The MCMC Module

mogp_emulator.MCMC.MCMC_step(loglikelihood, current_params, step_sizes, loglike_sign=1.0)

Method to take a weak prior Metropolis-Hastings MCMC step

Take a Metropolis-Hastings MCMC step with given log-likelihood function and step sizes. Method uses a multivariate normal distribution centered around the current parameter values from which the next point is drawn, and evaluates the log-likelihood for both points. If the next point has a larger log-likelihood, the step is always accepted, and if the log-likelihood is less for the proposed step it is accepted with a probability based on the difference between the two. If the method encounters an error when evalutaing the log-likelihood, the step is rejected. Returns the next point and a boolean indicating whether or not the step was accepted.

An optional parameter loglike_sign can be passed that must be a float with the value +/- 1. This is multiplied by the log-likelihood and thus allows methods that compute the negative log-likelihood to be used in this routine. If values other than +/- 1 are passed, the method will raise an error.

Parameters:
  • loglikelihood (function or other callable) – Log-likelihood function to be used in the MCMC step. Must be callable and must accept a single argument, which is the array holding the parameters. If this function computes the negative log-likelihood, pass loglike_sign = -1. to the function as well.
  • current_params (ndarray) – Current value of the parameters. Must be a 1D array.
  • step_sizes (ndarray) – Covariance matrix from which steps are drawn. Must be a 2D array with both dimensions the same length as current_params, and must be positive definite.
  • loglike_sign (float) – Sign for the log-likelihood function. If the provided loglikelihood function computes the negative log-likelihood, pass -1. for this parameter. Optional, default value is 1.
Returns:

Proposed next point and whether or not the point is accepted as a tuple. The first return item is the next point (as a 1D array with the same length as current_params) and the second item is a boolean indicating whether or not the step was accepted.
Return type:

tuple containing a ndarray and a bool
mogp_emulator.MCMC.MH_proposal(current_params, step_sizes)

Propose an MCMC step using a Metropolis-Hastings method

Proposes the next point in an MCMC sampler using the Metropolis-Hastings method. Inputs are the current point and a covariance matrix. The next point is drawn from a multivariate normal distribution centered around the current point. The covariance matrix must be a 2D n by n array, where n is the number of parameters, and must be positive definite. Returns a 1D array of length n holding the new proposed parameter values.

Parameters:
  • current_params (ndarray) – Current value of the parameters. Must be a 1D array.
  • step_sizes (ndarray) – Covariance matrix from which steps are drawn. Must be a 2D array with both dimensions the same length as current_params, and must be positive definite.
Returns:

New value of parameters, a 1D array with the same length as current_params
Return type:

ndarray
mogp_emulator.MCMC.autothin_samples(signal)

Automatically estimate thinning needed to obtain uncorrelated samples

This function attempts to estimate the thinning needed to obtain uncorrelated samples in an MCMC chain. For each separate parameter, the function computes the autocorrelation and estimates the lag needed to obtain uncorrelated samples. This is done by recognizing that the standard deviation of the autocorrelation of a random signal scales inversely with the square root of the number of samples, so when the autocorrelation drops below three times this, we use this as a guess to when the signal is uncorrelated. The maximum lag across all parameters is returned. If the chain contains fewer than 10 points, or the autocorrelation never drops below the target value, the method gives a warning and returns 1.

Parameters:signal (ndarray) – MCMC chain to be thinned. Must be a 1D or 2D array. If 2D, the first dimension indicates the MCMC samples, and the second dimension indicates the different parameter values. If 1D, the array is assumed to hold samples of a single parameter, and the array is reshaped to have a singleton second dimension.
Returns:Estimated stride needed to thin the samples to obtain uncorrelated samples. If the method does not succeed, gives a warning and returns 1.
Return type:int
mogp_emulator.MCMC.sample_MCMC(loglikelihood, start, step_sizes, n_samples=1000, thin=0, loglike_sign=1.0)

Draw MCMC samples for a given log-likelihood function with weak priors

Compute an MCMC chain for a given log-likelihood function with weak priors. Function requires the log-likelihood function, the starting point for the MCMC chain, and an array describing the step sizes. Optional parameters are the number of steps to take, how the thin the samples, and a sign for the log-likelihood function.

The log-likelihood function must be a function or other callable that accepts a single argument, which is a 1D array holding the current parameter values. The starting point must be a 1D array holding the starting parameter values, which must match the length of the input to the log-likelihood function. The step sizes must be a 2D array with each dimension having the same length as the number of parameters, and must be positive definite. The step size array is used as the covariance matrix for a multivariate normal distribution, from which steps are drawn.

Optional parameters are the number of steps to be taken (must be a positive integer). Note that if the chain is thinned, the number of points in the final MCMC chain will differ from the number of steps taken.

Thinning may be specified with a non-negative integer. If a positive integer is given, the chain will be thinned by only keeping every thin steps. Note that thin = 1 means that the chain will not be thinned. If thin = 0 is given (the default value), the chain will automatically be thinned by computing the autocorrelation of the chain for each parameter separately and estimating the value needed to eliminate correlations in the chain. If the autothinning method fails (usually occurrs if the posterior is multimodal), the chain will not be thinned and a warning will be given. More details on the autothinning procedure are described in the corresponding function.

An optional parameter loglike_sign can be passed that must be a float with the value +/- 1. This is multiplied by the log-likelihood and thus allows methods that compute the negative log-likelihood to be used in this routine.

Returns the final thinned MCMC chain (a 2D array, where the first dimension indicates the different samples and the second dimension indicates the different parameters), an array holding all rejected steps (also a 2D array like the MCMC chain, useful for diagnosing problems with convergence), the fraction of steps that are accepted (also useful for diagnosing problems with convergence), and the first lag autocorrelation of the thinned MCMC chain.

Parameters:
  • loglikelihood (function or other callable) – Log-likelihood function to be used in the MCMC step. Must be callable and must accept a single argument, which is the array holding the parameters. If this function computes the negative log-likelihood, pass loglike_sign = -1. to the function as well.
  • start (ndarray) – Starting value of the parameters. Must be a 1D array.
  • step_sizes (ndarray) – Covariance matrix from which steps are drawn. Must be a 2D array with both dimensions the same length as current_params, and must be positive definite.
  • n_samples (int) – Number of steps to be taken. Must be a positive integer. Optional, default value is 1000. Note that if the chain is thinned, the final MCMC chain will be shorter than this.
  • thin (int) – Integer describing how to thin the MCMC chain. A positive integer indicates manual thinning by keeping every thin steps (note that thin = 1 means the chain will not be thinned). A value of 0 will attempt to autothin the chain.
  • loglike_sign (float) – Sign for the log-likelihood function. If the provided loglikelihood function computes the negative log-likelihood, pass -1. for this parameter. Optional, default value is 1.
Returns:

MCMC chain (2D array), array of rejected points (2D array), acceptance rate (float), and first lag autocorrelation of the thinned MCMC chain (float)
Return type:

tuple containing (ndarray, ndarray, float, float)