All Study Guides Bayesian Statistics Unit 4
📊 Bayesian Statistics Unit 4 – Likelihood functionsLikelihood functions are a cornerstone of statistical inference, quantifying how well different parameter values explain observed data. They bridge the gap between theoretical models and empirical observations, enabling us to estimate parameters, compare hypotheses, and update our beliefs.
In Bayesian statistics, likelihood functions play a crucial role in combining prior knowledge with new data. They form the basis for updating prior distributions to posterior distributions, allowing for a principled approach to incorporating evidence and quantifying uncertainty in our statistical analyses.
What's a Likelihood Function?
Quantifies the plausibility of parameter values given observed data
Denoted as L ( θ ∣ x ) L(\theta|x) L ( θ ∣ x ) where θ \theta θ represents the parameter(s) and x x 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 ∣ θ ) P(x|\theta) P ( x ∣ θ )
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 (d a t a data d a t a , p r i o r b e l i e f s prior beliefs p r i or b e l i e f s , e x p e r t k n o w l e d g e expert knowledge e x p er t kn o wl e d g e )
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 G a u s s i a n Gaussian G a u ss ian , B i n o m i a l Binomial B in o mia l , P o i s s o n Poisson P o i sso n , E x p o n e n t i a l Exponential E x p o n e n t ia l
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 (i i d iid ii d ) data leads to the likelihood being a product of individual probabilities
Dependent data requires more complex likelihood formulations (t i m e s e r i e s time series t im eser i es , s p a t i a l m o d e l s spatial models s p a t ia l m o d e l s )
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 μ \mu μ and variance σ 2 \sigma^2 σ 2
Likelihood function: L ( μ , σ 2 ∣ x ) ∝ ∏ i = 1 n 1 2 π σ 2 exp ( − ( x i − μ ) 2 2 σ 2 ) 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) L ( μ , σ 2 ∣ x ) ∝ ∏ i = 1 n 2 π σ 2 1 exp ( − 2 σ 2 ( x i − μ ) 2 )
Binomial likelihood
Used for binary or success/failure data with a fixed number of trials
Parameterized by the success probability p p p and the number of trials n n n
Likelihood function: L ( p ∣ x ) ∝ p ∑ x i ( 1 − p ) n − ∑ x i L(p|x) \propto p^{\sum x_i} (1-p)^{n-\sum x_i} L ( p ∣ x ) ∝ p ∑ x i ( 1 − p ) n − ∑ x i
Poisson likelihood
Used for count data or rare events occurring in a fixed interval or space
Parameterized by the rate parameter λ \lambda λ
Likelihood function: L ( λ ∣ x ) ∝ ∏ i = 1 n λ x i e − λ x i ! L(\lambda|x) \propto \prod_{i=1}^n \frac{\lambda^{x_i} e^{-\lambda}}{x_i!} L ( λ ∣ x ) ∝ ∏ i = 1 n x i ! λ x i e − λ
Exponential likelihood
Used for positive continuous data with a constant hazard rate
Parameterized by the rate parameter λ \lambda λ
Likelihood function: L ( λ ∣ x ) ∝ λ n exp ( − λ ∑ i = 1 n x i ) L(\lambda|x) \propto \lambda^n \exp\left(-\lambda \sum_{i=1}^n x_i\right) L ( λ ∣ x ) ∝ λ n exp ( − λ ∑ i = 1 n x i )
Other common likelihood distributions include G a m m a Gamma G amma , B e t a Beta B e t a , M u l t i n o m i a l Multinomial M u lt in o mia l , D i r i c h l e t Dirichlet D i r i c h l e t
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 θ ^ M L E = arg max θ L ( θ ∣ x ) \hat{\theta}_{MLE} = \arg\max_{\theta} L(\theta|x) θ ^ M L E = arg max θ L ( θ ∣ 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 ∣ θ ) P(x|\theta) P ( x ∣ θ )
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 ( θ ∣ x ) L(\theta|x) L ( θ ∣ 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 ( θ ) p(\theta) p ( θ ) to form the posterior distribution p ( θ ∣ x ) p(\theta|x) p ( θ ∣ x )
Posterior distribution represents the updated beliefs about the parameters after observing the data
Obtained through Bayes' theorem: p ( θ ∣ x ) ∝ L ( θ ∣ x ) × p ( θ ) p(\theta|x) \propto L(\theta|x) \times p(\theta) p ( θ ∣ x ) ∝ L ( θ ∣ x ) × p ( θ )
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 (p o s t e r i o r m e a n posterior mean p os t er i or m e an , m e d i a n median m e d ian , m o d e mode m o d e )
Interval estimates provide a range of plausible parameter values (c r e d i b l e i n t e r v a l s credible intervals cre d ib l e in t er v a l s )
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 (e c o n o m i c s economics eco n o mi cs , b i o l o g y biology bi o l o g y , e n g i n e e r i n g engineering e n g in eer in g )
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 (E M EM 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 (A B C ABC A BC ) for models with intractable likelihoods
Synthetic likelihood for models where the likelihood function is difficult to evaluate
Applications in various domains such as f i n a n c e finance f inan ce , m a r k e t i n g marketing ma r k e t in g , e p i d e m i o l o g y epidemiology e p i d e mi o l o g y , g e n e t i c s genetics g e n e t i cs