11.3 Conjugate priors

Conjugate priors are a powerful tool in Bayesian statistics, simplifying the computation of posterior distributions. By choosing priors that match the likelihood function, analysts can obtain closed-form expressions for posteriors, making inference more tractable and efficient.

This topic explores common conjugate prior distributions, their advantages, and applications. It covers parameter estimation, predictive distributions, and hierarchical models, while also discussing limitations and alternatives to conjugate priors in Bayesian analysis.

Conjugate priors overview

• Conjugate priors are a class of prior distributions that, when combined with the likelihood function, result in a posterior distribution belonging to the same family as the prior
• Using conjugate priors simplifies the computation of the posterior distribution, making the analysis more tractable
• Conjugate priors allow for closed-form expressions of the posterior distribution, avoiding the need for complex numerical integration or sampling methods

Definition of conjugate priors

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• A prior distribution is said to be conjugate to a likelihood function if the resulting posterior distribution belongs to the same family as the prior
• The conjugacy property is determined by the mathematical form of the likelihood function and the prior distribution
• For example, if the likelihood is a binomial distribution and the prior is a beta distribution, the resulting posterior will also be a beta distribution

Advantages of using conjugate priors

• Conjugate priors lead to analytically tractable posterior distributions, simplifying the computation and interpretation of the results
• The use of conjugate priors allows for efficient updating of the posterior distribution as new data becomes available
• Conjugate priors provide a convenient way to incorporate prior knowledge or beliefs about the parameters of interest
• The resulting posterior distributions have well-known properties, making it easier to derive point estimates, credible intervals, and other quantities of interest

Common conjugate prior distributions

• Several conjugate prior distributions are commonly used in Bayesian analysis, each paired with a specific likelihood function
• The choice of the conjugate prior depends on the nature of the data and the parameters of interest
• Some common conjugate prior distributions include the beta-binomial, gamma-Poisson, Dirichlet-multinomial, normal-normal, and normal-inverse-gamma pairs

Beta-binomial conjugate priors

• The beta distribution is the conjugate prior for the binomial likelihood
• When the data follows a binomial distribution with parameter $\theta$, and the prior for $\theta$ is a beta distribution with parameters $\alpha$ and $\beta$, the posterior distribution is also a beta distribution with updated parameters $\alpha + x$ and $\beta + n - x$, where $x$ is the number of successes and $n$ is the total number of trials
• Beta-binomial conjugate priors are useful for modeling binary data or proportions (coin flips, survey responses)

Gamma-Poisson conjugate priors

• The gamma distribution is the conjugate prior for the Poisson likelihood
• When the data follows a Poisson distribution with rate parameter $\lambda$, and the prior for $\lambda$ is a gamma distribution with shape parameter $\alpha$ and rate parameter $\beta$, the posterior distribution is also a gamma distribution with updated parameters $\alpha + \sum_{i=1}^n x_i$ and $\beta + n$, where $x_i$ are the observed counts and $n$ is the number of observations
• Gamma-Poisson conjugate priors are useful for modeling count data (number of events in a fixed time interval, defects in a manufacturing process)

Dirichlet-multinomial conjugate priors

• The Dirichlet distribution is the conjugate prior for the multinomial likelihood
• When the data follows a multinomial distribution with parameter vector $\boldsymbol{\theta} = (\theta_1, \ldots, \theta_K)$, and the prior for $\boldsymbol{\theta}$ is a Dirichlet distribution with concentration parameters $\boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_K)$, the posterior distribution is also a Dirichlet distribution with updated parameters $\alpha_k + x_k$ for $k = 1, \ldots, K$, where $x_k$ is the count of observations in category $k$
• Dirichlet-multinomial conjugate priors are useful for modeling categorical data (survey responses with multiple options, document classification)

Normal-normal conjugate priors

• The normal distribution is the conjugate prior for the mean of a normal likelihood with known variance
• When the data follows a normal distribution with unknown mean $\mu$ and known variance $\sigma^2$, and the prior for $\mu$ is a normal distribution with mean $\mu_0$ and variance $\sigma_0^2$, the posterior distribution is also a normal distribution with updated mean $\frac{\frac{\mu_0}{\sigma_0^2} + \frac{n\bar{x}}{\sigma^2}}{\frac{1}{\sigma_0^2} + \frac{n}{\sigma^2}}$ and variance $\frac{1}{\frac{1}{\sigma_0^2} + \frac{n}{\sigma^2}}$, where $\bar{x}$ is the sample mean and $n$ is the sample size
• Normal-normal conjugate priors are useful for modeling continuous data with a known variance (heights, weights)

Normal-inverse-gamma conjugate priors

• The normal-inverse-gamma distribution is the conjugate prior for the mean and variance of a normal likelihood
• When the data follows a normal distribution with unknown mean $\mu$ and unknown variance $\sigma^2$, and the prior for $\mu$ and $\sigma^2$ is a normal-inverse-gamma distribution with parameters $\mu_0$, $\lambda_0$, $\alpha_0$, and $\beta_0$, the posterior distribution is also a normal-inverse-gamma distribution with updated parameters that depend on the data
• Normal-inverse-gamma conjugate priors are useful for modeling continuous data with unknown mean and variance (stock returns, measurement errors)

Posterior distribution derivation

• The posterior distribution is obtained by combining the prior distribution and the likelihood function using Bayes' theorem
• Conjugate priors allow for the derivation of the posterior distribution in closed form, without the need for numerical integration or sampling methods
• The posterior distribution represents the updated beliefs about the parameters after observing the data

Bayes' theorem in conjugate priors

• Bayes' theorem states that the posterior distribution is proportional to the product of the prior distribution and the likelihood function
• In the case of conjugate priors, the prior and the likelihood are chosen such that their product results in a posterior distribution belonging to the same family as the prior
• The posterior distribution is obtained by normalizing the product of the prior and the likelihood, ensuring that it integrates to 1

Updating prior beliefs with data

• The posterior distribution represents the updated beliefs about the parameters after observing the data
• As new data becomes available, the posterior distribution can be used as the prior for the next round of analysis, allowing for the continuous updating of beliefs
• Conjugate priors make this updating process computationally efficient, as the posterior distribution has the same form as the prior

Analytical solutions for posterior distributions

• Conjugate priors lead to analytically tractable posterior distributions, meaning that the posterior can be expressed in closed form
• The parameters of the posterior distribution are typically functions of the prior parameters and the observed data
• Having analytical solutions for the posterior distribution simplifies the computation of point estimates, credible intervals, and other quantities of interest (posterior mean, median, mode)

Parameter estimation with conjugate priors

• Conjugate priors allow for the estimation of the parameters of interest based on the posterior distribution
• Several methods can be used to obtain point estimates and uncertainty measures for the parameters
• These methods include maximum a posteriori (MAP) estimation, credible intervals, and comparison to maximum likelihood estimation

Maximum a posteriori (MAP) estimation

• MAP estimation involves finding the mode of the posterior distribution, which represents the most likely value of the parameter given the data and the prior
• For conjugate priors, the MAP estimate can often be obtained analytically by maximizing the posterior distribution
• MAP estimation provides a point estimate of the parameter that takes into account both the prior information and the observed data (beta-binomial: $\frac{\alpha - 1}{\alpha + \beta - 2}$, gamma-Poisson: $\frac{\alpha}{\beta}$)

Credible intervals for parameter estimates

• Credible intervals are the Bayesian counterpart to confidence intervals in frequentist statistics
• A credible interval represents the range of parameter values that contain a specified probability of the true parameter value, given the observed data and the prior
• For conjugate priors, credible intervals can often be obtained analytically using the quantiles of the posterior distribution (beta-binomial:

  qbeta(c(0.025, 0.975), alpha + x, beta + n - x)

  )

Comparison to maximum likelihood estimation

• Maximum likelihood estimation (MLE) is a frequentist approach to parameter estimation that relies solely on the likelihood function
• In contrast, Bayesian estimation with conjugate priors incorporates prior information in addition to the observed data
• When the sample size is large, the influence of the prior diminishes, and the Bayesian estimates converge to the MLE
• However, when the sample size is small or the prior information is strong, Bayesian estimation can provide more accurate and stable estimates than MLE

Predictive distribution

• The predictive distribution is the distribution of future observations given the observed data and the prior
• Conjugate priors allow for the analytical derivation of the predictive distribution, which is useful for making predictions and assessing model performance
• The predictive distribution can be used for model selection and Bayesian model averaging

Marginal likelihood for model selection

• The marginal likelihood, also known as the evidence, is the probability of the observed data given a specific model
• In the context of conjugate priors, the marginal likelihood can often be computed analytically by integrating out the parameters
• The marginal likelihood can be used for model selection, as it provides a measure of the overall fit of the model to the data while penalizing model complexity (beta-binomial:

  beta(alpha + x, beta + n - x) / beta(alpha, beta)

  )

Bayesian model averaging with conjugate priors

• Bayesian model averaging involves combining the predictions from multiple models, weighted by their posterior probabilities
• Conjugate priors facilitate the computation of the posterior probabilities of the models, as the marginal likelihoods can be obtained analytically
• Bayesian model averaging can improve predictive performance by accounting for model uncertainty and leveraging the strengths of different models

Conjugate priors in hierarchical models

• Hierarchical models are a class of Bayesian models that involve multiple levels of parameters, with priors specified on the parameters at each level
• Conjugate priors can be used in hierarchical models to simplify the computation of the posterior distribution and enable efficient inference
• Hierarchical models with conjugate priors are useful for modeling complex, structured data (students nested within schools, patients nested within hospitals)

Hyperparameter specification

• In hierarchical models, the priors on the parameters at each level are called hyperpriors, and their parameters are called hyperparameters
• The choice of hyperparameters can have a significant impact on the inference and should be carefully considered
• Conjugate priors allow for the specification of hyperparameters in a way that maintains the conjugacy property and enables analytical computation of the posterior distribution

Gibbs sampling for posterior inference

• Gibbs sampling is a Markov chain Monte Carlo (MCMC) method for sampling from the posterior distribution in hierarchical models
• When conjugate priors are used, the full conditional distributions of the parameters often have a known form, making Gibbs sampling particularly efficient
• Gibbs sampling involves iteratively sampling from the full conditional distributions of the parameters, which can be done analytically when conjugate priors are used

Limitations and alternatives

• While conjugate priors offer several advantages, they also have some limitations and may not always be the best choice for a given problem
• Alternative approaches, such as non-conjugate priors, MCMC methods, and empirical Bayes methods, can be used to address these limitations
• It is important to consider the specific requirements and characteristics of the problem at hand when choosing between conjugate priors and alternative methods

Non-conjugate priors and MCMC methods

• Non-conjugate priors are prior distributions that do not lead to analytically tractable posterior distributions when combined with the likelihood
• When non-conjugate priors are used, MCMC methods, such as Metropolis-Hastings or Hamiltonian Monte Carlo, can be employed to sample from the posterior distribution
• MCMC methods are more flexible than conjugate priors but can be computationally intensive and require careful tuning and convergence diagnostics

Sensitivity to prior choice

• The choice of the prior distribution can have a significant impact on the posterior inference, especially when the sample size is small
• Conjugate priors may not always accurately reflect the available prior information or the desired properties of the posterior distribution
• Sensitivity analysis should be conducted to assess the robustness of the results to different prior choices and to ensure that the prior is not overly influential

Empirical Bayes methods vs conjugate priors

• Empirical Bayes methods are a class of techniques that estimate the prior distribution from the observed data, rather than specifying it a priori
• Empirical Bayes methods can be used as an alternative to conjugate priors when the prior information is limited or when the conjugacy property is not satisfied
• However, empirical Bayes methods may not fully account for the uncertainty in the prior estimation and can be sensitive to the choice of the estimation procedure