chains.hpp

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#ifndef __STAN__MCMC__CHAINS_HPP__
#define __STAN__MCMC__CHAINS_HPP__
 
#include <algorithm>
#include <cmath>
#include <iostream>
#include <map>
#include <stdexcept>
#include <string>
#include <sstream>
#include <utility>
#include <vector>
#include <fstream>
#include <cstdlib>
 
#include <boost/accumulators/accumulators.hpp>
#include <boost/accumulators/statistics/stats.hpp>
#include <boost/accumulators/statistics/mean.hpp>
// #include <boost/accumulators/statistics/moment.hpp>
#include <boost/accumulators/statistics/tail_quantile.hpp>
#include <boost/accumulators/statistics/variance.hpp>
#include <boost/accumulators/statistics/covariance.hpp>
#include <boost/accumulators/statistics/variates/covariate.hpp>
 
 
#include <boost/random/uniform_int_distribution.hpp>
#include <boost/random/additive_combine.hpp>
 
#include <stan/math/matrix.hpp>
#include <stan/prob/autocorrelation.hpp>
#include <stan/prob/autocovariance.hpp>
 
namespace stan {  
 
  namespace mcmc {
 
    const std::vector<std::string>& 
    test_match_return_names(const std::vector<std::string>& names,
                            const std::vector<std::vector<size_t> >& dimss) {
      if (names.size() == dimss.size())
        return names;
      std::stringstream msg;
      msg << "names and dimss mismatch in size"
          << " names.size()=" << names.size()
          << " dimss.size()=" << dimss.size();
      throw std::invalid_argument(msg.str());
    }
 
    void validate_prob(double p) {
      // test this way so NaN fails
      if (p >= 0.0 && p <= 1.0) 
        return;
      std::stringstream msg;
      msg << "require probabilities to be finite between 0 and 1 inclusive."
          << " found p=" << p;
      throw std::invalid_argument(msg.str());
    }
 
    void validate_dims_idxs(const std::vector<size_t>& dims,
                            const std::vector<size_t>& idxs) {
      if (idxs.size() != dims.size()) {
        std::stringstream msg;
        msg << "index vector and dims vector must be same size."
            << "; idxs.size()=" << idxs.size()
            << "; dims.size()=" << dims.size();
        throw std::invalid_argument(msg.str());
      }
      for (size_t i = 0; i < idxs.size(); ++i) {
        if (idxs[i] >= dims[i]) {
          std::stringstream msg;
          msg << "indexes must be within bounds."
              << "; idxs[" << i << "]=" << idxs[i]
              << "; dims[" << i << "]=" << dims[i];
          throw std::out_of_range(msg.str());
        }
      }
    }
 
    size_t get_offset(const std::vector<size_t>& dims, 
                      const std::vector<size_t>& idxs) {
      validate_dims_idxs(dims,idxs);
      if (idxs.size() == 0)
        return 0;
      if (idxs.size() == 1)
        return idxs[0];
      size_t pos(0);
      // OK, stop at 1
      for (size_t i = idxs.size(); --i != 0; ) {
        pos += idxs[i];
        pos  *= dims[i-1];
      }
      return pos + idxs[0];
    }
 
    void
    increment_indexes(const std::vector<size_t>& dims,
                      std::vector<size_t>& idxs) {
      validate_dims_idxs(dims,idxs);
      for (size_t i = 0; i < dims.size(); ++i) {
        ++idxs[i];
        if (idxs[i] < dims[i]) 
          return;
        idxs[i] = 0;
      }
    }
 
    template <class RNG>
    void permutation(std::vector<size_t>& x,
                     size_t n,
                     RNG& rng) {
      x.resize(n);
      for (size_t i = 0; i < n; ++i)
        x[i] = i;
      if (x.size() < 2) return;
      for (int i = x.size(); --i != 0; ) {
        boost::random::uniform_int_distribution<size_t> uid(0,i);
        size_t j = uid(rng);
        size_t temp = x[i];
        x[i] = x[j];
        x[j] = temp;
      }
    }
    
 
    template <typename T>
    void permute(const std::vector<size_t>& pi,
                 const std::vector<T>& x_from,
                 std::vector<T>& x_to) {
      size_t N = pi.size();
      if (N != x_from.size()) {
        std::stringstream msg;
        msg << "Require permutation to be same size as source vector."
            << "; found pi.size()=" << pi.size()
            << "; x_from.size()=" << x_from.size();
      }
      x_to.resize(N);
      for (size_t i = 0; i < N; ++i)
        x_to[i] = x_from[pi[i]];
    }
 
    
    template <typename RNG = boost::random::ecuyer1988>
    class chains {
    private:
 
      size_t _warmup;
      const std::vector<std::string> _names;
      const std::vector<std::vector<size_t> > _dimss;
      const size_t _num_params; // total
      const std::vector<size_t> _starts;
      const std::map<std::string,size_t> _name_to_index;
      // [chain,param,sample]
      std::vector<std::vector<std::vector<double > > > _samples; 
      std::vector<size_t> _permutation;
      RNG _rng; // defaults to time-based init
 
      static size_t calc_num_params(const std::vector<size_t>& dims) {
        size_t num_params = 1;
        for (size_t i = 0;  i < dims.size(); ++i)
          num_params *= dims[i];
        return num_params;
      }
 
      static size_t 
      calc_total_num_params(const std::vector<std::vector<size_t> >& dimss) {
        int num_params = 0;
        for (size_t i = 0; i < dimss.size(); ++i)
          num_params += calc_num_params(dimss[i]);
        return num_params;
      }
 
      static std::vector<size_t> 
      calc_starts(const std::vector<std::vector<size_t> >& dimss) {
        std::vector<size_t> starts(dimss.size());
        starts[0] = 0;
        for (size_t i = 1; i < dimss.size(); ++i)
          starts[i] = starts[i - 1] + calc_num_params(dimss[i - 1]);
        return starts;
      }
 
      static std::map<std::string,size_t>
      calc_name_to_index(const std::vector<std::string> names) {
        std::map<std::string,size_t> name_to_index;
        for (size_t i = 0; i < names.size(); ++i)
          name_to_index[names[i]] = i;
        return name_to_index;
      }
 
      inline void validate_param_name_idx(size_t j) {
        if (j < num_param_names()) 
          return;
        std::stringstream msg;
        msg << "parameter name index must be less than number of params"
            << "; found j=" << j;
        throw std::out_of_range(msg.str());
      }
 
      inline void validate_param_idx(size_t n) {
        if (n < num_params())
          return;
        std::stringstream msg;
        msg << "parameter index must be less than number of params"
            << "; found n=" << n;
        throw std::out_of_range(msg.str());
      }
 
      inline void validate_chain_idx(size_t k) {
        if (k >= num_chains()) {
          std::stringstream msg;
          msg << "chain must be less than number of chains."
              << "; num chains=" << num_chains()
              << "; chain=" << k;
          throw std::out_of_range(msg.str());
        }
      }
 
      // k = chain idx, m = iteration
      void validate_iteration(size_t k,    
                              size_t m) {  
        validate_chain_idx(k);
        if (m >= _samples[k][0].size()) {
          std::stringstream msg;
          msg << "require sample index below number of samples"
              << "; sample index m=" << m
              << "; chain index k=" << k
              << "; num samples in chain" << k << "=" 
              << _samples[k][0].size();
          throw std::out_of_range(msg.str());
        }
      }
 
      void resize_permutation(size_t K) {
        if (_permutation.size() != K) 
          permutation(_permutation,K,_rng);
      }
 
    public:
 
      chains(const size_t num_chains,
             const std::vector<std::string>& names,
             const std::vector<std::vector<size_t> >& dimss) 
        : _warmup(0),
          // function call tests dimensionality match & returns names
          _names(names), // test_match_return_names(names,dimss)),
          _dimss(dimss),
          _num_params(calc_total_num_params(dimss)),
          _starts(calc_starts(dimss)),               // copy
          _name_to_index(calc_name_to_index(names)), // copy
          _samples(num_chains,std::vector<std::vector<double> >(_num_params))
      {  }
      
      inline size_t num_chains() {
        return _samples.size();
      }
 
      inline size_t num_params() { 
        return _num_params;
      }
 
      inline size_t num_param_names() {
        return _names.size();
      }
 
      const std::vector<std::string>& param_names() {
        return _names;
      }
 
      const std::string& param_name(size_t j) {
        validate_param_name_idx(j);
        return _names[j];
      }
 
      const std::vector<std::vector<size_t> >& param_dimss() {
        return _dimss;
      }
 
      const std::vector<size_t>& param_dims(size_t j) {
        validate_param_name_idx(j);
        return _dimss[j];
      }
 
      const std::vector<size_t>& param_starts() {
        return _starts;
      }
 
      size_t param_start(size_t j) {
        validate_param_name_idx(j);
        return _starts[j];
      }
 
      const std::vector<size_t> param_sizes() {
        std::vector<size_t> s(num_param_names());
        for (unsigned int j = 0; j < num_param_names(); ++j)
          s[j] = param_size(j); // could optimize tests in param_sizes() out
        return s;
      }
 
      size_t param_size(size_t j) {
        validate_param_name_idx(j);
        if (j + 1 < _starts.size()) 
          return _starts[j+1] - _starts[j];
        return num_params() - _starts[j];
      }
 
      size_t param_name_to_index(const std::string& name) {
        std::map<std::string,size_t>::const_iterator it
          = _name_to_index.find(name);
        if (it == _name_to_index.end()) {
          std::stringstream ss;
          ss << "unknown parameter name=" << name;
          throw std::out_of_range(ss.str());
        }
        return it->second;
      }
      
 
      size_t get_total_param_index(size_t j, // param id
                                   const std::vector<size_t>& idxs) {
        return get_offset(param_dims(j),idxs)
          + param_start(j);
      }
 
 
      void set_warmup(size_t warmup_iterations) {
        _warmup = warmup_iterations;
      }
 
 
      inline size_t warmup() {
        return _warmup;
      }
 
 
      void add(size_t chain,
               std::vector<double> theta) {
        validate_chain_idx(chain);
        if (theta.size() != _num_params) {
          std::stringstream msg;
          msg << "parameter vector size must match num params"
              << "; num params=" << _num_params
              << "; theta.size()=" << theta.size();
          throw std::invalid_argument(msg.str());
        }
        for (size_t i = 0; i < theta.size(); ++i)
          _samples[chain][i].push_back(theta[i]); // _samples very non-local
      }
      
      
 
 
 
      size_t num_warmup_samples(size_t k) {
        return std::min<size_t>(num_samples(k), warmup());
      }
 
      size_t num_warmup_samples() {
        size_t total = 0;
        for (size_t k = 0; k < num_chains(); ++k)
          total += num_warmup_samples(k);
        return total;
      }
 
      size_t num_kept_samples(size_t k) {
        if (num_samples(k) > warmup())
          return num_samples(k) - warmup();
        return 0U;
      }
 
 
      size_t num_kept_samples() {
        size_t total = 0;
        for (size_t k = 0; k < num_chains(); ++k)
          total += num_kept_samples(k);
        return total;
      }
 
      size_t num_samples() {
        size_t M = 0;
        for (size_t k = 0; k < num_chains(); ++k)
          M += num_samples(k);
        return M;
      }
 
      size_t num_samples(size_t k) {
        validate_chain_idx(k);
        return _samples[k][0].size();
      }
 
 
      void
      get_samples(size_t n,
                  std::vector<double>& samples) {
        validate_param_idx(n);
        samples.resize(0);
        samples.reserve(num_samples());
        for (size_t k = 0; k < num_chains(); ++k)
          samples.insert(samples.end(),
                         _samples[k][n].begin(), 
                         _samples[k][n].end());
      }
 
      void get_samples(size_t k, 
                       size_t n, 
                       std::vector<double>& samples) {  
        validate_chain_idx(k);
        validate_param_idx(n); 
        samples.resize(0);
        samples.reserve(num_samples(k));
        samples.insert(samples.end(),
                       _samples[k][n].begin(),
                       _samples[k][n].end());
      }
 
 
      void
      get_kept_samples_permuted(size_t n,
                                std::vector<double>& samples) {
        validate_param_idx(n);
        size_t M = num_kept_samples();
        samples.resize(M);
        resize_permutation(M);
        // const std::vector<size_t>& permutation = _permutation;
        size_t pos = 0;
        for (size_t k = 0; k < num_chains(); ++k) {
          // const std::vector<double>& samples_k_n = _samples[k][n];
          for (size_t m = warmup(); m < num_samples(k); ++m) {
            samples[_permutation[pos]] = _samples[k][n][m]; // _samples_k_n[m];
            ++pos;
          }
        }
      }
 
      template <typename F>
      void
      apply_kept_samples(size_t k,
                         size_t n,
                         F& f) {
        using std::vector;
        for (vector<double>::const_iterator it = _samples[k][n].begin() + warmup();
             it != _samples[k][n].end();
             ++it)
          f(*it);
      }
 
      template <typename F>
      void
      apply_kept_samples(size_t n,
                         F& f) {
        for (size_t k = 0; k < num_chains(); ++k)
          apply_kept_samples(k,n,f);
      }
 
      void
      get_kept_samples(size_t k,
                       size_t n,
                       std::vector<double>& samples) {
        validate_param_idx(n);
        samples.resize(0);
        samples.reserve(num_kept_samples(k));
        samples.insert(samples.end(),
                       _samples[k][n].begin() + warmup(),
                       _samples[k][n].end());
      }
 
 
 
      void
      get_warmup_samples(size_t n,
                         std::vector<double>& samples) {
        validate_param_idx(n);
        samples.resize(0);
        samples.reserve(num_warmup_samples());
        for (size_t k = 0; k < num_chains(); ++k) {
          if (num_warmup_samples(k) < warmup())
            samples.insert(samples.end(),
                           _samples[k][n].begin(),
                           _samples[k][n].end());
          else
            samples.insert(samples.end(),
                           _samples[k][n].begin(),
                           _samples[k][n].begin() + warmup());
        }
      }
 
      void
      get_warmup_samples(size_t k,
                         size_t n,
                         std::vector<double>& samples) {
        validate_param_idx(n);
        samples.resize(0);
        samples.reserve(num_warmup_samples(k));
        if (num_warmup_samples(k) < warmup())
            samples.insert(samples.end(),
                           _samples[k][n].begin(),
                           _samples[k][n].end());
        else
            samples.insert(samples.end(),
                           _samples[k][n].begin(),
                           _samples[k][n].begin() + warmup());
      }
 
 
      double mean(size_t k,
                  size_t n) {
        using boost::accumulators::accumulator_set;
        using boost::accumulators::stats;
        using boost::accumulators::tag::mean;
        validate_chain_idx(k);
        validate_param_idx(n);
        accumulator_set<double, stats<mean> > acc;
        apply_kept_samples(k,n,acc);
        return boost::accumulators::mean(acc);
      }
 
      double mean(size_t n) {
        using boost::accumulators::accumulator_set;
        using boost::accumulators::stats;
        using boost::accumulators::tag::mean;
        validate_param_idx(n);
        accumulator_set<double, stats<mean> > acc;
        apply_kept_samples(n,acc);
        return boost::accumulators::mean(acc);
      }
 
      double sd(size_t k,
                size_t n) {
        validate_chain_idx(k);
        validate_param_idx(n);
        using boost::accumulators::accumulator_set;
        using boost::accumulators::stats;
        using boost::accumulators::tag::variance;
        accumulator_set<double, stats<variance> > acc;
        apply_kept_samples(k,n,acc);
        double M = num_kept_samples(k);
        return std::sqrt((M / (M-1)) * boost::accumulators::variance(acc));
      }
 
      double sd(size_t n) {
        validate_param_idx(n);
        using boost::accumulators::accumulator_set;
        using boost::accumulators::stats;
        using boost::accumulators::tag::variance;
        accumulator_set<double, stats<variance> > acc;
        apply_kept_samples(n,acc);
        double M = num_kept_samples();
        return std::sqrt((M / (M-1)) * boost::accumulators::variance(acc));
      }
 
      double variance(size_t k,
                      size_t n) {
        validate_chain_idx(k);
        validate_param_idx(n);
        using boost::accumulators::accumulator_set;
        using boost::accumulators::stats;
        using boost::accumulators::tag::variance;
        accumulator_set<double, stats<variance> > acc;
        apply_kept_samples(k,n,acc);
        double M = num_kept_samples(k);
        return (M / (M-1)) *boost::accumulators::variance(acc);
      }
 
      double variance(size_t n) {
        validate_param_idx(n);
        using boost::accumulators::accumulator_set;
        using boost::accumulators::stats;
        using boost::accumulators::tag::variance;
        accumulator_set<double, stats<variance> > acc;
        apply_kept_samples(n,acc);
        double M = num_kept_samples(n);
        return (M / (M-1)) *boost::accumulators::variance(acc);
      }
 
      double covariance(size_t k,
                        size_t n1,
                        size_t n2) {
        validate_chain_idx(k);
        validate_param_idx(n1);
        validate_param_idx(n2);
        using boost::accumulators::accumulator_set;
        using boost::accumulators::stats;
        using boost::accumulators::tag::variance;
        using boost::accumulators::tag::covariance;
        using boost::accumulators::tag::covariate1;
        
        accumulator_set<double, stats<covariance<double, covariate1> > > acc;
        std::vector<double> samples1, samples2;
        this->get_kept_samples(k, n1, samples1);
        this->get_kept_samples(k, n2, samples2);
        for (size_t kk = 0; kk < this->num_kept_samples(k); kk++) {
          acc(samples1[kk], boost::accumulators::covariate1 = samples2[kk]);
        }
        double M = num_kept_samples(k);
        return (M / (M-1)) * boost::accumulators::covariance(acc);
      }
 
      double covariance(size_t n1, size_t n2) {
        validate_param_idx(n1);
        validate_param_idx(n2);
        using boost::accumulators::accumulator_set;
        using boost::accumulators::stats;
        using boost::accumulators::tag::variance;
        using boost::accumulators::tag::covariance;
        using boost::accumulators::tag::covariate1;
        
        accumulator_set<double, stats<covariance<double, covariate1> > > acc;
        for (size_t chain = 0; chain < this->num_chains(); chain++) {
          std::vector<double> samples1, samples2;
          this->get_kept_samples(chain, n1, samples1);
          this->get_kept_samples(chain, n2, samples2);
          for (size_t kk = 0; kk < this->num_kept_samples(chain); kk++) {
            acc(samples1[kk], boost::accumulators::covariate1 = samples2[kk]);
          }
        }
        double M = this->num_kept_samples();
        return (M / (M-1)) * boost::accumulators::covariance(acc);
      }
      
 
      double correlation(size_t k,
                        size_t n1,
                        size_t n2) {
        double cov = covariance(k, n1, n2);
        double sd1 = sd(k, n1);
        double sd2 = sd(k, n2);
                
        return cov / sd1 / sd2;
      }
 
      double correlation(size_t n1, size_t n2) {
        double cov = covariance(n1, n2);
        double sd1 = sd(n1);
        double sd2 = sd(n2);
 
        return cov / sd1 / sd2;
      }
      
 
      double quantile(size_t k,
                      size_t n,
                      double prob) {
        validate_chain_idx(k);
        validate_param_idx(n);
        using boost::accumulators::accumulator_set;
        using boost::accumulators::left;
        using boost::accumulators::quantile_probability;
        using boost::accumulators::right;
        using boost::accumulators::stats;
        using boost::accumulators::tag::tail;
        using boost::accumulators::tag::tail_quantile;
        double size = num_kept_samples(k);
        if (prob < 0.5) {
          std::size_t cs(2 + prob * size); // 2+ for interpolation
          accumulator_set<double, stats<tail_quantile<left> > > 
            acc(tail<left>::cache_size = cs);
          apply_kept_samples(k,n,acc);
          return boost::accumulators::quantile(acc, quantile_probability = prob);
        } else {
          std::size_t cs(2 + (1.0 - prob) * size);
          accumulator_set<double, stats<tail_quantile<right> > > 
            acc(tail<right>::cache_size = cs);
          apply_kept_samples(k,n,acc);
          return boost::accumulators::quantile(acc, quantile_probability = prob);
        }
      }
 
      double quantile(size_t n,
                      double prob) {
        validate_param_idx(n);
        using boost::accumulators::accumulator_set;
        using boost::accumulators::left;
        using boost::accumulators::quantile_probability;
        using boost::accumulators::right;
        using boost::accumulators::stats;
        using boost::accumulators::tag::tail;
        using boost::accumulators::tag::tail_quantile;
        double size = num_kept_samples();
        if (prob < 0.5) {
          std::size_t cs(2 + prob * size); // 2+ for interpolation
          accumulator_set<double, stats<tail_quantile<left> > > 
            acc(tail<left>::cache_size = cs);
          apply_kept_samples(n,acc);
          return boost::accumulators::quantile(acc, quantile_probability = prob);
        } else {
          std::size_t cs(2 + (1.0 - prob) * size);
          accumulator_set<double, stats<tail_quantile<right> > > 
            acc(tail<right>::cache_size = cs);
          apply_kept_samples(n,acc);
          return boost::accumulators::quantile(acc, quantile_probability = prob);
        }
      }
 
      void quantiles(size_t k,
                     size_t n,
                     const std::vector<double>& probs,
                     std::vector<double>& quantiles) {
        validate_chain_idx(k);
        validate_param_idx(n);
        for (size_t i = 0; i < probs.size(); ++i)
          validate_prob(probs[i]);
        quantiles.resize(probs.size());
 
        using boost::accumulators::accumulator_set;
        using boost::accumulators::quantile;
        using boost::accumulators::quantile_probability;
        using boost::accumulators::right;
        using boost::accumulators::stats;
        using boost::accumulators::tag::tail;
        using boost::accumulators::tag::tail_quantile;
        // keeps all (more efficient to have two for (min,0.5) and (0.5,max))
        accumulator_set<double, stats<tail_quantile<right> > >
          acc(tail<right>::cache_size = num_kept_samples(k));
        apply_kept_samples(k,n,acc);
        
        for (size_t i = 0; i < probs.size(); ++i)
          quantiles[i] = quantile(acc, quantile_probability = probs[i]);
      }
 
      void quantiles(size_t n,
                     const std::vector<double>& probs,
                     std::vector<double>& quantiles) {
        validate_param_idx(n);
        for (size_t i = 0; i < probs.size(); ++i)
          validate_prob(probs[i]);
        quantiles.resize(probs.size());
 
        using boost::accumulators::accumulator_set;
        using boost::accumulators::quantile;
        using boost::accumulators::quantile_probability;
        using boost::accumulators::right;
        using boost::accumulators::stats;
        using boost::accumulators::tag::tail;
        using boost::accumulators::tag::tail_quantile;
        accumulator_set<double, stats<tail_quantile<right> > >
          acc(tail<right>::cache_size = num_kept_samples());
        apply_kept_samples(n,acc);
        for (size_t i = 0; i < probs.size(); ++i)
          quantiles[i] = quantile(acc, quantile_probability = probs[i]);
      }
 
      std::pair<double,double>
      central_interval(size_t k,
                       size_t n,
                       double prob) {
        // validate k,n in calls to quantile
        validate_prob(prob);
        double low_prob = (1.0 - prob) / 2.0;
        double high_prob = 1.0 - low_prob;
        double low_quantile = quantile(k,n,low_prob);
        double high_quantile = quantile(k,n,high_prob);
        return std::pair<double,double>(low_quantile,high_quantile);
      }
 
      std::pair<double,double>
      central_interval(size_t n,
                       double prob) {
        // validate n in calls to quantile
        validate_prob(prob);
        double low_prob = (1.0 - prob) / 2.0;
        double high_prob = 1.0 - low_prob;
        double low_quantile = quantile(n,low_prob);
        double high_quantile = quantile(n,high_prob);
        return std::pair<double,double>(low_quantile,high_quantile);
      }
 
      void autocorrelation(const size_t k, const size_t n, 
                           std::vector<double>& ac) {
        std::vector<double> samples;
        get_kept_samples(k,n,samples);
        stan::prob::autocorrelation(samples,
                                    ac);
      }
      
      void autocovariance(const size_t k, const size_t n, 
                           std::vector<double>& acov) {
        std::vector<double> samples;
        get_kept_samples(k,n,samples);
        stan::prob::autocovariance(samples,
                                   acov);
      }
      
      // FIXME: reimplement using autocorrelation.
      double effective_sample_size(size_t n) {
        validate_param_idx(n);
        size_t m = this->num_chains();
        // need to generalize to each jagged samples per chain
        size_t n_samples = this->num_kept_samples(0U);
        for (size_t chain = 1; chain < m; chain++) {
          n_samples = std::min(n_samples, this->num_kept_samples(chain));
        }
 
        using std::vector;
        vector< vector<double> > acov;
        for (size_t chain = 0; chain < m; chain++) {
          vector<double> acov_chain;
          autocovariance(chain, n, acov_chain);
          acov.push_back(acov_chain);
        }
        
        vector<double> chain_mean;
        vector<double> chain_var;
        for (size_t chain = 0; chain < m; chain++) {
          double n_kept_samples = num_kept_samples(chain);
          chain_mean.push_back(this->mean(chain,n));
          chain_var.push_back(acov[chain][0]*n_kept_samples/(n_kept_samples-1));
        }
        double mean_var = stan::math::mean(chain_var);
        double var_plus = mean_var*(n_samples-1)/n_samples;
        if (m > 1) var_plus += stan::math::variance(chain_mean);
        vector<double> rho_hat_t;
        double rho_hat = 0;
        for (size_t t = 1; (t < n_samples && rho_hat >= 0); t++) {
          vector<double> acov_t(m);
          for (size_t chain = 0; chain < m; chain++) {
            acov_t[chain] = acov[chain][t];
          }
          rho_hat = 1 - (mean_var - stan::math::mean(acov_t)) / var_plus;
          if (rho_hat >= 0)
            rho_hat_t.push_back(rho_hat);
        }
        
        double ess = m*n_samples;
        if (rho_hat_t.size() > 0) {
          ess /= 1 + 2 * stan::math::sum(rho_hat_t);
        }
        return ess;
      }
      
      double split_potential_scale_reduction(size_t n) {
        size_t n_chains = this->num_chains();
        size_t n_samples = this->num_kept_samples(0U);
        for (size_t chain = 1; chain < n_chains; chain++) {
          n_samples = std::min(n_samples, this->num_kept_samples(chain));
        }
        if (n_samples % 2 == 1)
          n_samples--;
        
        std::vector<double> split_chain_mean;
        std::vector<double> split_chain_var;
 
        for (size_t chain = 0; chain < n_chains; chain++) {
          std::vector<double> samples(n_samples);
          this->get_kept_samples(chain, n, samples);
          
          std::vector<double> split_chain(n_samples/2);
          split_chain.assign(samples.begin(),
                             samples.begin()+n_samples/2);
          split_chain_mean.push_back(stan::math::mean(split_chain));
          split_chain_var.push_back(stan::math::variance(split_chain));
          
          split_chain.assign(samples.end()-n_samples/2, 
                             samples.end());
          split_chain_mean.push_back(stan::math::mean(split_chain));
          split_chain_var.push_back(stan::math::variance(split_chain));
        }
 
        double var_between = n_samples/2 * stan::math::variance(split_chain_mean);
        double var_within = stan::math::mean(split_chain_var);
        
        // rewrote [(n-1)*W/n + B/n]/W as (n-1+ B/W)/n
        return sqrt((var_between/var_within + n_samples/2 -1)/(n_samples/2));
      }
 
    };
 
    
    namespace {
      std::string read_header(std::fstream& file) {
        std::string header = "";
        // strip comments
        while (file.peek() == '#') {
          file.ignore(10000, '\n');
        }
        std::getline(file, header, '\n');
        return header;
      }
      
      void
      tokenize(const std::string& line, const char delimiter, 
               std::vector<std::string>& tokens) {
        tokens.clear();
        std::stringstream stream(line);
        std::string token;
        while (std::getline(stream,token,delimiter)) {
          tokens.push_back(token);
        }
      }
 
      void
      get_names(const std::vector<std::string>& tokens, 
                const size_t skip,
                std::vector<std::string>& names) {
        names.clear();
        for (size_t i = 0; i < tokens.size(); i++) {
          std::stringstream token(tokens[i]);
          std::string name;
          std::getline(token,name,'.');
          if (names.size() == 0 || names.back()!=name) {
            names.push_back(name);
          }
        }
        names.erase(names.begin(), names.begin()+skip);
      }
 
      void
      get_dimss(const std::vector<std::string>& tokens, 
                const size_t skip, 
                std::vector<std::vector<size_t> >& dimss) {
        dimss.clear();
        std::string last_name;
        std::vector<std::string> split;
        for (int i = tokens.size()-1; i >= 0; --i) {
          tokenize(tokens[i], '.', split);
          
          std::vector<size_t> dims;
          if (split.size() == 1) {
            dims.push_back(1);
            dimss.insert(dimss.begin(), dims);
          } else if (split.front() != last_name) {
            for (size_t j = 1; j < split.size(); j++) {
              dims.push_back((size_t)atol(split[j].c_str()));
            }
            dimss.insert(dimss.begin(), dims);
          }
          last_name = split.front();
        }
        dimss.erase(dimss.begin(), dimss.begin()+skip);
      }
      
      void
      read_values(std::fstream& file, const size_t num_values,
                  std::vector<std::vector<double> >& thetas) {
        thetas.clear();
        read_header(file); // ignore header
        std::vector<double> theta;
        std::string line;
        std::vector<std::string> tokens;
        while (file.peek() != std::istream::traits_type::eof()) {
          while (file.peek() == '#') { // ignore comments
            file.ignore(10000, '\n');
          }
          std::getline(file, line, '\n');
          tokenize(line, ',', tokens);
          theta.clear();
          for (size_t i = tokens.size()-num_values; i < tokens.size(); i++) {
            theta.push_back(atof(tokens[i].c_str()));
          }
          if (theta.size() > 0) {
            thetas.push_back(theta);
          }
        }
      }
      
      void
      read_values(std::fstream& file, 
                  std::vector<std::vector<double> >& thetas) {
        thetas.clear();
        std::string header = read_header(file);
        int num_values = 1;
        for (const char* header_ptr = header.c_str(); 
             *header_ptr != '\n' && *header_ptr != 0; header_ptr++)
          num_values += *header_ptr == ',';
 
        std::vector<double> theta;
        std::string line;
        std::vector<std::string> tokens;
        while (file.peek() != std::istream::traits_type::eof()) {
          while (file.peek() == '#') { // ignore comments
            file.ignore(10000, '\n');
          }
          std::getline(file, line, '\n');
          tokenize(line, ',', tokens);
          theta.clear();
          for (size_t i = tokens.size()-num_values; i < tokens.size(); i++) {
            theta.push_back(atof(tokens[i].c_str()));
          }
          if (theta.size() > 0) {
            thetas.push_back(theta);
          }
        }
      }
      
      void
      reorder_values(std::vector<std::vector<double> >& thetas,
                     const std::vector<size_t>& from,
                     const std::vector<size_t>& to) {
        std::vector<double> temp(from.size());
        for (size_t ii = 0; ii < thetas.size(); ii++) {
          for (size_t jj = 0; jj < from.size(); jj++) {
            temp[jj] = thetas[ii][from[jj]];
          }
          for (size_t jj = 0; jj < to.size(); jj++) {
            thetas[ii][to[jj]] = temp[jj];
          }
        }
      }
 
      void
      get_reordering(const std::vector<std::vector<size_t> >& dimss,
                     std::vector<size_t>& from,
                     std::vector<size_t>& to) {
        from.clear();
        to.clear();
        
        size_t offset = 0;
        for (size_t ii = 0; ii < dimss.size(); ii++) {
          size_t curr_size = dimss[ii][0];
          if (dimss[ii].size() > 1) {
            for (size_t jj = 1; jj < dimss[ii].size(); jj++)
              curr_size *= dimss[ii][jj];
            
            std::vector<size_t> idxs;
            for (size_t jj = 0; jj < dimss[ii].size(); jj++) {
              idxs.push_back(0);
            }
            size_t from_index = 0;
            for (size_t to_index = 0; to_index < curr_size; to_index++) {
              from_index = 0;
              size_t offset_temp = 1;
              for (size_t kk = idxs.size(); kk > 0; --kk) {
                from_index += idxs[kk-1] * offset_temp;
                offset_temp *= dimss[ii][kk-1];
              }
              if (from_index != to_index) {
                from.push_back(offset+from_index);
                to.push_back(offset+to_index);
              }
              increment_indexes(dimss[ii], idxs);
            }
          }
          offset += curr_size;
        }
      }
    }
 
    void
    read_variables(const std::string filename, const size_t skip, 
                   std::vector<std::string>& names,
                   std::vector<std::vector<size_t> >& dimss) {
      names.clear();
      dimss.clear();
      std::fstream csv_output_file(filename.c_str(), std::fstream::in);
      if (!csv_output_file.is_open()) {
        throw std::runtime_error("Could not open " + filename);
      }
      std::string header = read_header(csv_output_file);
      csv_output_file.close();
 
      std::vector<std::string> tokens;
      tokenize(header, ',', tokens);
      get_names(tokens, skip, names);
      get_dimss(tokens, skip, dimss);
    }
    
    template <typename RNG>
    size_t add_chain(stan::mcmc::chains<RNG>& chains, 
                     const size_t chain, 
                     const std::string filename,
                     const size_t skip) {
      std::fstream csv_output_file(filename.c_str(), std::fstream::in);
      if (!csv_output_file.is_open()) {
        throw std::runtime_error("Could not open " + filename);
      }
      std::vector<std::vector<double> > thetas;
      read_values(csv_output_file, chains.num_params(), thetas);
      csv_output_file.close();
      
      std::vector<size_t> from, to;
      get_reordering(chains.param_dimss(), from, to);
      reorder_values(thetas, from, to);
      for (size_t i = 0; i < thetas.size(); i++) {
        chains.add(chain, thetas[i]);
      }
      return thetas.size();
    }
 
  }
}
 
 
#endif
 
 
/*
 
 
pair<double,double> smallest_interval(size_t n,
                                      double prob);
 
double potential_scale_reduction(size_t n)
 
double mcmc_error_mean(size_t n);
                   
void print(ostream&);
 
ostream& operator<<(ostream&, const chains&);
*/