Infr provides a set of functions for performing Bayesian Inference on systems which can be described as Markov Chains. Given a model, a data series and a prior distribution it can compute estimates for the marginal and posterior probabilities for the parameters.
Infr operates on Markov Chain models which map data points to random variables. A model should be functions of the form (f beta x) -> distribution, where distribution is a lisp-stat distribution object.
Load the system with asdf.
(asdf:load-system :infr)
For testing purposes one can generate a vector containing a realisation of a process defined by the model function, f:
(defvar y (generate-markov-chain f n :xi (x0 x1 ...))).
To compute the log posterior probability density for beta=(2.0) given the data sequence y and the model f, with a normal prior:
(log-posterior f (vector (r-normal 0d0 0.5d0)) (vector 2d0) y).
Infr explots the fact the the joint probability of a realization of a markov chain is equal to the prodduct of the probability of each step. 𝑃 (𝑥𝑛, 𝑥𝑛−1, …𝑥0) = 𝑃 (𝑥𝑛|𝑥𝑛−1)𝑃 (𝑥𝑛−1 | 𝑥𝑛−2)…𝑃 (𝑥1|𝑥0)𝑃 (𝑥0) This reduces computing the Bayesian Likelihood to a simple product.
The marginal likelihood is estimated by a one-step particle filter method. A sample of particles is drawn from the prior distribution and used to estimate the integral.
The method is explained fully in this article.
generate-markov-chain f n &key (xi #(0d0)) => vector
- f - Function (f beta x) => distribution
- n - integer
- xi - vector
Generate a realization of n elements of the Markov chain defined by f, beginning with xi.
estimate-parameters f prior y &key (offset 2) (sample-count 100) => vector
- f - model function
- prior - distribution vector
- y - data vector
- offset - integer - Offset from start of y to allow for the 'memory' of the model.
- Number of samples taken from the prior distribution to estimate beta.
Find a Monte Carlo estimate of the parameter of a model. We draw a large number of samples from 'prior. Then for each sampled value of beta we find P(y|model,beta). Finally we take the mean of the samples using P(y|model,beta) as weights, to find an expected value for beta.
log-marginal (model prior y &key (offset 2) (sample-count 100) => double-float
Estimate the log of the marginal probability for y. We take a random sample of beta and use it to estimate the integral of P(y|beta)P(beta) over beta. The result can be used to find the Bayes factor to compare different models.
log-posterior (model prior beta y &key (offset 2) (sample-count 100) => double-float
- beta - parameter vector
Calculate the posterior probability density of beta: log P(beta| model, y)