PinNUTS is not No-U-Turn-Sampler

PinNUTS🥜 is an improved Hamiltonian Monte Carlo algorithm with high acceptance rates. It benefits from avoiding trajectories from u-turning and a multinomial acceptance scheme. It is not NUTS.

PinNUTS is dynamic euclidean HMC with multinomial trajectory sampling, but DEHMCMTS doesn't sound like peanuts🥜🥜🥜.

Content

The package mainly contains:

• pinnuts.pinnuts - return samples using dyHMC
• emcee_pinnuts.PinNUTSSampler - emcee PinNUTS sampler, a derived class from emcee.Sampler

This package is forked from https://github.com/mfouesneau/NUTS

A few words about NUTS and dyHMC

Hamiltonian Monte Carlo or Hybrid Monte Carlo (HMC) is a Markov chain Monte Carlo (MCMC) algorithm that avoids the random walk behavior and sensitivity to correlated parameters, biggest weakness of many MCMC methods. Instead, it takes a series of steps informed by first-order gradient information.

This feature allows it to converge much more quickly to high-dimensional target distributions compared to simpler methods such as Metropolis, Gibbs sampling (and derivatives).

However, HMC's performance is highly sensitive to two user-specified parameters: a step size, and a desired number of steps. In particular, if the number of steps is too small then the algorithm will just exhibit random walk behavior, whereas if it is too large it will waste computations.

Hoffman & Gelman introduced NUTS or the No-U-Turn Sampler, an extension to HMC that eliminates the need to set a number of steps. NUTS uses a recursive algorithm to find likely candidate points that automatically stops when it starts to double back and retrace its steps. Empirically, NUTS perform at least as effciently as and sometimes more effciently than a well tuned standard HMC method, without requiring user intervention or costly tuning runs.

Moreover, Hoffman & Gelman derived a method for adapting the step size parameter on the fly based on primal-dual averaging. NUTS can thus be used with no hand-tuning at all.

In practice, the implementation still requires a number of steps, a burning period and a stepsize. However, the stepsize will be optimized during the burning period, and the final values of all the user-defined values will be revised by the algorithm.

More recently, the Stan team made improvements over NUTS by altering the method of recursively selecting points in a trajectory from slice sampling to multinomial sampling, which increases the acceptance rate and makes distant moves more frequent. This improvement is implemented here. These improvements are nicely described in Betancourt (2016). Additionally, Stan implements a generalised U-turning criterion. However, this does not seem to have a strong impact.

References:

• http://arxiv.org/abs/1111.4246: Matthew D. Hoffman & Andrew Gelman, 2011, "_The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo"
• http://arxiv.org/abs/1701.02434: Michael Betancourt, 2016, "A conceptual introduction to Hamiltonian Monte Carlo"

Example Usage

sampling a 2d highly correlated Gaussian distribution see pinnuts.pinnuts

• define a log-likelihood and gradient function:

  def correlated_normal(theta):
      """ Example of a target distribution that could be sampled from using NUTS.  (Doesn't include the normalizing constant.)
      Note: 
      cov = np.asarray([[1, 1.98],
                        [1.98, 4]])
      """
  
      #A = np.linalg.inv( cov )
      A = np.asarray([[50.251256, -24.874372],
                      [-24.874372, 12.562814]])
  
      grad = -np.dot(theta, A)
      logp = 0.5 * np.dot(grad, theta.T)
      return logp, grad

• set your initial conditions: number of dimensions, _number of steps, number of adaptation/burning steps, initial guess, and initial step size.:

  D = 2
  M = 5000
  Madapt = 5000
  theta0 = np.random.normal(0, 1, D)
  delta = 0.2
  
  mean = np.zeros(2)
  cov = np.asarray([[1, 1.98], 
                    [1.98, 4]])

• run the sampling:

  samples, lnprob, epsilon = pinnuts(correlated_normal, M, Madapt, theta0, delta)

• some statistics: expecting mean = (0, 0) and std = (1., 4.):

  samples = samples[1::10, :]
  print('Mean: {}'.format(np.mean(samples, axis=0)))
  print('Stddev: {}'.format(np.std(samples, axis=0)))

• a quick plot:

  import pylab as plt
  temp = np.random.multivariate_normal(mean, cov, size=500)
  plt.plot(temp[:, 0], temp[:, 1], '.')
  plt.plot(samples[:, 0], samples[:, 1], 'r+')
  plt.show()

Example usage as an EMCEE sampler

see emcee_pinnuts.test_sampler

• define a log-likelihood function:

  def lnprobfn(theta):
      return correlated_normal(theta)[0]

• define a gradient function (if not numerical estimates are made, but slower):

  def gradfn(theta):
      return correlated_normal(theta)[1]

• set your initial conditions: number of dimensions, _number of steps, number of adaptation/burning steps, initial guess, and initial step size:

  D = 2
  M = 5000
  Madapt = 5000
  theta0 = np.random.normal(0, 1, D)
  delta = 0.2
  
  mean = np.zeros(2)
  cov = np.asarray([[1, 1.98],
                    [1.98, 4]])

• run the sampling:

  sampler = PinNUTSSampler(D, lnprobfn, gradfn)
  samples = sampler.run_mcmc( theta0, M, Madapt, delta )