Summary  
The chapter explains how to implement a likelihood function that evaluates the plausibility of model parameters given fixed observed data and how to combine that function with a prior distribution to update parameter beliefs via Bayes’ theorem.

General domain of usage  
Statistical modeling

The **likelihood function** is a central concept in Bayesian inference. It quantifies how plausible the observed data are for different values of the model parameters. Formally, if you have observed data $$x$$ and are considering a parameter $$θ$$, the likelihood function is written as $$L(θ; x) = P(x | θ)$$. This means you treat the data as fixed and view the likelihood as a function of the parameter.

Intuitively, the likelihood answers the question: **"Given the data I have seen, how plausible is each possible value of the parameter?"** In Bayesian inference, the likelihood is combined with the prior distribution to update beliefs about the parameter after seeing the data, according to Bayes' theorem. Unlike probability, which often describes the chance of future data given known parameters, likelihood is used to evaluate which parameter values best explain the data you have already observed.

If you observe $$n$$ coin flips with $$k$$ heads and want to estimate the probability of heads $$p$$, the likelihood function is:

$$
L(p | k | n) = p^k * (1-p)^(n-k),
$$  

where $$p$$ is between 0 and 1.

Bernoulli Distribution

For data points $$x_1, ..., x_n$$ assumed to come from a normal distribution with unknown mean $$μ$$ and known variance $$σ^2$$, the likelihood function is:

$$
L(\mu \mid x, \sigma^2)
= \prod_{i=1}^{n}
\frac{1}{\sqrt{2\pi\sigma^2}}
\exp^{\left(
-\frac{(x_i - \mu)^2}{2\sigma^2}
\right)}
$$

which is a function of $$μ$$ given the observed data.

Normal Distribution

If you observe $$n$$ counts from a Poisson process with unknown rate $$λ$$, the likelihood is:

$$
L(\lambda \mid x)
= \prod_{i=1}^{n}
\frac{e^{-\lambda}\lambda^{x_i}}{x_i!}
$$


viewed as a function of $$λ$$ for your observed counts.

Poisson Distribution

Which statement best describes the role of the likelihood function in Bayesian inference?

Which of the following best distinguishes likelihood from probability in statistical modeling?

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Likelihood Functions in Bayesian Inference

1. Which statement best describes the role of the likelihood function in Bayesian inference?

2. Which of the following best distinguishes likelihood from probability in statistical modeling?