# pylint: disable=too-many-lines
"""Statistical functions in ArviZ."""
import warnings
import logging
from collections import OrderedDict
from copy import deepcopy
from typing import Optional, List, Union
import numpy as np
import pandas as pd
import scipy.stats as st
from scipy.optimize import minimize
import xarray as xr
from ..data import convert_to_inference_data, convert_to_dataset, InferenceData, CoordSpec, DimSpec
from .diagnostics import _multichain_statistics, _mc_error, ess
from .stats_utils import (
make_ufunc as _make_ufunc,
wrap_xarray_ufunc as _wrap_xarray_ufunc,
logsumexp as _logsumexp,
ELPDData,
stats_variance_2d as svar,
_circular_standard_deviation,
get_log_likelihood as _get_log_likelihood,
)
from ..numeric_utils import _fast_kde, histogram, get_bins
from ..utils import _var_names, Numba, _numba_var, get_coords, credible_interval_warning
from ..rcparams import rcParams
_log = logging.getLogger(__name__)
__all__ = [
"apply_test_function",
"compare",
"hdi",
"hpd",
"loo",
"loo_pit",
"psislw",
"r2_score",
"summary",
"waic",
]
def compare(
dataset_dict, ic=None, method="BB-pseudo-BMA", b_samples=1000, alpha=1, seed=None, scale=None
):
r"""Compare models based on PSIS-LOO `loo` or WAIC `waic` cross-validation.
LOO is leave-one-out (PSIS-LOO `loo`) cross-validation and
WAIC is the widely applicable information criterion.
Read more theory here - in a paper by some of the leading authorities
on model selection dx.doi.org/10.1111/1467-9868.00353
Parameters
----------
dataset_dict: dict[str] -> InferenceData
A dictionary of model names and InferenceData objects
ic: str
Information Criterion (PSIS-LOO `loo` or WAIC `waic`) used to compare models. Defaults to
``rcParams["stats.information_criterion"]``.
method: str
Method used to estimate the weights for each model. Available options are:
- 'stacking' : stacking of predictive distributions.
- 'BB-pseudo-BMA' : (default) pseudo-Bayesian Model averaging using Akaike-type
weighting. The weights are stabilized using the Bayesian bootstrap.
- 'pseudo-BMA': pseudo-Bayesian Model averaging using Akaike-type
weighting, without Bootstrap stabilization (not recommended).
For more information read https://arxiv.org/abs/1704.02030
b_samples: int
Number of samples taken by the Bayesian bootstrap estimation.
Only useful when method = 'BB-pseudo-BMA'.
alpha: float
The shape parameter in the Dirichlet distribution used for the Bayesian bootstrap. Only
useful when method = 'BB-pseudo-BMA'. When alpha=1 (default), the distribution is uniform
on the simplex. A smaller alpha will keeps the final weights more away from 0 and 1.
seed: int or np.random.RandomState instance
If int or RandomState, use it for seeding Bayesian bootstrap. Only
useful when method = 'BB-pseudo-BMA'. Default None the global
np.random state is used.
scale: str
Output scale for IC. Available options are:
- `log` : (default) log-score (after Vehtari et al. (2017))
- `negative_log` : -1 * (log-score)
- `deviance` : -2 * (log-score)
A higher log-score (or a lower deviance) indicates a model with better predictive
accuracy.
Returns
-------
A DataFrame, ordered from best to worst model (measured by information criteria).
The index reflects the key with which the models are passed to this function. The columns are:
rank: The rank-order of the models. 0 is the best.
IC: Information Criteria (PSIS-LOO `loo` or WAIC `waic`).
Higher IC indicates higher out-of-sample predictive fit ("better" model). Default LOO.
If `scale` is `deviance` or `negative_log` smaller IC indicates
higher out-of-sample predictive fit ("better" model).
pIC: Estimated effective number of parameters.
dIC: Relative difference between each IC (PSIS-LOO `loo` or WAIC `waic`)
and the lowest IC (PSIS-LOO `loo` or WAIC `waic`).
The top-ranked model is always 0.
weight: Relative weight for each model.
This can be loosely interpreted as the probability of each model (among the compared model)
given the data. By default the uncertainty in the weights estimation is considered using
Bayesian bootstrap.
SE: Standard error of the IC estimate.
If method = BB-pseudo-BMA these values are estimated using Bayesian bootstrap.
dSE: Standard error of the difference in IC between each model and the top-ranked model.
It's always 0 for the top-ranked model.
warning: A value of 1 indicates that the computation of the IC may not be reliable.
This could be indication of WAIC/LOO starting to fail see
http://arxiv.org/abs/1507.04544 for details.
scale: Scale used for the IC.
Examples
--------
Compare the centered and non centered models of the eight school problem:
.. ipython::
In [1]: import arviz as az
...: data1 = az.load_arviz_data("non_centered_eight")
...: data2 = az.load_arviz_data("centered_eight")
...: compare_dict = {"non centered": data1, "centered": data2}
...: az.compare(compare_dict)
Compare the models using LOO-CV, returning the IC in log scale and calculating the
weights using the stacking method.
.. ipython::
In [1]: az.compare(compare_dict, ic="loo", method="stacking", scale="log")
See Also
--------
loo : Compute the Pareto Smoothed importance sampling Leave One Out cross-validation.
waic : Compute the widely applicable information criterion.
"""
names = list(dataset_dict.keys())
scale = rcParams["stats.ic_scale"] if scale is None else scale.lower()
if scale == "log":
scale_value = 1
ascending = False
warnings.warn(
"\nThe scale is now log by default. Use 'scale' argument or "
"'stats.ic_scale' rcParam if you rely on a specific value.\nA higher "
"log-score (or a lower deviance) indicates a model with better predictive "
"accuracy."
)
else:
if scale == "negative_log":
scale_value = -1
else:
scale_value = -2
ascending = True
ic = rcParams["stats.information_criterion"] if ic is None else ic.lower()
if ic == "loo":
ic_func = loo
df_comp = pd.DataFrame(
index=names,
columns=[
"rank",
"loo",
"p_loo",
"d_loo",
"weight",
"se",
"dse",
"warning",
"loo_scale",
],
)
scale_col = "loo_scale"
elif ic == "waic":
ic_func = waic
df_comp = pd.DataFrame(
index=names,
columns=[
"rank",
"waic",
"p_waic",
"d_waic",
"weight",
"se",
"dse",
"warning",
"waic_scale",
],
)
scale_col = "waic_scale"
else:
raise NotImplementedError("The information criterion {} is not supported.".format(ic))
if method.lower() not in ["stacking", "bb-pseudo-bma", "pseudo-bma"]:
raise ValueError("The method {}, to compute weights, is not supported.".format(method))
ic_se = "{}_se".format(ic)
p_ic = "p_{}".format(ic)
ic_i = "{}_i".format(ic)
ics = pd.DataFrame()
names = []
for name, dataset in dataset_dict.items():
names.append(name)
ics = ics.append([ic_func(dataset, pointwise=True, scale=scale)])
ics.index = names
ics.sort_values(by=ic, inplace=True, ascending=ascending)
ics[ic_i] = ics[ic_i].apply(lambda x: x.values.flatten())
if method.lower() == "stacking":
rows, cols, ic_i_val = _ic_matrix(ics, ic_i)
exp_ic_i = np.exp(ic_i_val / scale_value)
last_col = cols - 1
def w_fuller(weights):
return np.concatenate((weights, [max(1.0 - np.sum(weights), 0.0)]))
def log_score(weights):
w_full = w_fuller(weights)
score = 0.0
for i in range(rows):
score += np.log(np.dot(exp_ic_i[i], w_full))
return -score
def gradient(weights):
w_full = w_fuller(weights)
grad = np.zeros(last_col)
for k in range(last_col - 1):
for i in range(rows):
grad[k] += (exp_ic_i[i, k] - exp_ic_i[i, last_col]) / np.dot(
exp_ic_i[i], w_full
)
return -grad
theta = np.full(last_col, 1.0 / cols)
bounds = [(0.0, 1.0) for _ in range(last_col)]
constraints = [
{"type": "ineq", "fun": lambda x: 1.0 - np.sum(x)},
{"type": "ineq", "fun": np.sum},
]
weights = minimize(
fun=log_score, x0=theta, jac=gradient, bounds=bounds, constraints=constraints
)
weights = w_fuller(weights["x"])
ses = ics[ic_se]
elif method.lower() == "bb-pseudo-bma":
rows, cols, ic_i_val = _ic_matrix(ics, ic_i)
ic_i_val = ic_i_val * rows
b_weighting = st.dirichlet.rvs(alpha=[alpha] * rows, size=b_samples, random_state=seed)
weights = np.zeros((b_samples, cols))
z_bs = np.zeros_like(weights)
for i in range(b_samples):
z_b = np.dot(b_weighting[i], ic_i_val)
u_weights = np.exp((z_b - np.min(z_b)) / scale_value)
z_bs[i] = z_b # pylint: disable=unsupported-assignment-operation
weights[i] = u_weights / np.sum(u_weights)
weights = weights.mean(axis=0)
ses = pd.Series(z_bs.std(axis=0), index=names) # pylint: disable=no-member
elif method.lower() == "pseudo-bma":
min_ic = ics.iloc[0][ic]
z_rv = np.exp((ics[ic] - min_ic) / scale_value)
weights = z_rv / np.sum(z_rv)
ses = ics[ic_se]
if np.any(weights):
min_ic_i_val = ics[ic_i].iloc[0]
for idx, val in enumerate(ics.index):
res = ics.loc[val]
if scale_value < 0:
diff = res[ic_i] - min_ic_i_val
else:
diff = min_ic_i_val - res[ic_i]
d_ic = np.sum(diff)
d_std_err = np.sqrt(len(diff) * np.var(diff))
std_err = ses.loc[val]
weight = weights[idx]
df_comp.at[val] = (
idx,
res[ic],
res[p_ic],
d_ic,
weight,
std_err,
d_std_err,
res["warning"],
res[scale_col],
)
return df_comp.sort_values(by=ic, ascending=ascending)
def _ic_matrix(ics, ic_i):
"""Store the previously computed pointwise predictive accuracy values (ics) in a 2D matrix."""
cols, _ = ics.shape
rows = len(ics[ic_i].iloc[0])
ic_i_val = np.zeros((rows, cols))
for idx, val in enumerate(ics.index):
ic = ics.loc[val][ic_i]
if len(ic) != rows:
raise ValueError("The number of observations should be the same across all models")
ic_i_val[:, idx] = ic
return rows, cols, ic_i_val
def hpd(
# pylint: disable=unused-argument
ary,
hdi_prob=None,
circular=False,
multimodal=False,
skipna=False,
group="posterior",
var_names=None,
filter_vars=None,
coords=None,
max_modes=10,
**kwargs,
):
"""Pending deprecation. Please refer to :func:`~arviz.hdi`."""
# pylint: enable=unused-argument
warnings.warn(("hpd will be deprecated " "Please replace hdi"),)
return hdi(
ary,
hdi_prob,
circular,
multimodal,
skipna,
group,
var_names,
filter_vars,
coords,
max_modes,
**kwargs,
)
def hdi(
ary,
hdi_prob=None,
circular=False,
multimodal=False,
skipna=False,
group="posterior",
var_names=None,
filter_vars=None,
coords=None,
max_modes=10,
**kwargs,
):
"""
Calculate highest density interval (HDI) of array for given probability.
The HDI is the minimum width Bayesian credible interval (BCI).
Parameters
----------
ary: obj
object containing posterior samples.
Any object that can be converted to an az.InferenceData object.
Refer to documentation of az.convert_to_dataset for details.
hdi_prob: float, optional
HDI prob for which interval will be computed. Defaults to ``stats.hdi_prob`` rcParam.
circular: bool, optional
Whether to compute the hdi taking into account `x` is a circular variable
(in the range [-np.pi, np.pi]) or not. Defaults to False (i.e non-circular variables).
Only works if multimodal is False.
multimodal: bool, optional
If true it may compute more than one hdi interval if the distribution is multimodal and the
modes are well separated.
skipna: bool, optional
If true ignores nan values when computing the hdi interval. Defaults to false.
group: str, optional
Specifies which InferenceData group should be used to calculate hdi.
Defaults to 'posterior'
var_names: list, optional
Names of variables to include in the hdi report. Prefix the variables by `~`
when you want to exclude them from the report: `["~beta"]` instead of `["beta"]`
(see `az.summary` for more details).
filter_vars: {None, "like", "regex"}, optional, default=None
If `None` (default), interpret var_names as the real variables names. If "like",
interpret var_names as substrings of the real variables names. If "regex",
interpret var_names as regular expressions on the real variables names. A la
`pandas.filter`.
coords: mapping, optional
Specifies the subset over to calculate hdi.
max_modes: int, optional
Specifies the maximum number of modes for multimodal case.
kwargs: dict, optional
Additional keywords passed to :func:`~arviz.wrap_xarray_ufunc`.
Returns
-------
np.ndarray or xarray.Dataset, depending upon input
lower(s) and upper(s) values of the interval(s).
See Also
--------
plot_hdi : Plot HDI intervals for regression data.
xarray.Dataset.quantile : Calculate quantiles of array for given probabilities.
Examples
--------
Calculate the HDI of a Normal random variable:
.. ipython::
In [1]: import arviz as az
...: import numpy as np
...: data = np.random.normal(size=2000)
...: az.hdi(data, hdi_prob=.68)
Calculate the HDI of a dataset:
.. ipython::
In [1]: import arviz as az
...: data = az.load_arviz_data('centered_eight')
...: az.hdi(data)
We can also calculate the HDI of some of the variables of dataset:
.. ipython::
In [1]: az.hdi(data, var_names=["mu", "theta"])
If we want to calculate the HDI over specified dimension of dataset,
we can pass `input_core_dims` by kwargs:
.. ipython::
In [1]: az.hdi(data, input_core_dims = [["chain"]])
We can also calculate the hdi over a particular selection over all groups:
.. ipython::
In [1]: az.hdi(data, coords={"chain":[0, 1, 3]}, input_core_dims = [["draw"]])
"""
if hdi_prob is None:
hdi_prob = rcParams["stats.hdi_prob"]
else:
if not 1 >= hdi_prob > 0:
raise ValueError("The value of hdi_prob should be in the interval (0, 1]")
func_kwargs = {
"hdi_prob": hdi_prob,
"skipna": skipna,
"out_shape": (max_modes, 2) if multimodal else (2,),
}
kwargs.setdefault("output_core_dims", [["hdi", "mode"] if multimodal else ["hdi"]])
if not multimodal:
func_kwargs["circular"] = circular
else:
func_kwargs["max_modes"] = max_modes
func = _hdi_multimodal if multimodal else _hdi
isarray = isinstance(ary, np.ndarray)
if isarray and ary.ndim <= 1:
func_kwargs.pop("out_shape")
hdi_data = func(ary, **func_kwargs) # pylint: disable=unexpected-keyword-arg
return hdi_data[~np.isnan(hdi_data).all(axis=1), :] if multimodal else hdi_data
if isarray and ary.ndim == 2:
warnings.warn(
"hdi currently interprets 2d data as (draw, shape) but this will change in "
"a future release to (chain, draw) for coherence with other functions",
FutureWarning,
)
ary = np.expand_dims(ary, 0)
ary = convert_to_dataset(ary, group=group)
if coords is not None:
ary = get_coords(ary, coords)
var_names = _var_names(var_names, ary, filter_vars)
ary = ary[var_names] if var_names else ary
hdi_coord = xr.DataArray(["lower", "higher"], dims=["hdi"], attrs=dict(hdi_prob=hdi_prob))
hdi_data = _wrap_xarray_ufunc(func, ary, func_kwargs=func_kwargs, **kwargs).assign_coords(
{"hdi": hdi_coord}
)
hdi_data = hdi_data.dropna("mode", how="all") if multimodal else hdi_data
return hdi_data.x.values if isarray else hdi_data
def _hdi(ary, hdi_prob, circular, skipna):
"""Compute hpi over the flattened array."""
ary = ary.flatten()
if skipna:
nans = np.isnan(ary)
if not nans.all():
ary = ary[~nans]
n = len(ary)
if circular:
mean = st.circmean(ary, high=np.pi, low=-np.pi)
ary = ary - mean
ary = np.arctan2(np.sin(ary), np.cos(ary))
ary = np.sort(ary)
interval_idx_inc = int(np.floor(hdi_prob * n))
n_intervals = n - interval_idx_inc
interval_width = ary[interval_idx_inc:] - ary[:n_intervals]
if len(interval_width) == 0:
raise ValueError("Too few elements for interval calculation. ")
min_idx = np.argmin(interval_width)
hdi_min = ary[min_idx]
hdi_max = ary[min_idx + interval_idx_inc]
if circular:
hdi_min = hdi_min + mean
hdi_max = hdi_max + mean
hdi_min = np.arctan2(np.sin(hdi_min), np.cos(hdi_min))
hdi_max = np.arctan2(np.sin(hdi_max), np.cos(hdi_max))
hdi_interval = np.array([hdi_min, hdi_max])
return hdi_interval
def _hdi_multimodal(ary, hdi_prob, skipna, max_modes):
"""Compute HDI if the distribution is multimodal."""
ary = ary.flatten()
if skipna:
ary = ary[~np.isnan(ary)]
if ary.dtype.kind == "f":
density, lower, upper = _fast_kde(ary)
range_x = upper - lower
dx = range_x / len(density)
bins = np.linspace(lower, upper, len(density))
else:
bins = get_bins(ary)
_, density, _ = histogram(ary, bins=bins)
dx = np.diff(bins)[0]
density *= dx
idx = np.argsort(-density)
intervals = bins[idx][density[idx].cumsum() <= hdi_prob]
intervals.sort()
intervals_splitted = np.split(intervals, np.where(np.diff(intervals) >= dx * 1.1)[0] + 1)
hdi_intervals = np.full((max_modes, 2), np.nan)
for i, interval in enumerate(intervals_splitted):
if i == max_modes:
warnings.warn(
"found more modes than {0}, returning only the first {0} modes", max_modes
)
break
if interval.size == 0:
hdi_intervals[i] = np.asarray([bins[0], bins[0]])
else:
hdi_intervals[i] = np.asarray([interval[0], interval[-1]])
return np.array(hdi_intervals)
def loo(data, pointwise=None, var_name=None, reff=None, scale=None):
"""Compute Pareto-smoothed importance sampling leave-one-out cross-validation (PSIS-LOO-CV).
Estimates the expected log pointwise predictive density (elpd) using Pareto-smoothed
importance sampling leave-one-out cross-validation (PSIS-LOO-CV). Also calculates LOO's
standard error and the effective number of parameters. Read more theory here
https://arxiv.org/abs/1507.04544 and here https://arxiv.org/abs/1507.02646
Parameters
----------
data: obj
Any object that can be converted to an az.InferenceData object. Refer to documentation of
az.convert_to_inference_data for details
pointwise: bool, optional
If True the pointwise predictive accuracy will be returned. Defaults to
``stats.ic_pointwise`` rcParam.
var_name : str, optional
The name of the variable in log_likelihood groups storing the pointwise log
likelihood data to use for loo computation.
reff: float, optional
Relative MCMC efficiency, `ess / n` i.e. number of effective samples divided by the number
of actual samples. Computed from trace by default.
scale: str
Output scale for loo. Available options are:
- `log` : (default) log-score
- `negative_log` : -1 * log-score
- `deviance` : -2 * log-score
A higher log-score (or a lower deviance or negative log_score) indicates a model with
better predictive accuracy.
Returns
-------
ELPDData object (inherits from panda.Series) with the following row/attributes:
loo: approximated expected log pointwise predictive density (elpd)
loo_se: standard error of loo
p_loo: effective number of parameters
shape_warn: bool
True if the estimated shape parameter of
Pareto distribution is greater than 0.7 for one or more samples
loo_i: array of pointwise predictive accuracy, only if pointwise True
pareto_k: array of Pareto shape values, only if pointwise True
loo_scale: scale of the loo results
The returned object has a custom print method that overrides pd.Series method.
Examples
--------
Calculate LOO of a model:
.. ipython::
In [1]: import arviz as az
...: data = az.load_arviz_data("centered_eight")
...: az.loo(data)
Calculate LOO of a model and return the pointwise values:
.. ipython::
In [2]: data_loo = az.loo(data, pointwise=True)
...: data_loo.loo_i
"""
inference_data = convert_to_inference_data(data)
log_likelihood = _get_log_likelihood(inference_data, var_name=var_name)
pointwise = rcParams["stats.ic_pointwise"] if pointwise is None else pointwise
log_likelihood = log_likelihood.stack(sample=("chain", "draw"))
shape = log_likelihood.shape
n_samples = shape[-1]
n_data_points = np.product(shape[:-1])
scale = rcParams["stats.ic_scale"] if scale is None else scale.lower()
if scale == "deviance":
scale_value = -2
elif scale == "log":
scale_value = 1
elif scale == "negative_log":
scale_value = -1
else:
raise TypeError('Valid scale values are "deviance", "log", "negative_log"')
if reff is None:
if not hasattr(inference_data, "posterior"):
raise TypeError("Must be able to extract a posterior group from data.")
posterior = inference_data.posterior
n_chains = len(posterior.chain)
if n_chains == 1:
reff = 1.0
else:
ess_p = ess(posterior, method="mean")
# this mean is over all data variables
reff = (
np.hstack([ess_p[v].values.flatten() for v in ess_p.data_vars]).mean() / n_samples
)
log_weights, pareto_shape = psislw(-log_likelihood, reff)
log_weights += log_likelihood
warn_mg = False
if np.any(pareto_shape > 0.7):
warnings.warn(
"Estimated shape parameter of Pareto distribution is greater than 0.7 for "
"one or more samples. You should consider using a more robust model, this is because "
"importance sampling is less likely to work well if the marginal posterior and "
"LOO posterior are very different. This is more likely to happen with a non-robust "
"model and highly influential observations."
)
warn_mg = True
ufunc_kwargs = {"n_dims": 1, "ravel": False}
kwargs = {"input_core_dims": [["sample"]]}
loo_lppd_i = scale_value * _wrap_xarray_ufunc(
_logsumexp, log_weights, ufunc_kwargs=ufunc_kwargs, **kwargs
)
loo_lppd = loo_lppd_i.values.sum()
loo_lppd_se = (n_data_points * np.var(loo_lppd_i.values)) ** 0.5
lppd = np.sum(
_wrap_xarray_ufunc(
_logsumexp,
log_likelihood,
func_kwargs={"b_inv": n_samples},
ufunc_kwargs=ufunc_kwargs,
**kwargs,
).values
)
p_loo = lppd - loo_lppd / scale_value
if pointwise:
if np.equal(loo_lppd, loo_lppd_i).all(): # pylint: disable=no-member
warnings.warn(
"The point-wise LOO is the same with the sum LOO, please double check "
"the Observed RV in your model to make sure it returns element-wise logp."
)
return ELPDData(
data=[
loo_lppd,
loo_lppd_se,
p_loo,
n_samples,
n_data_points,
warn_mg,
loo_lppd_i.rename("loo_i"),
pareto_shape,
scale,
],
index=[
"loo",
"loo_se",
"p_loo",
"n_samples",
"n_data_points",
"warning",
"loo_i",
"pareto_k",
"loo_scale",
],
)
else:
return ELPDData(
data=[loo_lppd, loo_lppd_se, p_loo, n_samples, n_data_points, warn_mg, scale],
index=["loo", "loo_se", "p_loo", "n_samples", "n_data_points", "warning", "loo_scale"],
)
def psislw(log_weights, reff=1.0):
"""
Pareto smoothed importance sampling (PSIS).
Parameters
----------
log_weights: array
Array of size (n_observations, n_samples)
reff: float
relative MCMC efficiency, `ess / n`
Returns
-------
lw_out: array
Smoothed log weights
kss: array
Pareto tail indices
References
----------
* Vehtari et al. (2015) see https://arxiv.org/abs/1507.02646
Examples
--------
Get Pareto smoothed importance sampling (PSIS) log weights:
.. ipython::
In [1]: import arviz as az
...: data = az.load_arviz_data("centered_eight")
...: log_likelihood = data.sample_stats.log_likelihood.stack(sample=("chain", "draw"))
...: az.psislw(-log_likelihood, reff=0.8)
"""
if hasattr(log_weights, "sample"):
n_samples = len(log_weights.sample)
shape = [size for size, dim in zip(log_weights.shape, log_weights.dims) if dim != "sample"]
else:
n_samples = log_weights.shape[-1]
shape = log_weights.shape[:-1]
# precalculate constants
cutoff_ind = -int(np.ceil(min(n_samples / 5.0, 3 * (n_samples / reff) ** 0.5))) - 1
cutoffmin = np.log(np.finfo(float).tiny) # pylint: disable=no-member, assignment-from-no-return
k_min = 1.0 / 3
# create output array with proper dimensions
out = tuple([np.empty_like(log_weights), np.empty(shape)])
# define kwargs
func_kwargs = {"cutoff_ind": cutoff_ind, "cutoffmin": cutoffmin, "k_min": k_min, "out": out}
ufunc_kwargs = {"n_dims": 1, "n_output": 2, "ravel": False, "check_shape": False}
kwargs = {"input_core_dims": [["sample"]], "output_core_dims": [["sample"], []]}
log_weights, pareto_shape = _wrap_xarray_ufunc(
_psislw, log_weights, ufunc_kwargs=ufunc_kwargs, func_kwargs=func_kwargs, **kwargs
)
if isinstance(log_weights, xr.DataArray):
log_weights = log_weights.rename("log_weights").rename(sample="sample")
if isinstance(pareto_shape, xr.DataArray):
pareto_shape = pareto_shape.rename("pareto_shape")
return log_weights, pareto_shape
def _psislw(log_weights, cutoff_ind, cutoffmin, k_min=1.0 / 3):
"""
Pareto smoothed importance sampling (PSIS) for a 1D vector.
Parameters
----------
log_weights: array
Array of length n_observations
cutoff_ind: int
cutoffmin: float
k_min: float
Returns
-------
lw_out: array
Smoothed log weights
kss: float
Pareto tail index
"""
x = np.asarray(log_weights)
# improve numerical accuracy
x -= np.max(x)
# sort the array
x_sort_ind = np.argsort(x)
# divide log weights into body and right tail
xcutoff = max(x[x_sort_ind[cutoff_ind]], cutoffmin)
expxcutoff = np.exp(xcutoff)
(tailinds,) = np.where(x > xcutoff) # pylint: disable=unbalanced-tuple-unpacking
x_tail = x[tailinds]
tail_len = len(x_tail)
if tail_len <= 4:
# not enough tail samples for gpdfit
k = np.inf
else:
# order of tail samples
x_tail_si = np.argsort(x_tail)
# fit generalized Pareto distribution to the right tail samples
x_tail = np.exp(x_tail) - expxcutoff
k, sigma = _gpdfit(x_tail[x_tail_si])
if k >= k_min:
# no smoothing if short tail or GPD fit failed
# compute ordered statistic for the fit
sti = np.arange(0.5, tail_len) / tail_len
smoothed_tail = _gpinv(sti, k, sigma)
smoothed_tail = np.log( # pylint: disable=assignment-from-no-return
smoothed_tail + expxcutoff
)
# place the smoothed tail into the output array
x[tailinds[x_tail_si]] = smoothed_tail
# truncate smoothed values to the largest raw weight 0
x[x > 0] = 0
# renormalize weights
x -= _logsumexp(x)
return x, k
def _gpdfit(ary):
"""Estimate the parameters for the Generalized Pareto Distribution (GPD).
Empirical Bayes estimate for the parameters of the generalized Pareto
distribution given the data.
Parameters
----------
ary: array
sorted 1D data array
Returns
-------
k: float
estimated shape parameter
sigma: float
estimated scale parameter
"""
prior_bs = 3
prior_k = 10
n = len(ary)
m_est = 30 + int(n ** 0.5)
b_ary = 1 - np.sqrt(m_est / (np.arange(1, m_est + 1, dtype=float) - 0.5))
b_ary /= prior_bs * ary[int(n / 4 + 0.5) - 1]
b_ary += 1 / ary[-1]
k_ary = np.log1p(-b_ary[:, None] * ary).mean(axis=1) # pylint: disable=no-member
len_scale = n * (np.log(-(b_ary / k_ary)) - k_ary - 1)
weights = 1 / np.exp(len_scale - len_scale[:, None]).sum(axis=1)
# remove negligible weights
real_idxs = weights >= 10 * np.finfo(float).eps
if not np.all(real_idxs):
weights = weights[real_idxs]
b_ary = b_ary[real_idxs]
# normalise weights
weights /= weights.sum()
# posterior mean for b
b_post = np.sum(b_ary * weights)
# estimate for k
k_post = np.log1p(-b_post * ary).mean() # pylint: disable=invalid-unary-operand-type,no-member
# add prior for k_post
k_post = (n * k_post + prior_k * 0.5) / (n + prior_k)
sigma = -k_post / b_post
return k_post, sigma
def _gpinv(probs, kappa, sigma):
"""Inverse Generalized Pareto distribution function."""
# pylint: disable=unsupported-assignment-operation, invalid-unary-operand-type
x = np.full_like(probs, np.nan)
if sigma <= 0:
return x
ok = (probs > 0) & (probs < 1)
if np.all(ok):
if np.abs(kappa) < np.finfo(float).eps:
x = -np.log1p(-probs)
else:
x = np.expm1(-kappa * np.log1p(-probs)) / kappa
x *= sigma
else:
if np.abs(kappa) < np.finfo(float).eps:
x[ok] = -np.log1p(-probs[ok])
else:
x[ok] = np.expm1(-kappa * np.log1p(-probs[ok])) / kappa
x *= sigma
x[probs == 0] = 0
if kappa >= 0:
x[probs == 1] = np.inf
else:
x[probs == 1] = -sigma / kappa
return x
def r2_score(y_true, y_pred):
"""R² for Bayesian regression models. Only valid for linear models.
Parameters
----------
y_true: array-like of shape = (n_samples) or (n_samples, n_outputs)
Ground truth (correct) target values.
y_pred: array-like of shape = (n_samples) or (n_samples, n_outputs)
Estimated target values.
Returns
-------
Pandas Series with the following indices:
r2: Bayesian R²
r2_std: standard deviation of the Bayesian R².
Examples
--------
Calculate R² for Bayesian regression models :
.. ipython::
In [1]: import arviz as az
...: data = az.load_arviz_data('regression1d')
...: y_true = data.observed_data["y"].values
...: y_pred = data.posterior_predictive.stack(sample=("chain", "draw"))["y"].values.T
...: az.r2_score(y_true, y_pred)
"""
_numba_flag = Numba.numba_flag
if y_pred.ndim == 1:
var_y_est = _numba_var(svar, np.var, y_pred)
var_e = _numba_var(svar, np.var, (y_true - y_pred))
else:
var_y_est = _numba_var(svar, np.var, y_pred.mean(0))
var_e = _numba_var(svar, np.var, (y_true - y_pred), axis=0)
r_squared = var_y_est / (var_y_est + var_e)
return pd.Series([np.mean(r_squared), np.std(r_squared)], index=["r2", "r2_std"])
def summary(
data,
var_names: Optional[List[str]] = None,
filter_vars=None,
fmt: str = "wide",
kind: str = "all",
round_to=None,
include_circ=None,
stat_funcs=None,
extend=True,
hdi_prob=None,
order="C",
index_origin=None,
skipna=False,
coords: Optional[CoordSpec] = None,
dims: Optional[DimSpec] = None,
credible_interval=None,
) -> Union[pd.DataFrame, xr.Dataset]:
"""Create a data frame with summary statistics.
Parameters
----------
data: obj
Any object that can be converted to an az.InferenceData object
Refer to documentation of az.convert_to_dataset for details
var_names: list
Names of variables to include in summary. Prefix the variables by `~` when you
want to exclude them from the summary: `["~beta"]` instead of `["beta"]` (see
examples below).
filter_vars: {None, "like", "regex"}, optional, default=None
If `None` (default), interpret var_names as the real variables names. If "like",
interpret var_names as substrings of the real variables names. If "regex",
interpret var_names as regular expressions on the real variables names. A la
`pandas.filter`.
fmt: {'wide', 'long', 'xarray'}
Return format is either pandas.DataFrame {'wide', 'long'} or xarray.Dataset {'xarray'}.
kind: {'all', 'stats', 'diagnostics'}
Whether to include the `stats`: `mean`, `sd`, `hdi_3%`, `hdi_97%`, or the `diagnostics`:
`mcse_mean`, `mcse_sd`, `ess_bulk`, `ess_tail`, and `r_hat`. Default to include `all` of
them.
round_to: int
Number of decimals used to round results. Defaults to 2. Use "none" to return raw numbers.
include_circ: bool
Whether to include circular statistics
stat_funcs: dict
A list of functions or a dict of functions with function names as keys used to calculate
statistics. By default, the mean, standard deviation, simulation standard error, and
highest posterior density intervals are included.
The functions will be given one argument, the samples for a variable as an nD array,
The functions should be in the style of a ufunc and return a single number. For example,
`np.mean`, or `scipy.stats.var` would both work.
extend: boolean
If True, use the statistics returned by ``stat_funcs`` in addition to, rather than in place
of, the default statistics. This is only meaningful when ``stat_funcs`` is not None.
hdi_prob: float, optional
HDI interval to compute. Defaults to 0.94. This is only meaningful when ``stat_funcs`` is
None.
order: {"C", "F"}
If fmt is "wide", use either C or F unpacking order. Defaults to C.
index_origin: int
If fmt is "wide, select n-based indexing for multivariate parameters.
Defaults to rcParam data.index.origin, which is 0.
skipna: bool
If true ignores nan values when computing the summary statistics, it does not affect the
behaviour of the functions passed to ``stat_funcs``. Defaults to false.
coords: Dict[str, List[Any]], optional
Coordinates specification to be used if the ``fmt`` is ``'xarray'``.
dims: Dict[str, List[str]], optional
Dimensions specification for the variables to be used if the ``fmt`` is ``'xarray'``.
credible_interval: float, optional
deprecated: Please see hdi_prob
Returns
-------
pandas.DataFrame or xarray.Dataset
Return type dicated by `fmt` argument.
Return value will contain summary statistics for each variable. Default statistics are:
`mean`, `sd`, `hdi_3%`, `hdi_97%`, `mcse_mean`, `mcse_sd`, `ess_bulk`, `ess_tail`, and
`r_hat`.
`r_hat` is only computed for traces with 2 or more chains.
Examples
--------
.. ipython::
In [1]: import arviz as az
...: data = az.load_arviz_data("centered_eight")
...: az.summary(data, var_names=["mu", "tau"])
You can use `filter_vars` to select variables without having to specify all the exact
names. Use `filter_vars="like"` to select based on partial naming:
.. ipython::
In [1]: az.summary(data, var_names=["the"], filter_vars="like")
Use `filter_vars="regex"` to select based on regular expressions, and prefix the variables
you want to exclude by `~`. Here, we exclude from the summary all the variables
starting with the letter t:
.. ipython::
In [1]: az.summary(data, var_names=["~^t"], filter_vars="regex")
Other statistics can be calculated by passing a list of functions
or a dictionary with key, function pairs.
.. ipython::
In [1]: import numpy as np
...: def median_sd(x):
...: median = np.percentile(x, 50)
...: sd = np.sqrt(np.mean((x-median)**2))
...: return sd
...:
...: func_dict = {
...: "std": np.std,
...: "median_std": median_sd,
...: "5%": lambda x: np.percentile(x, 5),
...: "median": lambda x: np.percentile(x, 50),
...: "95%": lambda x: np.percentile(x, 95),
...: }
...: az.summary(
...: data,
...: var_names=["mu", "tau"],
...: stat_funcs=func_dict,
...: extend=False
...: )
"""
if credible_interval:
hdi_prob = credible_interval_warning(hdi_prob, hdi_prob)
extra_args = {} # type: Dict[str, Any]
if coords is not None:
extra_args["coords"] = coords
if dims is not None:
extra_args["dims"] = dims
if index_origin is None:
index_origin = rcParams["data.index_origin"]
if hdi_prob is None:
hdi_prob = rcParams["stats.hdi_prob"]
else:
if not 1 >= hdi_prob > 0:
raise ValueError("The value of hdi_prob should be in the interval (0, 1]")
posterior = convert_to_dataset(data, group="posterior", **extra_args)
var_names = _var_names(var_names, posterior, filter_vars)
posterior = posterior if var_names is None else posterior[var_names]
fmt_group = ("wide", "long", "xarray")
if not isinstance(fmt, str) or (fmt.lower() not in fmt_group):
raise TypeError("Invalid format: '{}'. Formatting options are: {}".format(fmt, fmt_group))
unpack_order_group = ("C", "F")
if not isinstance(order, str) or (order.upper() not in unpack_order_group):
raise TypeError(
"Invalid order: '{}'. Unpacking options are: {}".format(order, unpack_order_group)
)
alpha = 1 - hdi_prob
extra_metrics = []
extra_metric_names = []
if stat_funcs is not None:
if isinstance(stat_funcs, dict):
for stat_func_name, stat_func in stat_funcs.items():
extra_metrics.append(
xr.apply_ufunc(
_make_ufunc(stat_func), posterior, input_core_dims=(("chain", "draw"),)
)
)
extra_metric_names.append(stat_func_name)
else:
for stat_func in stat_funcs:
extra_metrics.append(
xr.apply_ufunc(
_make_ufunc(stat_func), posterior, input_core_dims=(("chain", "draw"),)
)
)
extra_metric_names.append(stat_func.__name__)
if extend and kind in ["all", "stats"]:
mean = posterior.mean(dim=("chain", "draw"), skipna=skipna)
sd = posterior.std(dim=("chain", "draw"), ddof=1, skipna=skipna)
hdi_post = hdi(posterior, hdi_prob=hdi_prob, multimodal=False, skipna=skipna)
hdi_lower = hdi_post.sel(hdi="lower", drop=True)
hdi_higher = hdi_post.sel(hdi="higher", drop=True)
if include_circ:
nan_policy = "omit" if skipna else "propagate"
circ_mean = xr.apply_ufunc(
_make_ufunc(st.circmean),
posterior,
kwargs=dict(high=np.pi, low=-np.pi, nan_policy=nan_policy),
input_core_dims=(("chain", "draw"),),
)
_numba_flag = Numba.numba_flag
func = None
if _numba_flag:
func = _circular_standard_deviation
kwargs_circ_std = dict(high=np.pi, low=-np.pi, skipna=skipna)
else:
func = st.circstd
kwargs_circ_std = dict(high=np.pi, low=-np.pi, nan_policy=nan_policy)
circ_sd = xr.apply_ufunc(
_make_ufunc(func),
posterior,
kwargs=kwargs_circ_std,
input_core_dims=(("chain", "draw"),),
)
circ_mcse = xr.apply_ufunc(
_make_ufunc(_mc_error),
posterior,
kwargs=dict(circular=True),
input_core_dims=(("chain", "draw"),),
)
circ_hdi = hdi(posterior, hdi_prob=hdi_prob, circular=True, skipna=skipna)
circ_hdi_lower = circ_hdi.sel(hdi="lower", drop=True)
circ_hdi_higher = circ_hdi.sel(hdi="higher", drop=True)
if kind in ["all", "diagnostics"]:
mcse_mean, mcse_sd, ess_mean, ess_sd, ess_bulk, ess_tail, r_hat = xr.apply_ufunc(
_make_ufunc(_multichain_statistics, n_output=7, ravel=False),
posterior,
input_core_dims=(("chain", "draw"),),
output_core_dims=tuple([] for _ in range(7)),
)
# Combine metrics
metrics = []
metric_names = []
if extend:
metrics_names_ = (
"mean",
"sd",
"hdi_{:g}%".format(100 * alpha / 2),
"hdi_{:g}%".format(100 * (1 - alpha / 2)),
"mcse_mean",
"mcse_sd",
"ess_mean",
"ess_sd",
"ess_bulk",
"ess_tail",
"r_hat",
)
if kind == "all":
metrics_ = (
mean,
sd,
hdi_lower,
hdi_higher,
mcse_mean,
mcse_sd,
ess_mean,
ess_sd,
ess_bulk,
ess_tail,
r_hat,
)
elif kind == "stats":
metrics_ = (mean, sd, hdi_lower, hdi_higher)
metrics_names_ = metrics_names_[:4]
elif kind == "diagnostics":
metrics_ = (mcse_mean, mcse_sd, ess_mean, ess_sd, ess_bulk, ess_tail, r_hat)
metrics_names_ = metrics_names_[4:]
metrics.extend(metrics_)
metric_names.extend(metrics_names_)
if include_circ:
metrics.extend((circ_mean, circ_sd, circ_hdi_lower, circ_hdi_higher, circ_mcse))
metric_names.extend(
(
"circular_mean",
"circular_sd",
"circular_hdi_{:g}%".format(100 * alpha / 2),
"circular_hdi_{:g}%".format(100 * (1 - alpha / 2)),
"circular_mcse",
)
)
metrics.extend(extra_metrics)
metric_names.extend(extra_metric_names)
joined = (
xr.concat(metrics, dim="metric").assign_coords(metric=metric_names).reset_coords(drop=True)
)
if fmt.lower() == "wide":
dfs = []
for var_name, values in joined.data_vars.items():
if len(values.shape[1:]):
metric = list(values.metric.values)
data_dict = OrderedDict()
for idx in np.ndindex(values.shape[1:] if order == "C" else values.shape[1:][::-1]):
if order == "F":
idx = tuple(idx[::-1])
ser = pd.Series(values[(Ellipsis, *idx)].values, index=metric)
key_index = ",".join(map(str, (i + index_origin for i in idx)))
key = "{}[{}]".format(var_name, key_index)
data_dict[key] = ser
df = pd.DataFrame.from_dict(data_dict, orient="index")
df = df.loc[list(data_dict.keys())]
else:
df = values.to_dataframe()
df.index = list(df.index)
df = df.T
dfs.append(df)
summary_df = pd.concat(dfs, sort=False)
elif fmt.lower() == "long":
df = joined.to_dataframe().reset_index().set_index("metric")
df.index = list(df.index)
summary_df = df
else:
# format is 'xarray'
summary_df = joined
if (round_to is not None) and (round_to not in ("None", "none")):
summary_df = summary_df.round(round_to)
elif round_to not in ("None", "none") and (fmt.lower() in ("long", "wide")):
# Don't round xarray object by default (even with "none")
decimals = {
col: 3
if col not in {"ess_mean", "ess_sd", "ess_bulk", "ess_tail", "r_hat"}
else 2
if col == "r_hat"
else 0
for col in summary_df.columns
}
summary_df = summary_df.round(decimals)
return summary_df
def waic(data, pointwise=None, var_name=None, scale=None):
"""Compute the widely applicable information criterion.
Estimates the expected log pointwise predictive density (elpd) using WAIC. Also calculates the
WAIC's standard error and the effective number of parameters.
Read more theory here https://arxiv.org/abs/1507.04544 and here https://arxiv.org/abs/1004.2316
Parameters
----------
data: obj
Any object that can be converted to an az.InferenceData object. Refer to documentation of
``az.convert_to_inference_data`` for details
pointwise: bool
If True the pointwise predictive accuracy will be returned. Defaults to
``stats.ic_pointwise`` rcParam.
var_name : str, optional
The name of the variable in log_likelihood groups storing the pointwise log
likelihood data to use for waic computation.
scale: str
Output scale for WAIC. Available options are:
- `log` : (default) log-score
- `negative_log` : -1 * log-score
- `deviance` : -2 * log-score
A higher log-score (or a lower deviance or negative log_score) indicates a model with
better predictive accuracy.
Returns
-------
ELPDData object (inherits from panda.Series) with the following row/attributes:
waic: approximated expected log pointwise predictive density (elpd)
waic_se: standard error of waic
p_waic: effective number parameters
var_warn: bool
True if posterior variance of the log predictive densities exceeds 0.4
waic_i: xarray.DataArray with the pointwise predictive accuracy, only if pointwise=True
waic_scale: scale of the reported waic results
The returned object has a custom print method that overrides pd.Series method.
Examples
--------
Calculate WAIC of a model:
.. ipython::
In [1]: import arviz as az
...: data = az.load_arviz_data("centered_eight")
...: az.waic(data)
Calculate WAIC of a model and return the pointwise values:
.. ipython::
In [2]: data_waic = az.waic(data, pointwise=True)
...: data_waic.waic_i
"""
inference_data = convert_to_inference_data(data)
log_likelihood = _get_log_likelihood(inference_data, var_name=var_name)
scale = rcParams["stats.ic_scale"] if scale is None else scale.lower()
pointwise = rcParams["stats.ic_pointwise"] if pointwise is None else pointwise
if scale == "deviance":
scale_value = -2
elif scale == "log":
scale_value = 1
elif scale == "negative_log":
scale_value = -1
else:
raise TypeError('Valid scale values are "deviance", "log", "negative_log"')
log_likelihood = log_likelihood.stack(sample=("chain", "draw"))
shape = log_likelihood.shape
n_samples = shape[-1]
n_data_points = np.product(shape[:-1])
ufunc_kwargs = {"n_dims": 1, "ravel": False}
kwargs = {"input_core_dims": [["sample"]]}
lppd_i = _wrap_xarray_ufunc(
_logsumexp,
log_likelihood,
func_kwargs={"b_inv": n_samples},
ufunc_kwargs=ufunc_kwargs,
**kwargs,
)
vars_lpd = log_likelihood.var(dim="sample")
warn_mg = False
if np.any(vars_lpd > 0.4):
warnings.warn(
(
"For one or more samples the posterior variance of the log predictive "
"densities exceeds 0.4. This could be indication of WAIC starting to fail. \n"
"See http://arxiv.org/abs/1507.04544 for details"
)
)
warn_mg = True
waic_i = scale_value * (lppd_i - vars_lpd)
waic_se = (n_data_points * np.var(waic_i.values)) ** 0.5
waic_sum = np.sum(waic_i.values)
p_waic = np.sum(vars_lpd.values)
if pointwise:
if np.equal(waic_sum, waic_i).all(): # pylint: disable=no-member
warnings.warn(
"""The point-wise WAIC is the same with the sum WAIC, please double check
the Observed RV in your model to make sure it returns element-wise logp.
"""
)
return ELPDData(
data=[
waic_sum,
waic_se,
p_waic,
n_samples,
n_data_points,
warn_mg,
waic_i.rename("waic_i"),
scale,
],
index=[
"waic",
"waic_se",
"p_waic",
"n_samples",
"n_data_points",
"warning",
"waic_i",
"waic_scale",
],
)
else:
return ELPDData(
data=[waic_sum, waic_se, p_waic, n_samples, n_data_points, warn_mg, scale],
index=[
"waic",
"waic_se",
"p_waic",
"n_samples",
"n_data_points",
"warning",
"waic_scale",
],
)
def loo_pit(idata=None, *, y=None, y_hat=None, log_weights=None):
"""Compute leave one out (PSIS-LOO) probability integral transform (PIT) values.
Parameters
----------
idata: InferenceData
InferenceData object.
y: array, DataArray or str
Observed data. If str, idata must be present and contain the observed data group
y_hat: array, DataArray or str
Posterior predictive samples for ``y``. It must have the same shape as y plus an
extra dimension at the end of size n_samples (chains and draws stacked). If str or
None, idata must contain the posterior predictive group. If None, y_hat is taken
equal to y, thus, y must be str too.
log_weights: array or DataArray
Smoothed log_weights. It must have the same shape as ``y_hat``
Returns
-------
loo_pit: array or DataArray
Value of the LOO-PIT at each observed data point.
Examples
--------
Calculate LOO-PIT values using as test quantity the observed values themselves.
.. ipython::
In [1]: import arviz as az
...: data = az.load_arviz_data("centered_eight")
...: az.loo_pit(idata=data, y="obs")
Calculate LOO-PIT values using as test quantity the square of the difference between
each observation and `mu`. Both ``y`` and ``y_hat`` inputs will be array-like,
but ``idata`` will still be passed in order to calculate the ``log_weights`` from
there.
.. ipython::
In [1]: T = data.observed_data.obs - data.posterior.mu.median(dim=("chain", "draw"))
...: T_hat = data.posterior_predictive.obs - data.posterior.mu
...: T_hat = T_hat.stack(sample=("chain", "draw"))
...: az.loo_pit(idata=data, y=T**2, y_hat=T_hat**2)
"""
y_str = ""
if idata is not None and not isinstance(idata, InferenceData):
raise ValueError("idata must be of type InferenceData or None")
if idata is None:
if not all(isinstance(arg, (np.ndarray, xr.DataArray)) for arg in (y, y_hat, log_weights)):
raise ValueError(
"all 3 y, y_hat and log_weights must be array or DataArray when idata is None "
"but they are of types {}".format([type(arg) for arg in (y, y_hat, log_weights)])
)
else:
if y_hat is None and isinstance(y, str):
y_hat = y
elif y_hat is None:
raise ValueError("y_hat cannot be None if y is not a str")
if isinstance(y, str):
y_str = y
y = idata.observed_data[y].values
elif not isinstance(y, (np.ndarray, xr.DataArray)):
raise ValueError("y must be of types array, DataArray or str, not {}".format(type(y)))
if isinstance(y_hat, str):
y_hat = idata.posterior_predictive[y_hat].stack(sample=("chain", "draw")).values
elif not isinstance(y_hat, (np.ndarray, xr.DataArray)):
raise ValueError(
"y_hat must be of types array, DataArray or str, not {}".format(type(y_hat))
)
if log_weights is None:
if y_str:
try:
log_likelihood = _get_log_likelihood(idata, var_name=y)
except TypeError:
log_likelihood = _get_log_likelihood(idata)
else:
log_likelihood = _get_log_likelihood(idata)
log_likelihood = log_likelihood.stack(sample=("chain", "draw"))
posterior = convert_to_dataset(idata, group="posterior")
n_chains = len(posterior.chain)
n_samples = len(log_likelihood.sample)
ess_p = ess(posterior, method="mean")
# this mean is over all data variables
reff = (
(np.hstack([ess_p[v].values.flatten() for v in ess_p.data_vars]).mean() / n_samples)
if n_chains > 1
else 1
)
log_weights = psislw(-log_likelihood, reff=reff)[0].values
elif not isinstance(log_weights, (np.ndarray, xr.DataArray)):
raise ValueError(
"log_weights must be None or of types array or DataArray, not {}".format(
type(log_weights)
)
)
if len(y.shape) + 1 != len(y_hat.shape):
raise ValueError(
"y_hat must have 1 more dimension than y, but y_hat has {} dims and y has "
"{} dims".format(len(y.shape), len(y_hat.shape))
)
if y.shape != y_hat.shape[:-1]:
raise ValueError(
"y has shape: {} which should be equal to y_hat shape (omitting the last "
"dimension): {}".format(y.shape, y_hat.shape)
)
if y_hat.shape != log_weights.shape:
raise ValueError(
"y_hat and log_weights must have the same shape but have shapes {} and {}".format(
y_hat.shape, log_weights.shape
)
)
kwargs = {
"input_core_dims": [[], ["sample"], ["sample"]],
"output_core_dims": [[]],
"join": "left",
}
ufunc_kwargs = {"n_dims": 1}
return _wrap_xarray_ufunc(_loo_pit, y, y_hat, log_weights, ufunc_kwargs=ufunc_kwargs, **kwargs)
def _loo_pit(y, y_hat, log_weights):
"""Compute LOO-PIT values."""
sel = y_hat <= y
if np.sum(sel) > 0:
value = np.exp(_logsumexp(log_weights[sel]))
return min(1, value)
else:
return 0
def apply_test_function(
idata,
func,
group="both",
var_names=None,
pointwise=False,
out_data_shape=None,
out_pp_shape=None,
out_name_data="T",
out_name_pp=None,
func_args=None,
func_kwargs=None,
ufunc_kwargs=None,
wrap_data_kwargs=None,
wrap_pp_kwargs=None,
inplace=True,
overwrite=None,
):
"""Apply a Bayesian test function to an InferenceData object.
Parameters
----------
idata: InferenceData
InferenceData object on which to apply the test function. This function will add
new variables to the InferenceData object to store the result without modifying the
existing ones.
func: callable
Callable that calculates the test function. It must have the following call signature
``func(y, theta, *args, **kwargs)`` (where ``y`` is the observed data or posterior
predictive and ``theta`` the model parameters) even if not all the arguments are
used.
group: str, optional
Group on which to apply the test function. Can be observed_data, posterior_predictive
or both.
var_names: dict group -> var_names, optional
Mapping from group name to the variables to be passed to func. It can be a dict of
strings or lists of strings. There is also the option of using ``both`` as key,
in which case, the same variables are used in observed data and posterior predictive
groups
pointwise: bool, optional
If True, apply the test function to each observation and sample, otherwise, apply
test function to each sample.
out_data_shape, out_pp_shape: tuple, optional
Output shape of the test function applied to the observed/posterior predictive data.
If None, the default depends on the value of pointwise.
out_name_data, out_name_pp: str, optional
Name of the variables to add to the observed_data and posterior_predictive datasets
respectively. ``out_name_pp`` can be ``None``, in which case will be taken equal to
``out_name_data``.
func_args: sequence, optional
Passed as is to ``func``
func_kwargs: mapping, optional
Passed as is to ``func``
wrap_data_kwargs, wrap_pp_kwargs: mapping, optional
kwargs passed to ``az.stats.wrap_xarray_ufunc``. By default, some suitable input_core_dims
are used.
inplace: bool, optional
If True, add the variables inplace, othewise, return a copy of idata with the variables
added.
overwrite: bool, optional
Overwrite data in case ``out_name_data`` or ``out_name_pp`` are already variables in
dataset. If ``None`` it will be the opposite of inplace.
Returns
-------
idata: InferenceData
Output InferenceData object. If ``inplace=True``, it is the same input object modified
inplace.
Notes
-----
This function is provided for convenience to wrap scalar or functions working on low
dims to inference data object. It is not optimized to be faster nor as fast as vectorized
computations.
Examples
--------
Use ``apply_test_function`` to wrap ``np.min`` for illustration purposes. And plot the
results.
.. plot::
:context: close-figs
>>> import arviz as az
>>> idata = az.load_arviz_data("centered_eight")
>>> az.apply_test_function(idata, lambda y, theta: np.min(y))
>>> T = np.asscalar(idata.observed_data.T)
>>> az.plot_posterior(idata, var_names=["T"], group="posterior_predictive", ref_val=T)
"""
out = idata if inplace else deepcopy(idata)
valid_groups = ("observed_data", "posterior_predictive", "both")
if group not in valid_groups:
raise ValueError(
"Invalid group argument. Must be one of {} not {}.".format(valid_groups, group)
)
if overwrite is None:
overwrite = not inplace
if out_name_pp is None:
out_name_pp = out_name_data
if func_args is None:
func_args = tuple()
if func_kwargs is None:
func_kwargs = {}
if ufunc_kwargs is None:
ufunc_kwargs = {}
ufunc_kwargs.setdefault("check_shape", False)
ufunc_kwargs.setdefault("ravel", False)
if wrap_data_kwargs is None:
wrap_data_kwargs = {}
if wrap_pp_kwargs is None:
wrap_pp_kwargs = {}
if var_names is None:
var_names = {}
both_var_names = var_names.pop("both", None)
var_names.setdefault("posterior", list(out.posterior.data_vars))
in_posterior = out.posterior[var_names["posterior"]]
if isinstance(in_posterior, xr.Dataset):
in_posterior = in_posterior.to_array().squeeze()
groups = ("posterior_predictive", "observed_data") if group == "both" else [group]
for grp in groups:
out_group_shape = out_data_shape if grp == "observed_data" else out_pp_shape
out_name_group = out_name_data if grp == "observed_data" else out_name_pp
wrap_group_kwargs = wrap_data_kwargs if grp == "observed_data" else wrap_pp_kwargs
if not hasattr(out, grp):
raise ValueError("InferenceData object must have {} group".format(grp))
if not overwrite and out_name_group in getattr(out, grp).data_vars:
raise ValueError(
"Should overwrite: {} variable present in group {}, but overwrite is False".format(
out_name_group, grp
)
)
var_names.setdefault(
grp, list(getattr(out, grp).data_vars) if both_var_names is None else both_var_names
)
in_group = getattr(out, grp)[var_names[grp]]
if isinstance(in_group, xr.Dataset):
in_group = in_group.to_array(dim="{}_var".format(grp)).squeeze()
if pointwise:
out_group_shape = in_group.shape if out_group_shape is None else out_group_shape
elif grp == "observed_data":
out_group_shape = () if out_group_shape is None else out_group_shape
elif grp == "posterior_predictive":
out_group_shape = in_group.shape[:2] if out_group_shape is None else out_group_shape
loop_dims = in_group.dims[: len(out_group_shape)]
wrap_group_kwargs.setdefault(
"input_core_dims",
[
[dim for dim in dataset.dims if dim not in loop_dims]
for dataset in [in_group, in_posterior]
],
)
func_kwargs["out"] = np.empty(out_group_shape)
out_group = getattr(out, grp)
try:
out_group[out_name_group] = _wrap_xarray_ufunc(
func,
in_group.values,
in_posterior.values,
func_args=func_args,
func_kwargs=func_kwargs,
ufunc_kwargs=ufunc_kwargs,
**wrap_group_kwargs,
)
except IndexError:
excluded_dims = set(
wrap_group_kwargs["input_core_dims"][0] + wrap_group_kwargs["input_core_dims"][1]
)
out_group[out_name_group] = _wrap_xarray_ufunc(
func,
*xr.broadcast(in_group, in_posterior, exclude=excluded_dims),
func_args=func_args,
func_kwargs=func_kwargs,
ufunc_kwargs=ufunc_kwargs,
**wrap_group_kwargs,
)
setattr(out, grp, out_group)
return out
