# uber/pyro

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 from __future__ import absolute_import, division, print_function import numbers import torch def _compute_chain_variance_stats(input): # compute within-chain variance and variance estimator # input has shape N x C x sample_shape N = input.size(0) chain_var = input.var(dim=0) var_within = chain_var.mean(dim=0) var_estimator = (N - 1) / N * var_within if input.size(1) > 1: chain_mean = input.mean(dim=0) var_between = chain_mean.var(dim=0) var_estimator = var_estimator + var_between return var_within, var_estimator def gelman_rubin(input, chain_dim=0, sample_dim=1): """ Computes R-hat over chains of samples. It is required that ``input.size(sample_dim) >= 2`` and ``input.size(chain_dim) >= 2``. :param torch.Tensor input: the input tensor. :param int chain_dim: the chain dimension. :param int sample_dim: the sample dimension. :returns torch.Tensor: R-hat of ``input``. """ assert input.dim() >= 2 assert input.size(sample_dim) >= 2 assert input.size(chain_dim) >= 2 # change input.shape to 1 x 1 x input.shape # then transpose sample_dim with 0, chain_dim with 1 sample_dim = input.dim() + sample_dim if sample_dim < 0 else sample_dim chain_dim = input.dim() + chain_dim if chain_dim < 0 else chain_dim assert chain_dim != sample_dim input = input.reshape((1, 1) + input.shape) input = input.transpose(0, sample_dim + 2).transpose(1, chain_dim + 2) var_within, var_estimator = _compute_chain_variance_stats(input) rhat = (var_estimator / var_within).sqrt() return rhat.squeeze(max(sample_dim, chain_dim)).squeeze(min(sample_dim, chain_dim)) def split_gelman_rubin(input, chain_dim=0, sample_dim=1): """ Computes R-hat over chains of samples. It is required that ``input.size(sample_dim) >= 4``. :param torch.Tensor input: the input tensor. :param int chain_dim: the chain dimension. :param int sample_dim: the sample dimension. :returns torch.Tensor: split R-hat of ``input``. """ assert input.dim() >= 2 assert input.size(sample_dim) >= 4 # change input.shape to 1 x 1 x input.shape # then transpose chain_dim with 0, sample_dim with 1 sample_dim = input.dim() + sample_dim if sample_dim < 0 else sample_dim chain_dim = input.dim() + chain_dim if chain_dim < 0 else chain_dim assert chain_dim != sample_dim input = input.reshape((1, 1) + input.shape) input = input.transpose(0, chain_dim + 2).transpose(1, sample_dim + 2) N_half = input.size(1) // 2 new_input = torch.stack([input[:, :N_half], input[:, -N_half:]], dim=1) new_input = new_input.reshape((-1, N_half) + input.shape[2:]) split_rhat = gelman_rubin(new_input) return split_rhat.squeeze(max(sample_dim, chain_dim)).squeeze(min(sample_dim, chain_dim)) def _fft_next_good_size(N): # find the smallest number >= N such that the only divisors are 2, 3, 5 if N <= 2: return 2 while True: m = N while m % 2 == 0: m //= 2 while m % 3 == 0: m //= 3 while m % 5 == 0: m //= 5 if m == 1: return N N += 1 def autocorrelation(input, dim=0): """ Computes the autocorrelation of samples at dimension ``dim``. Reference: https://en.wikipedia.org/wiki/Autocorrelation#Efficient_computation :param torch.Tensor input: the input tensor. :param int dim: the dimension to calculate autocorrelation. :returns torch.Tensor: autocorrelation of ``input``. """ if (not input.is_cuda) and (not torch.backends.mkl.is_available()): raise NotImplementedError("For CPU tensor, this method is only supported " "with MKL installed.") # Adapted from Stan implementation # https://github.com/stan-dev/math/blob/develop/stan/math/prim/mat/fun/autocorrelation.hpp N = input.size(dim) M = _fft_next_good_size(N) M2 = 2 * M # transpose dim with -1 for Fourier transform input = input.transpose(dim, -1) # centering and padding x centered_signal = input - input.mean(dim=-1, keepdim=True) pad = input.new_zeros(input.shape[:-1] + (M2 - N,)) centered_signal = torch.cat([centered_signal, pad], dim=-1) # Fourier transform freqvec = torch.rfft(centered_signal, signal_ndim=1, onesided=False) # take square of magnitude of freqvec (or freqvec x freqvec*) freqvec_gram = freqvec.pow(2).sum(-1, keepdim=True) freqvec_gram = torch.cat([freqvec_gram, input.new_zeros(freqvec_gram.shape)], dim=-1) # inverse Fourier transform autocorr = torch.irfft(freqvec_gram, signal_ndim=1, onesided=False) # truncate and normalize the result, then transpose back to original shape autocorr = autocorr[..., :N] autocorr = autocorr / input.new_tensor(range(N, 0, -1)) autocorr = autocorr / autocorr[..., :1] return autocorr.transpose(dim, -1) def autocovariance(input, dim=0): """ Computes the autocovariance of samples at dimension ``dim``. :param torch.Tensor input: the input tensor. :param int dim: the dimension to calculate autocorrelation. :returns torch.Tensor: autocorrelation of ``input``. """ return autocorrelation(input, dim) * input.var(dim, unbiased=False, keepdim=True) def _cummin(input): """ Computes cummulative minimum of input at dimension ``dim=0``. :param torch.Tensor input: the input tensor. :returns torch.Tensor: accumulate min of `input` at dimension `dim=0`. """ # FIXME: is there a better trick to find accumulate min of a sequence? N = input.size(0) input_tril = input.unsqueeze(0).repeat((N,) + (1,) * input.dim()) triu_mask = input.new_ones(N, N).triu(diagonal=1).reshape((N, N) + (1,) * (input.dim() - 1)) triu_mask = triu_mask.expand((N, N) + input.shape[1:]) > 0.5 input_tril.masked_fill_(triu_mask, input.max()) return input_tril.min(dim=1)[0] def effective_sample_size(input, chain_dim=0, sample_dim=1): """ Computes effective sample size of input. Reference: [1] `Introduction to Markov Chain Monte Carlo`, Charles J. Geyer [2] `Stan Reference Manual version 2.18`, Stan Development Team :param torch.Tensor input: the input tensor. :param int chain_dim: the chain dimension. :param int sample_dim: the sample dimension. :returns torch.Tensor: effective sample size of ``input``. """ assert input.dim() >= 2 assert input.size(sample_dim) >= 2 # change input.shape to 1 x 1 x input.shape # then transpose sample_dim with 0, chain_dim with 1 sample_dim = input.dim() + sample_dim if sample_dim < 0 else sample_dim chain_dim = input.dim() + chain_dim if chain_dim < 0 else chain_dim assert chain_dim != sample_dim input = input.reshape((1, 1) + input.shape) input = input.transpose(0, sample_dim + 2).transpose(1, chain_dim + 2) N, C = input.size(0), input.size(1) # find autocovariance for each chain at lag k gamma_k_c = autocovariance(input, dim=0) # N x C x sample_shape # find autocorrelation at lag k (from Stan reference) var_within, var_estimator = _compute_chain_variance_stats(input) rho_k = (var_estimator - var_within + gamma_k_c.mean(dim=1)) / var_estimator rho_k[0] = 1 # correlation at lag 0 is always 1 # initial positive sequence (formula 1.18 in [1]) applied for autocorrelation Rho_k = rho_k if N % 2 == 0 else rho_k[:-1] Rho_k = Rho_k.reshape((N // 2, 2) + Rho_k.shape[1:]).sum(dim=1) # separate the first index Rho_init = Rho_k[0] if Rho_k.size(0) > 1: # Theoretically, Rho_k is positive, but due to noise of correlation computation, # Rho_k might not be positive at some point. So we need to truncate (ignore first index). Rho_positive = Rho_k[1:].clamp(min=0) # Now we make the initial monotone (decreasing) sequence. Rho_monotone = _cummin(Rho_positive) # Formula 1.19 in [1] tau = -1 + 2 * Rho_init + 2 * Rho_monotone.sum(dim=0) else: tau = -1 + 2 * Rho_init n_eff = C * N / tau return n_eff.squeeze(max(sample_dim, chain_dim)).squeeze(min(sample_dim, chain_dim)) def resample(input, num_samples, dim=0, replacement=False): """ Draws ``num_samples`` samples from ``input`` at dimension ``dim``. :param torch.Tensor input: the input tensor. :param int num_samples: the number of samples to draw from ``input``. :param int dim: dimension to draw from ``input``. :returns torch.Tensor: samples drawn randomly from ``input``. """ weights = input.new_ones(input.size(dim)) indices = torch.multinomial(weights, num_samples, replacement) return input.index_select(dim, indices) def quantile(input, probs, dim=0): """ Computes quantiles of ``input`` at ``probs``. If ``probs`` is a scalar, the output will be squeezed at ``dim``. :param torch.Tensor input: the input tensor. :param list probs: quantile positions. :param int dim: dimension to take quantiles from ``input``. :returns torch.Tensor: quantiles of ``input`` at ``probs``. """ if isinstance(probs, (numbers.Number, list, tuple)): probs = input.new_tensor(probs) sorted_input = input.sort(dim)[0] max_index = input.size(dim) - 1 indices = probs * max_index # because indices is float, we interpolate the quantiles linearly from nearby points indices_below = indices.long() indices_above = (indices_below + 1).clamp(max=max_index) quantiles_above = sorted_input.index_select(dim, indices_above) quantiles_below = sorted_input.index_select(dim, indices_below) shape_to_broadcast = [1] * input.dim() shape_to_broadcast[dim] = indices.numel() weights_above = indices - indices_below.type_as(indices) weights_above = weights_above.reshape(shape_to_broadcast) weights_below = 1 - weights_above quantiles = weights_below * quantiles_below + weights_above * quantiles_above return quantiles if probs.shape != torch.Size([]) else quantiles.squeeze(dim) def pi(input, prob, dim=0): """ Computes percentile interval which assigns equal probability mass to each tail of the interval. :param torch.Tensor input: the input tensor. :param float prob: the probability mass of samples within the interval. :param int dim: dimension to calculate percentile interval from ``input``. :returns torch.Tensor: quantiles of ``input`` at ``probs``. """ return quantile(input, [(1 - prob) / 2, (1 + prob) / 2], dim) def hpdi(input, prob, dim=0): """ Computes "highest posterior density interval" which is the narrowest interval with probability mass ``prob``. :param torch.Tensor input: the input tensor. :param float prob: the probability mass of samples within the interval. :param int dim: dimension to calculate percentile interval from ``input``. :returns torch.Tensor: quantiles of ``input`` at ``probs``. """ sorted_input = input.sort(dim)[0] mass = input.size(dim) index_length = int(prob * mass) intervals_left = sorted_input.index_select( dim, input.new_tensor(range(mass - index_length), dtype=torch.long)) intervals_right = sorted_input.index_select( dim, input.new_tensor(range(index_length, mass), dtype=torch.long)) intervals_length = intervals_right - intervals_left index_start = intervals_length.argmin(dim) indices = torch.stack([index_start, index_start + index_length], dim) return torch.gather(sorted_input, dim, indices) def _weighted_mean(input, log_weights, dim=0, keepdim=False): dim = input.dim() + dim if dim < 0 else dim log_weights = log_weights.reshape([-1] + (input.dim() - dim - 1) * [1]) max_log_weight = log_weights.max(dim=0)[0] relative_probs = (log_weights - max_log_weight).exp() return (input * relative_probs).sum(dim=dim, keepdim=keepdim) / relative_probs.sum() def _weighted_variance(input, log_weights, dim=0, keepdim=False, unbiased=True): # Ref: https://en.wikipedia.org/wiki/Weighted_arithmetic_mean#Frequency_weights deviation_squared = (input - _weighted_mean(input, log_weights, dim, keepdim=True)).pow(2) correction = log_weights.size(0) / (log_weights.size(0) - 1.) if unbiased else 1. return _weighted_mean(deviation_squared, log_weights, dim, keepdim) * correction def waic(input, log_weights=None, pointwise=False, dim=0): """ Computes "Widely Applicable/Watanabe-Akaike Information Criterion" (WAIC) and its corresponding effective number of parameters. Reference: [1] `WAIC and cross-validation in Stan`, Aki Vehtari, Andrew Gelman :param torch.Tensor input: the input tensor, which is log likelihood of a model. :param torch.Tensor log_weights: weights of samples along ``dim``. :param int dim: the sample dimension of ``input``. :returns tuple: tuple of WAIC and effective number of parameters. """ log_weights = input.new_zeros(input.size(dim)) if log_weights is None else log_weights # computes log pointwise predictive density: formula (3) of [1] dim = input.dim() + dim if dim < 0 else dim weighted_input = input + log_weights.reshape([-1] + (input.dim() - dim - 1) * [1]) lpd = torch.logsumexp(weighted_input, dim=dim) - torch.logsumexp(log_weights, dim=0) # computes the effective number of parameters: formula (6) of [1] p_waic = _weighted_variance(input, log_weights, dim) # computes expected log pointwise predictive density: formula (4) of [1] elpd = lpd - p_waic waic = -2 * elpd return (waic, p_waic) if pointwise else (waic.sum(), p_waic.sum())