The function evidence_CTI uses ZV-CV on the controlled thermodynamic integration estimator for the normalising constant.
evidence_CTI(samples, loglike, der_loglike, der_logprior, temperatures,
temperatures_all, most_recent, obs_estim_choose, obs_estim, options)evidence_SMC(samples, loglike, der_loglike, der_logprior, temperatures,
temperatures_all, most_recent, obs_estim_choose, obs_estim, options)
An \(N\) by \(d\) by \(T\) matrix of samples from the \(T\) power posteriors, where \(N\) is the number of samples and \(d\) is the dimension of the target distribution
An \(N\) by \(T\) matrix of log likelihood values corresponding to samples
An \(N\) by \(d\) by \(T\) matrix of the derivatives of the log likelihood with respect to the parameters, with parameter values corresponding to samples
An \(N\) by \(d\) by \(T\) matrix of the derivatives of the log prior with respect to the parameters, with parameter values corresponding to samples
A vector of length \(T\) of temperatures for the power posterior temperatures
An adjusted vector of length \(tau\) of temperatures. Better performance should be obtained with a more conservative temperature schedule. See Expand_Temperatures for a function to adjust the temperatures.
A vector of length \(tau\) which gives the indices in the original temperatures that the new temperatures correspond to.
See zvcv.
See zvcv.
See zvcv.
The function evidence_CTI returns a list, containing the following components:
log_evidence_PS1: The 1st order quadrature estimate for the log normalising constant
log_evidence_PS2: The 2nd order quadrature estimate for the log normalising constant
regression_LL: The set of \(tau\) zvcv type returns for the 1st order quadrature expectations
regression_vLL: The set of \(tau\) zvcv type returns for the 2nd order quadrature expectations
The function evidence_SMC returns a list, containing the following components:
log_evidence: The logged SMC estimate for the normalising constant
regression_SMC: The set of \(tau\) zvcv type returns for the expectations
Mira, A., Solgi, R., & Imparato, D. (2013). Zero variance Markov chain Monte Carlo for Bayesian estimators. Statistics and Computing, 23(5), 653-662.
South, L. F., Oates, C. J., Mira, A., & Drovandi, C. (2019). Regularised zero variance control variates for high-dimensional variance reduction. https://arxiv.org/abs/1811.05073
See Expand_Temperatures for a function that can be used to find stricter (or less stricter) temperature schedules based on the conditional effective sample size. See an example at VDP and see ZVCV for more package details.