prompt: We are currently solving the following issue within our repository. Here is the issue text:
  --- BEGIN ISSUE ---
  Rename `netket.stats.Stats.R` to `Rhat`?
  As we have seen in the discussion on #365, the `stats.R` parameter is ambiguously named; it is sometimes called "Gelman-Rubin" statistic or "potential scale reduction", but I think the most common way to refer to it is $\hat R$ in LaTeX math or "Rhat" in words (this name is used, e.g., by the [stan](https://mc-stan.org/rstan/reference/Rhat.html) framework). So I propose we also call it `Rhat` starting in 3.0.
  
  What do you think?
  
  --- END ISSUE ---
  
  Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
  --- BEGIN FILES ---
  Path: `netket/stats/mc_stats.py`
  Content:
  ```
  1 from numba import jit
  2 import numpy as _np
  3 from . import mean as _mean
  4 from . import var as _var
  5 from . import total_size as _total_size
  6 
  7 
  8 def _format_decimal(value, std):
  9     decimals = max(int(_np.ceil(-_np.log10(std))), 0)
  10     return (
  11         "{0:.{1}f}".format(value, decimals),
  12         "{0:.{1}f}".format(std, decimals + 1),
  13     )
  14 
  15 
  16 class Stats:
  17     """A dict-compatible class containing the result of the statistics function."""
  18 
  19     _NaN = float("NaN")
  20 
  21     def __init__(
  22         self, mean=_NaN, error_of_mean=_NaN, variance=_NaN, tau_corr=_NaN, R=_NaN
  23     ):
  24         self.mean = mean
  25         self.error_of_mean = error_of_mean
  26         self.variance = variance
  27         self.tau_corr = tau_corr
  28         self.R = R
  29 
  30     def to_json(self):
  31         jsd = {}
  32         jsd["Mean"] = self.mean.real
  33         jsd["Variance"] = self.variance
  34         jsd["Sigma"] = self.error_of_mean
  35         jsd["R"] = self.R
  36         jsd["TauCorr"] = self.tau_corr
  37         return jsd
  38 
  39     def __repr__(self):
  40         mean, err = _format_decimal(self.mean, self.error_of_mean)
  41         return "{} ± {} [var={:.1e}, R={:.4f}]".format(
  42             mean, err, self.variance, self.R
  43         )
  44 
  45     def __getitem__(self, name):
  46         if name is "mean" or "Mean":
  47             return self.mean
  48         elif name is "variance" or "Variance":
  49             return self.variance
  50         elif name is "error_of_mean" or "Sigma":
  51             return self.error_of_mean
  52         elif name is "R":
  53             return self.R
  54         elif name is "tau_corr" or "TauCorr":
  55             return self.tau_corr
  56 
  57 
  58 @jit(nopython=True)
  59 def _get_blocks(data, l):
  60     n_blocks = int(_np.floor(data.shape[1] / float(l)))
  61     blocks = _np.empty(data.shape[0] * n_blocks, dtype=data.dtype)
  62     k = 0
  63     for i in range(data.shape[0]):
  64         for b in range(n_blocks):
  65             blocks[k] = data[i, b * l : (b + 1) * l].mean()
  66             k += 1
  67     return blocks
  68 
  69 
  70 def _block_variance(data, l):
  71     blocks = _get_blocks(data, l)
  72     ts = _total_size(blocks)
  73     if ts > 0:
  74         return _var(blocks) / float(ts), ts
  75     else:
  76         return _np.nan, 0
  77 
  78 
  79 def _batch_variance(data):
  80     b_means = _np.mean(data, axis=1)
  81     ts = _total_size(b_means)
  82     return _var(b_means), ts
  83 
  84 
  85 def statistics(data):
  86     r"""
  87     Returns statistics of a given array (or matrix, see below) containing a stream of data.
  88     This is particularly useful to analyze Markov Chain data, but it can be used
  89     also for other type of time series.
  90 
  91     Args:
  92         data (vector or matrix): The input data. It can be real or complex valued.
  93                                 * if a vector, it is assumed that this is a time
  94                                   series of data (not necessarily independent).
  95                                 * if a matrix, it is assumed that that rows data[i]
  96                                   contain independent time series.
  97 
  98     Returns:
  99        Stats: A dictionary-compatible class containing the average (mean),
  100              the variance (variance),
  101              the error of the mean (error_of_mean), and an estimate of the
  102              autocorrelation time (tau_corr). In addition to accessing the elements with the standard
  103              dict sintax (e.g. res['mean']), one can also access them directly with the dot operator
  104              (e.g. res.mean).
  105     """
  106 
  107     stats = Stats()
  108     data = _np.atleast_1d(data)
  109     if data.ndim == 1:
  110         data = data.reshape((1, -1))
  111 
  112     if data.ndim > 2:
  113         raise NotImplementedError("Statistics are implemented only for ndim<=2")
  114 
  115     stats.mean = _mean(data)
  116     stats.variance = _var(data)
  117 
  118     ts = _total_size(data)
  119 
  120     bare_var = stats.variance / ts
  121 
  122     batch_var, n_batches = _batch_variance(data)
  123 
  124     b_s = 32
  125     l_block = max(1, data.shape[1] // b_s)
  126 
  127     block_var, n_blocks = _block_variance(data, l_block)
  128 
  129     tau_batch = (batch_var / bare_var - 1) * 0.5
  130     tau_block = (block_var / bare_var - 1) * 0.5
  131 
  132     block_good = tau_block < 6 * l_block and n_blocks >= b_s
  133     batch_good = tau_batch < 6 * data.shape[1] and n_batches >= b_s
  134 
  135     if batch_good:
  136         stats.error_of_mean = _np.sqrt(batch_var / ts)
  137         stats.tau_corr = max(0, tau_batch)
  138     elif block_good:
  139         stats.error_of_mean = _np.sqrt(block_var)
  140         stats.tau_corr = max(0, tau_block)
  141     else:
  142         stats.error_of_mean = _np.nan
  143         stats.tau_corr = _np.nan
  144 
  145     if n_batches > 1:
  146         N = data.shape[-1]
  147         stats.R = _np.sqrt((N - 1) / N + batch_var / stats.variance)
  148 
  149     return stats
  150
  ```
  --- END FILES ---
  
  Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.
  
  Here is an example:
  
  ```diff
  diff --git a/examples/server_async.py b/examples/server_async.py
  --- a/examples/server_async.py
  +++ b/examples/server_async.py
  @@ -313,4 +313,4 @@
   
   
   if __name__ == "__main__":
  -    asyncio.run(run_async_server("."), debug=True)
  +    asyncio.run(run_async_server(), debug=True)
  diff --git a/examples/server_sync.py b/examples/server_sync.py
  --- a/examples/server_sync.py
  +++ b/examples/server_sync.py
  @@ -313,5 +313,5 @@
   
   
   if __name__ == "__main__":
  -    server = run_sync_server(".")
  +    server = run_sync_server()
       server.shutdown()
  
  ```
  
golden_diff: diff --git a/netket/stats/mc_stats.py b/netket/stats/mc_stats.py
  --- a/netket/stats/mc_stats.py
  +++ b/netket/stats/mc_stats.py
  @@ -19,27 +19,32 @@
       _NaN = float("NaN")
   
       def __init__(
  -        self, mean=_NaN, error_of_mean=_NaN, variance=_NaN, tau_corr=_NaN, R=_NaN
  +        self,
  +        mean=_NaN,
  +        error_of_mean=_NaN,
  +        variance=_NaN,
  +        tau_corr=_NaN,
  +        R_hat=_NaN,
       ):
           self.mean = mean
           self.error_of_mean = error_of_mean
           self.variance = variance
           self.tau_corr = tau_corr
  -        self.R = R
  +        self.R_hat = R_hat
   
       def to_json(self):
           jsd = {}
           jsd["Mean"] = self.mean.real
           jsd["Variance"] = self.variance
           jsd["Sigma"] = self.error_of_mean
  -        jsd["R"] = self.R
  +        jsd["R_hat"] = self.R_hat
           jsd["TauCorr"] = self.tau_corr
           return jsd
   
       def __repr__(self):
           mean, err = _format_decimal(self.mean, self.error_of_mean)
  -        return "{} ± {} [var={:.1e}, R={:.4f}]".format(
  -            mean, err, self.variance, self.R
  +        return "{} ± {} [var={:.1e}, R_hat={:.4f}]".format(
  +            mean, err, self.variance, self.R_hat
           )
   
       def __getitem__(self, name):
  @@ -49,8 +54,8 @@
               return self.variance
           elif name is "error_of_mean" or "Sigma":
               return self.error_of_mean
  -        elif name is "R":
  -            return self.R
  +        elif name is "R_hat":
  +            return self.R_hat
           elif name is "tau_corr" or "TauCorr":
               return self.tau_corr
   
  @@ -144,6 +149,13 @@
   
       if n_batches > 1:
           N = data.shape[-1]
  -        stats.R = _np.sqrt((N - 1) / N + batch_var / stats.variance)
  +
  +        # V_loc = _np.var(data, axis=-1, ddof=0)
  +        # W_loc = _np.mean(V_loc)
  +        # W = _mean(W_loc)
  +        # # This approximation seems to hold well enough for larger n_samples
  +        W = stats.variance
  +
  +        stats.R_hat = _np.sqrt((N - 1) / N + batch_var / W)
   
       return stats
  
verification_info: {"golden_diff": "diff --git a/netket/stats/mc_stats.py b/netket/stats/mc_stats.py\n--- a/netket/stats/mc_stats.py\n+++ b/netket/stats/mc_stats.py\n@@ -19,27 +19,32 @@\n     _NaN = float(\"NaN\")\n \n     def __init__(\n-        self, mean=_NaN, error_of_mean=_NaN, variance=_NaN, tau_corr=_NaN, R=_NaN\n+        self,\n+        mean=_NaN,\n+        error_of_mean=_NaN,\n+        variance=_NaN,\n+        tau_corr=_NaN,\n+        R_hat=_NaN,\n     ):\n         self.mean = mean\n         self.error_of_mean = error_of_mean\n         self.variance = variance\n         self.tau_corr = tau_corr\n-        self.R = R\n+        self.R_hat = R_hat\n \n     def to_json(self):\n         jsd = {}\n         jsd[\"Mean\"] = self.mean.real\n         jsd[\"Variance\"] = self.variance\n         jsd[\"Sigma\"] = self.error_of_mean\n-        jsd[\"R\"] = self.R\n+        jsd[\"R_hat\"] = self.R_hat\n         jsd[\"TauCorr\"] = self.tau_corr\n         return jsd\n \n     def __repr__(self):\n         mean, err = _format_decimal(self.mean, self.error_of_mean)\n-        return \"{} \u00b1 {} [var={:.1e}, R={:.4f}]\".format(\n-            mean, err, self.variance, self.R\n+        return \"{} \u00b1 {} [var={:.1e}, R_hat={:.4f}]\".format(\n+            mean, err, self.variance, self.R_hat\n         )\n \n     def __getitem__(self, name):\n@@ -49,8 +54,8 @@\n             return self.variance\n         elif name is \"error_of_mean\" or \"Sigma\":\n             return self.error_of_mean\n-        elif name is \"R\":\n-            return self.R\n+        elif name is \"R_hat\":\n+            return self.R_hat\n         elif name is \"tau_corr\" or \"TauCorr\":\n             return self.tau_corr\n \n@@ -144,6 +149,13 @@\n \n     if n_batches > 1:\n         N = data.shape[-1]\n-        stats.R = _np.sqrt((N - 1) / N + batch_var / stats.variance)\n+\n+        # V_loc = _np.var(data, axis=-1, ddof=0)\n+        # W_loc = _np.mean(V_loc)\n+        # W = _mean(W_loc)\n+        # # This approximation seems to hold well enough for larger n_samples\n+        W = stats.variance\n+\n+        stats.R_hat = _np.sqrt((N - 1) / N + batch_var / W)\n \n     return stats\n", "issue": "Rename `netket.stats.Stats.R` to `Rhat`?\nAs we have seen in the discussion on #365, the `stats.R` parameter is ambiguously named; it is sometimes called \"Gelman-Rubin\" statistic or \"potential scale reduction\", but I think the most common way to refer to it is $\\hat R$ in LaTeX math or \"Rhat\" in words (this name is used, e.g., by the [stan](https://mc-stan.org/rstan/reference/Rhat.html) framework). So I propose we also call it `Rhat` starting in 3.0.\r\n\r\nWhat do you think?\n", "before_files": [{"content": "from numba import jit\nimport numpy as _np\nfrom . import mean as _mean\nfrom . import var as _var\nfrom . import total_size as _total_size\n\n\ndef _format_decimal(value, std):\n    decimals = max(int(_np.ceil(-_np.log10(std))), 0)\n    return (\n        \"{0:.{1}f}\".format(value, decimals),\n        \"{0:.{1}f}\".format(std, decimals + 1),\n    )\n\n\nclass Stats:\n    \"\"\"A dict-compatible class containing the result of the statistics function.\"\"\"\n\n    _NaN = float(\"NaN\")\n\n    def __init__(\n        self, mean=_NaN, error_of_mean=_NaN, variance=_NaN, tau_corr=_NaN, R=_NaN\n    ):\n        self.mean = mean\n        self.error_of_mean = error_of_mean\n        self.variance = variance\n        self.tau_corr = tau_corr\n        self.R = R\n\n    def to_json(self):\n        jsd = {}\n        jsd[\"Mean\"] = self.mean.real\n        jsd[\"Variance\"] = self.variance\n        jsd[\"Sigma\"] = self.error_of_mean\n        jsd[\"R\"] = self.R\n        jsd[\"TauCorr\"] = self.tau_corr\n        return jsd\n\n    def __repr__(self):\n        mean, err = _format_decimal(self.mean, self.error_of_mean)\n        return \"{} \u00b1 {} [var={:.1e}, R={:.4f}]\".format(\n            mean, err, self.variance, self.R\n        )\n\n    def __getitem__(self, name):\n        if name is \"mean\" or \"Mean\":\n            return self.mean\n        elif name is \"variance\" or \"Variance\":\n            return self.variance\n        elif name is \"error_of_mean\" or \"Sigma\":\n            return self.error_of_mean\n        elif name is \"R\":\n            return self.R\n        elif name is \"tau_corr\" or \"TauCorr\":\n            return self.tau_corr\n\n\n@jit(nopython=True)\ndef _get_blocks(data, l):\n    n_blocks = int(_np.floor(data.shape[1] / float(l)))\n    blocks = _np.empty(data.shape[0] * n_blocks, dtype=data.dtype)\n    k = 0\n    for i in range(data.shape[0]):\n        for b in range(n_blocks):\n            blocks[k] = data[i, b * l : (b + 1) * l].mean()\n            k += 1\n    return blocks\n\n\ndef _block_variance(data, l):\n    blocks = _get_blocks(data, l)\n    ts = _total_size(blocks)\n    if ts > 0:\n        return _var(blocks) / float(ts), ts\n    else:\n        return _np.nan, 0\n\n\ndef _batch_variance(data):\n    b_means = _np.mean(data, axis=1)\n    ts = _total_size(b_means)\n    return _var(b_means), ts\n\n\ndef statistics(data):\n    r\"\"\"\n    Returns statistics of a given array (or matrix, see below) containing a stream of data.\n    This is particularly useful to analyze Markov Chain data, but it can be used\n    also for other type of time series.\n\n    Args:\n        data (vector or matrix): The input data. It can be real or complex valued.\n                                * if a vector, it is assumed that this is a time\n                                  series of data (not necessarily independent).\n                                * if a matrix, it is assumed that that rows data[i]\n                                  contain independent time series.\n\n    Returns:\n       Stats: A dictionary-compatible class containing the average (mean),\n             the variance (variance),\n             the error of the mean (error_of_mean), and an estimate of the\n             autocorrelation time (tau_corr). In addition to accessing the elements with the standard\n             dict sintax (e.g. res['mean']), one can also access them directly with the dot operator\n             (e.g. res.mean).\n    \"\"\"\n\n    stats = Stats()\n    data = _np.atleast_1d(data)\n    if data.ndim == 1:\n        data = data.reshape((1, -1))\n\n    if data.ndim > 2:\n        raise NotImplementedError(\"Statistics are implemented only for ndim<=2\")\n\n    stats.mean = _mean(data)\n    stats.variance = _var(data)\n\n    ts = _total_size(data)\n\n    bare_var = stats.variance / ts\n\n    batch_var, n_batches = _batch_variance(data)\n\n    b_s = 32\n    l_block = max(1, data.shape[1] // b_s)\n\n    block_var, n_blocks = _block_variance(data, l_block)\n\n    tau_batch = (batch_var / bare_var - 1) * 0.5\n    tau_block = (block_var / bare_var - 1) * 0.5\n\n    block_good = tau_block < 6 * l_block and n_blocks >= b_s\n    batch_good = tau_batch < 6 * data.shape[1] and n_batches >= b_s\n\n    if batch_good:\n        stats.error_of_mean = _np.sqrt(batch_var / ts)\n        stats.tau_corr = max(0, tau_batch)\n    elif block_good:\n        stats.error_of_mean = _np.sqrt(block_var)\n        stats.tau_corr = max(0, tau_block)\n    else:\n        stats.error_of_mean = _np.nan\n        stats.tau_corr = _np.nan\n\n    if n_batches > 1:\n        N = data.shape[-1]\n        stats.R = _np.sqrt((N - 1) / N + batch_var / stats.variance)\n\n    return stats\n", "path": "netket/stats/mc_stats.py"}], "after_files": [{"content": "from numba import jit\nimport numpy as _np\nfrom . import mean as _mean\nfrom . import var as _var\nfrom . import total_size as _total_size\n\n\ndef _format_decimal(value, std):\n    decimals = max(int(_np.ceil(-_np.log10(std))), 0)\n    return (\n        \"{0:.{1}f}\".format(value, decimals),\n        \"{0:.{1}f}\".format(std, decimals + 1),\n    )\n\n\nclass Stats:\n    \"\"\"A dict-compatible class containing the result of the statistics function.\"\"\"\n\n    _NaN = float(\"NaN\")\n\n    def __init__(\n        self,\n        mean=_NaN,\n        error_of_mean=_NaN,\n        variance=_NaN,\n        tau_corr=_NaN,\n        R_hat=_NaN,\n    ):\n        self.mean = mean\n        self.error_of_mean = error_of_mean\n        self.variance = variance\n        self.tau_corr = tau_corr\n        self.R_hat = R_hat\n\n    def to_json(self):\n        jsd = {}\n        jsd[\"Mean\"] = self.mean.real\n        jsd[\"Variance\"] = self.variance\n        jsd[\"Sigma\"] = self.error_of_mean\n        jsd[\"R_hat\"] = self.R_hat\n        jsd[\"TauCorr\"] = self.tau_corr\n        return jsd\n\n    def __repr__(self):\n        mean, err = _format_decimal(self.mean, self.error_of_mean)\n        return \"{} \u00b1 {} [var={:.1e}, R_hat={:.4f}]\".format(\n            mean, err, self.variance, self.R_hat\n        )\n\n    def __getitem__(self, name):\n        if name is \"mean\" or \"Mean\":\n            return self.mean\n        elif name is \"variance\" or \"Variance\":\n            return self.variance\n        elif name is \"error_of_mean\" or \"Sigma\":\n            return self.error_of_mean\n        elif name is \"R_hat\":\n            return self.R_hat\n        elif name is \"tau_corr\" or \"TauCorr\":\n            return self.tau_corr\n\n\n@jit(nopython=True)\ndef _get_blocks(data, l):\n    n_blocks = int(_np.floor(data.shape[1] / float(l)))\n    blocks = _np.empty(data.shape[0] * n_blocks, dtype=data.dtype)\n    k = 0\n    for i in range(data.shape[0]):\n        for b in range(n_blocks):\n            blocks[k] = data[i, b * l : (b + 1) * l].mean()\n            k += 1\n    return blocks\n\n\ndef _block_variance(data, l):\n    blocks = _get_blocks(data, l)\n    ts = _total_size(blocks)\n    if ts > 0:\n        return _var(blocks) / float(ts), ts\n    else:\n        return _np.nan, 0\n\n\ndef _batch_variance(data):\n    b_means = _np.mean(data, axis=1)\n    ts = _total_size(b_means)\n    return _var(b_means), ts\n\n\ndef statistics(data):\n    r\"\"\"\n    Returns statistics of a given array (or matrix, see below) containing a stream of data.\n    This is particularly useful to analyze Markov Chain data, but it can be used\n    also for other type of time series.\n\n    Args:\n        data (vector or matrix): The input data. It can be real or complex valued.\n                                * if a vector, it is assumed that this is a time\n                                  series of data (not necessarily independent).\n                                * if a matrix, it is assumed that that rows data[i]\n                                  contain independent time series.\n\n    Returns:\n       Stats: A dictionary-compatible class containing the average (mean),\n             the variance (variance),\n             the error of the mean (error_of_mean), and an estimate of the\n             autocorrelation time (tau_corr). In addition to accessing the elements with the standard\n             dict sintax (e.g. res['mean']), one can also access them directly with the dot operator\n             (e.g. res.mean).\n    \"\"\"\n\n    stats = Stats()\n    data = _np.atleast_1d(data)\n    if data.ndim == 1:\n        data = data.reshape((1, -1))\n\n    if data.ndim > 2:\n        raise NotImplementedError(\"Statistics are implemented only for ndim<=2\")\n\n    stats.mean = _mean(data)\n    stats.variance = _var(data)\n\n    ts = _total_size(data)\n\n    bare_var = stats.variance / ts\n\n    batch_var, n_batches = _batch_variance(data)\n\n    b_s = 32\n    l_block = max(1, data.shape[1] // b_s)\n\n    block_var, n_blocks = _block_variance(data, l_block)\n\n    tau_batch = (batch_var / bare_var - 1) * 0.5\n    tau_block = (block_var / bare_var - 1) * 0.5\n\n    block_good = tau_block < 6 * l_block and n_blocks >= b_s\n    batch_good = tau_batch < 6 * data.shape[1] and n_batches >= b_s\n\n    if batch_good:\n        stats.error_of_mean = _np.sqrt(batch_var / ts)\n        stats.tau_corr = max(0, tau_batch)\n    elif block_good:\n        stats.error_of_mean = _np.sqrt(block_var)\n        stats.tau_corr = max(0, tau_block)\n    else:\n        stats.error_of_mean = _np.nan\n        stats.tau_corr = _np.nan\n\n    if n_batches > 1:\n        N = data.shape[-1]\n\n        # V_loc = _np.var(data, axis=-1, ddof=0)\n        # W_loc = _np.mean(V_loc)\n        # W = _mean(W_loc)\n        # # This approximation seems to hold well enough for larger n_samples\n        W = stats.variance\n\n        stats.R_hat = _np.sqrt((N - 1) / N + batch_var / W)\n\n    return stats\n", "path": "netket/stats/mc_stats.py"}]}