical evidence and provides a unifying framework for statistical inference.
Likelihood
The likelihood function is one of the most basic concepts in statistical inference. Entire theories of inference
have been constructed based on it. We use scalar data using the notation x, vector data using the notation x, a
scalar parameter using the notation and a vector parameter using the notation . But it makes no difference
in likelihood inference if the data x is a vector. Nor does it make a difference in the fundamental definitions if
the parameter is a vector. Moreover, the term ‘probability density’ is used to denote both discrete and
continuous models in this document. The purpose of the likelihood function is to convey the information about
the unknown quantities. Likelihood inferences are based only on the data x and the model
Ω
which
represents the entire set of possible probability measure under consideration.
Having observed x, the likelihood function for the given sample with a probability density
is denoted by
. Now likelihood
defined on the parameter space taking values in
, defined by
Since,
is just the probability of obtaining the data x when the true value of the parameter is , on , we
prefer
as the true value of the parameter over
whenever
.
Assuming a discrete model parameterized by a fixed unknown , the likelihood
is the probability of the
observed x considered as a function of . Uncertainty in the data is conveyed through the likelihood function
and provides relative preferences for various parameter values. It describes how likely the observed data are for
each value of θ. The interpretation of the likelihood is the probability of x assuming the true value of the
parameter is but not the probability of for given x. Thus, likelihood is considered as a function of , for fixed
x, whereas the density is considered as a function of x for fixed
For example, consider a binomial experiment with n=5, the likelihood function is
=
and for x=0,1,2,3,4 and 5, the plot of
are given below.
Fig-1
Obvious interpretation of the graph is that, when x=0, the likelihood is concentrated near zero indicating that the
true value of is close to zero. When x=1, the likelihood is concentrated near 0.2 indicating that the true value
of is close to 0.2 and so on. Hence, likelihood provides a relative preference for various parameter values. Also
note that, for a typical value of x=1 out of n=5 tosses, the appropriate model for the observed data is the Binomial
(5, model with =
with the corresponding likelihood
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Likelihood function for different x
θ
Likelihood
x=0
x=1
x=2
x=3
x=4
x=5
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=
which represented by the blue colour in Fig-1. It shows that the likelihood is peaked at = 0.2 and takes a
maximum of 0.4.
In continuous case, we can define
Or, if
, assuming the continuity of
>
The above result shows that for the comparison of within
the likelihood is required up to
, and
hence we can ignore .
In general, the likelihood for a model can be expressed as
where h(x) is any strictly positive valued function of x that does not contain the parameter and hence, for the
purpose of making inference about it does not matter if multiplicative terms not containing unknown
parameters are dropped from the likelihood function. In other words, a likelihood function and any positive
multiple of it are equivalent as far as likelihood-based inferences are concerned.
Example -1: Suppose that
are i.i.d. samples from
distribution where, is
unknown and
is known. The likelihood function is given by
=
Therefore, we need only to consider the part
For example, suppose n= 50,
, the plot of the likelihood function for different values of = 0, 1, 2 and 3
are shown below.
We can observe from the above figure that the likelihood function concentrates near to the mean value.
Hence, the entire likelihood function is the carrier of information on under x and not the maximiser of
-1 0 1 2 3 4
0.0 0.2 0.4 0.6 0.8 1.0
Likelihood function of N(θ,1) for different xbar
θ
Likelihood
xbar=0
xbar=1
xbar=2
xbar=3
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the parameter. Further, if
and
are two independent data sets with probabilities
and
, then likelihood of the combined data set is
*
=
, where
and
are the likelihoods of
the individual data sets.
Example-2:Suppose that a coin is tossed 10 times and the head occurred less than 4 times. Then the likelihood
function is
Example-3:The medical team identifies the 50
th
sampled person having the first AB blood group. The
likelihood for observing an AB blood group is
Example-4:Based on a random sample of size 4 from exponential distribution with parameter , it is only
reported the maximum of the sample value
=10 and the second sample of size n=3 reported the sum s=8,
then
For the maximum, the likelihood function is given by
For n=4 and
=10, we have
For the sum, the distribution is Gamma (3, , then
Therefore, the likelihood of the combined data
likelihood curves have been normalized to have maximum value 1, so their shapes can be compared directly
without scale effects.
likelihood curves have been normalized to have maximum value 1, so their shapes can be compared directly
without scale effects.
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is more left-shifted and relatively sharper, indicating that the extreme observation favors smaller θ.
is broader and peaks near θ=3/8, reflecting information from the total exposure time.
, is the most
concentrated, illustrating how combining independent summaries reduces uncertainty.
Likelihood function and Bayesian
Bayes rule is stated as
, where c is the normalizing constant and the likelihood of
the sample is
. The distribution of
is known as the prior distribution, which we can call as
prior likelihood.
This means that the effect of Bayesian method is the same as likelihood method as we can consider posterior
likelihood as the combining of prior likelihood and current likelihood. That is
Posterior Likelihood Prior likelihood * Current likelihood
If we do not know anything about , we can take
=1, the likelihood function expresses the current
information on , after observing x, and in such cases posterior density and likelihood functions would be the
equivalent.
Likelihood Principle (Alan Birnbaum (1962))
If two experiments yield data such that their likelihood functions are proportional, then those two sets of data
represent equivalent instances of statistical evidence. Likelihood principle asserts that the likelihood function
contains all the necessary information and hence if two model and data combinations yield equivalent likelihood
functions, then inferences about the unknown parameter must be the same. According to this principle, the
likelihood is the most compact summary of the data without loose of information and, any inference regarding
the unknown must be based on
alone. In other words, when two likelihood functions are equal or even
proportional to each other, both are equivalent and must yield same parameter and variance estimates. Principle
has the effect of concluding equivalent strength of evidence for proportional likelihood functions.
Example-1
Case-1: Suppose that a coin is tossed 10 times and that x=4 heads observed resulting in x is a Binomial(10, )
with =
. The likelihood function is given by
=
.
Case-2: Suppose a coin is tossed independently until four heads are obtained and the
number of tails observed until the fourth head is x=6. Then x is a Negative-Binomial(4, ) with =
.
The likelihood function is given by
=
.
Note that in both cases the likelihoods are proportional. So, the likelihood principle asserts that we must ignore
the fact that the data were obtained in entirely different ways and these two model and data combinations must
yield the same inferences about the unknown . But there are many situations in which we might consider
additional information regarding the data beyond the likelihood function and might take different inferences for
the two situations. For example, while using a moment method of estimation or testing the null hypothesis of
we make use of the sampling method and additional model features beyond the observed data and
we may reach entirely different inference about for the two situations.
Example-2 (Basu,1975)
Consider four different coin tossing experiments
(1) Toss the coin exactly10 times;
(2) Continue tossing until 6 heads appear;
(3) Continue tossing until 3 consecutive heads appear;
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(4) Continue tossing until the accumulated number of heads exceeds that of tails by exactly 2
Suppose that all four experiments have the same outcome x=(T, H, T, T, H, H, T, H, H, H).
We may feel that the evidence for θ, the probability of heads, is the same in every case. Once the sequence of
heads and tails is known, the intentions of the original experimenter are immaterial to inference about the
probability of heads.
Stopping Rule Principle
The intentions of the experimenter, represented by τ, are irrelevant for making inferences about θ, once the
observations (
) are known. This means, once the data is observed, we can ignore the sampling
plan.
CONCLUSION
The likelihood function provides a fundamental framework for statistical inference by summarizing the
information contained in observed data regarding unknown parameters. Unlike probability distributions, which
describe uncertainty in future observations, likelihood functions quantify the relative support that observed data
provide for different parameter values. Through a series of examples, this paper has demonstrated how likelihood
functions can be constructed, interpreted, and combined across independent sources of information. The
discussion highlights that the entire shape of the likelihood function carries inferential information and not
merely its maximum point. The Likelihood Principle emphasizes that all relevant evidence concerning a
parameter is contained in the likelihood function, while the Stopping Rule Principle suggests that inference
should depend only on the observed data and not on the sampling intentions. Furthermore, the close relationship
between likelihood and Bayesian inference illustrates how prior and current information can be integrated within
a unified inferential framework. Overall, likelihood-based inference offers a coherent and powerful approach for
estimation, hypothesis testing, and evidence evaluation, making it one of the cornerstones of modern statistical
inference.
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