📊Bayesian Statistics Unit 4 – Likelihood functions

What's a Likelihood Function?

• Quantifies the plausibility of parameter values given observed data
• Denoted as $L(\theta|x)$ where $\theta$ represents the parameter(s) and $x$ represents the observed data
• Differs from probability as it treats the parameter(s) as variable(s) and the data as fixed
• Proportional to the probability of observing the data given the parameter values $P(x|\theta)$
• Plays a crucial role in parameter estimation and model selection
  • Used to determine the most plausible parameter values that explain the observed data
  • Allows for comparing the relative support for different models or hypotheses
• Essential concept in Bayesian inference for updating prior beliefs about parameters
• Can be used to construct confidence intervals and hypothesis tests

Why Likelihood Matters in Bayesian Stats

• Fundamental component of Bayesian inference alongside the prior distribution
• Allows updating prior beliefs about parameters based on observed data
• Combines with the prior distribution to form the posterior distribution
  • Posterior distribution represents the updated beliefs about the parameters after considering the data
  • Obtained by multiplying the likelihood function and the prior distribution
• Enables a principled approach to incorporate prior knowledge and data into parameter estimation
• Provides a framework for model comparison and selection
  • Bayes factors compare the likelihood of the data under different models
  • Allows for assessing the relative evidence for competing hypotheses
• Crucial for making probabilistic statements and quantifying uncertainty in Bayesian analysis
• Facilitates the integration of multiple sources of information ($data$, $prior beliefs$, $expert knowledge$)

Building Likelihood Functions

• Specify the probability distribution that generates the observed data given the parameters
• Determine the functional form of the likelihood based on the assumed data generating process
  • Commonly used distributions include $Gaussian$, $Binomial$, $Poisson$, $Exponential$
  • Choice depends on the nature of the data and the underlying scientific context
• Treat the observed data as fixed and express the likelihood as a function of the parameters
• Incorporate any relevant assumptions or constraints on the parameters
• Consider the independence structure of the data points
  • Independent and identically distributed ($iid$) data leads to the likelihood being a product of individual probabilities
  • Dependent data requires more complex likelihood formulations ($time series$, $spatial models$)
• Normalize the likelihood function to ensure it integrates to a constant
• Verify that the likelihood function is mathematically valid and computationally tractable

Common Likelihood Distributions

• Gaussian (Normal) likelihood
  • Used for continuous data that is symmetric and unimodal
  • Parameterized by the mean $\mu$ and variance $\sigma^2$
  • Likelihood function: $L(\mu, \sigma^2|x) \propto \prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x_i-\mu)^2}{2\sigma^2}\right)$
• Binomial likelihood
  • Used for binary or success/failure data with a fixed number of trials
  • Parameterized by the success probability $p$ and the number of trials $n$
  • Likelihood function: $L(p|x) \propto p^{\sum x_i} (1-p)^{n-\sum x_i}$
• Poisson likelihood
  • Used for count data or rare events occurring in a fixed interval or space
  • Parameterized by the rate parameter $\lambda$
  • Likelihood function: $L(\lambda|x) \propto \prod_{i=1}^n \frac{\lambda^{x_i} e^{-\lambda}}{x_i!}$
• Exponential likelihood
  • Used for positive continuous data with a constant hazard rate
  • Parameterized by the rate parameter $\lambda$
  • Likelihood function: $L(\lambda|x) \propto \lambda^n \exp\left(-\lambda \sum_{i=1}^n x_i\right)$
• Other common likelihood distributions include $Gamma$, $Beta$, $Multinomial$, $Dirichlet$

Maximum Likelihood Estimation

• Method for estimating the parameters of a statistical model by maximizing the likelihood function
• Finds the parameter values that make the observed data most probable under the assumed model
• Denoted as $\hat{\theta}_{MLE} = \arg\max_{\theta} L(\theta|x)$
• Computed by solving the likelihood equations obtained by setting the derivatives of the log-likelihood to zero
  • Log-likelihood is often used for mathematical convenience and numerical stability
  • Maximum of the log-likelihood corresponds to the maximum of the likelihood function
• Provides point estimates of the parameters without incorporating prior information
• Asymptotically unbiased, consistent, and efficient under certain regularity conditions
• Can be used as a starting point for Bayesian inference by combining with prior distributions
• Limitations include potential overfitting, sensitivity to model misspecification, and lack of uncertainty quantification

Likelihood vs. Probability

• Likelihood and probability are related but distinct concepts
• Probability quantifies the plausibility of an event or outcome before it occurs
  • Probability is a function of the data given fixed parameter values $P(x|\theta)$
  • Probabilities sum or integrate to one over the sample space of possible outcomes
• Likelihood quantifies the plausibility of parameter values given observed data
  • Likelihood is a function of the parameters given fixed observed data $L(\theta|x)$
  • Likelihoods do not necessarily sum or integrate to one over the parameter space
• Likelihood is not a probability distribution over the parameters
  • It does not obey the axioms of probability
  • Cannot be directly interpreted as the probability of the parameters being true
• Likelihood ratios compare the relative plausibility of different parameter values
  • Ratios of likelihoods are meaningful and interpretable
  • Allows for assessing the relative support for different hypotheses or models
• Likelihood principle states that all relevant information for inference is contained in the likelihood function
  • Implies that inferences should be based on the likelihood rather than the probability of the data

Likelihood in Bayesian Inference

• Likelihood function is a key component of Bayesian inference
• Combines with the prior distribution $p(\theta)$ to form the posterior distribution $p(\theta|x)$
  • Posterior distribution represents the updated beliefs about the parameters after observing the data
  • Obtained through Bayes' theorem: $p(\theta|x) \propto L(\theta|x) \times p(\theta)$
• Likelihood function determines how the data influences the posterior distribution
  • Stronger likelihood concentrates the posterior around the most plausible parameter values
  • Weaker likelihood results in a more diffuse posterior distribution
• Bayesian inference incorporates prior knowledge and uncertainty about the parameters
  • Prior distribution represents the initial beliefs before observing the data
  • Likelihood function updates the prior beliefs based on the observed data
• Posterior distribution is used for parameter estimation, prediction, and decision making
  • Point estimates can be obtained using summary statistics ($posterior mean$, $median$, $mode$)
  • Interval estimates provide a range of plausible parameter values ($credible intervals$)
• Bayesian model comparison and selection rely on the likelihood function
  • Bayes factors compare the marginal likelihood of the data under different models
  • Marginal likelihood integrates the likelihood over the prior distribution of the parameters

Practical Applications

• Parameter estimation in various fields ($economics$, $biology$, $engineering$)
  • Estimating the parameters of regression models, time series models, or machine learning algorithms
  • Determining the most plausible values of physical constants or biological rates
• Hypothesis testing and model selection
  • Comparing the relative support for different scientific hypotheses or theories
  • Selecting the best model among a set of competing models based on their likelihood
• Bayesian inference in decision making and uncertainty quantification
  • Updating prior beliefs about parameters or hypotheses based on observed data
  • Incorporating expert knowledge and prior information into the analysis
• Likelihood-based methods for handling missing data or measurement errors
  • Expectation-Maximization ($EM$) algorithm for parameter estimation with incomplete data
  • Likelihood-based imputation methods for handling missing values
• Likelihood-free inference methods for complex models
  • Approximate Bayesian Computation ($ABC$) for models with intractable likelihoods
  • Synthetic likelihood for models where the likelihood function is difficult to evaluate
• Applications in various domains such as $finance$, $marketing$, $epidemiology$, $genetics$