# InferenceData schema specification#

The InferenceData schema approach defines a data structure compatible with NetCDF. Its purpose is to serve the following three goals:

1. Usefulness in the analysis of Bayesian inference results.

2. Reproducibility of Bayesian inference analysis.

3. Interoperability between different inference backends and programming languages.

Currently there are two beta implementations of this design:

## Terminology#

The terminology used in this specification is based on xarray’s terminology, however, no xarray knowledge is assumed in this description. There are also some extensions particular to the InferenceData case.

• Variable: NetCDF-like variables are multidimensional labeled arrays representing a single quantity. Variables and their dimensions must be named. They can also have attributes describing it. Relevant terms related to InferenceData variables are following:

• variable_name

• values (its data)

• dimensions

• coordinates

• attributes

• Dimension: The dimensions of an object are its named axes. A variable containing 3D data can have dimensions [chain, draw, dim0], i.e., its 0th-dimension is chain, its 1st-dimension is draw, and so on. Every dimension present in an InferenceData variable must share names with a coordinate. Given that dimensions must be named, dimension and dimension name are used equivalents.

• Coordinate: A named array that labels a dimension. A coordinate named chain with values [0, 1, 2, 3] would label the chain dimension. Coordinate names and values can be loosely thought of as labels and tick labels along a dimension, respectively.

• Attribute: An ordered dictionary that can store arbitrary metadata.

• Group: Dataset containing one or several variables with a conceptual link between them. Variables inside a group will generally share some dimensions too. For example, the posterior group contains a representation of the posterior distribution conditioned on the observations in the observed_data group.

• Matching samples: Two variables (or groups) will be called to have matching samples if they are generated with the same set of samples. Therefore, they will share dimensions and coordinates corresponding to the sampling process. Sample dimensions (generally (chain, draw)) are the ones introduced by the sampling process.

• Matching variables: Two groups with matching variables are groups that conceptually share variables, variable dimensions and coordinates of the variable dimensions but do not necessarily share variable names nor sample dimensions. Variable dimensions are the ones intrinsic to the data and model as opposed to sample dimensions which are the ones relative to the sampling process. When talking about specific variables, this same idea is expressed as one variable being the counterpart of the other.

## Current design#

InferenceData stores all quantities that are relevant to fulfilling its goals in different groups. Different groups generally distinguish conceptually different quantities in Bayesian inference, however, convenience in creation and usage of InferenceData objects also plays a role. In general, each quantity (such as posterior distribution or observed data) will be represented by several multidimensional labeled variables.

### Rules#

Following are a few rules which should be followed:

• Each group should have one entry per variable and each variable should be named.

• When relevant, the first two dimensions of each variable should be the sample identifier (chain, draw).

• For groups like observed_data or constant_data, the two initial dimensions(chain, draw) are omitted.

• Dimensions must be named and share name with a coordinate specifying the index values, called coordinate values.

• Coordinate values can be repeated and should not necessarily be numerical values.

• Variables must not share names with dimensions.

• Moreover, each group contains the following attributes:

• created_at: the date of creation of the group.

• inference_library: the library used to run the inference.

• inference_library_version: version of the inference library used.

### Relations#

InferenceData data objects contain any combination of the groups described below. There are also some relations (detailed below) between the variables and dimensions of different groups. Hence, whenever related groups are present they should comply with these relations.

#### posterior#

Samples from the posterior distribution p(theta|y).

#### sample_stats#

Information and diagnostics for each posterior sample, provided by the inference backend. It may vary depending on the algorithm used by the backend (i.e. an affine invariant sampler has no energy associated). Therefore none of these parameters should be assumed to be present in the sample_stats group. The convention below serves to ensure that if a variable is present with one of these names it will correspond to the definition given in front of it.

The name convention used for sample_stats variables is the following:

• lp: The joint log posterior density for the model (up to an additive constant).

• acceptance_rate: The average acceptance probabilities of all possible samples in the proposed tree.

• step_size: The current integration step size.

• step_size_nom: The nominal integration step size. The step_size may differ from this for example, if the step size is jittered. It should only be present if step_size is also present and it varies between samples (i.e. step size is jittered).

• tree_depth: The number of tree doublings in the balanced binary tree.

• n_steps: The number of leapfrog steps computed. It is related to tree_depth with n_steps <= 2^tree_dept.

• diverging: (boolean) Indicates the presence of leapfrog transitions with large energy deviation from starting and subsequent termination of the trajectory. “large” is defined as max_energy_error going over a threshold.

• energy: The value of the Hamiltonian energy for the accepted proposal (up to an additive constant).

• energy_error: The difference in the Hamiltonian energy between the initial point and the accepted proposal.

• max_energy_error: The maximum absolute difference in Hamiltonian energy between the initial point and all possible samples in the proposed tree.

• int_time: The total integration time (static HMC sampler)

#### log_likelihood#

Pointwise log likelihood data. Samples should match with posterior ones and its variables should match observed_data variables. The observed_data counterpart variable may have a different name. Moreover, some cases such as a multivariate normal may require some dimensions or coordinates to be different.

#### posterior_predictive#

Posterior predictive samples p(y|y) corresponding to the posterior predictive distribution evaluated at the observed_data. Samples should match with posterior ones and its variables should match observed_data variables. The observed_data counterpart variable may have a different name.

#### observed_data#

Observed data on which the posterior is conditional. It should only contain data which is modeled as a random variable. Each variable should have a counterpart in posterior_predictive, however, the posterior_predictive counterpart variable may have a different name.

#### constant_data#

Model constants, data included in the model which is not modeled as a random variable. It should be the data used to generate samples in all the groups except the predictions groups.

#### prior#

Samples from the prior distribution p(theta). Samples do not need to match posterior samples. However, this group will still follow the convention on chain and draw as first dimensions. It should have matching variables with the posterior group.

#### sample_stats_prior#

Information and diagnostics for the samples in the prior group, provided by the inference backend. It may vary depending on the algorithm used by the backend. Variable names follow the same convention defined in sample_stats.

#### prior_predictive#

Samples from the prior predictive distribution. Samples should match prior samples and each variable should have a counterpart in posterior_predictive/observed_data.

#### predictions#

Out of sample posterior predictive samples p(y’|y). Samples should match posterior samples. Its variables should have a counterpart in posterior_predictive. However, variables in predictions and their counterpart in posterior_predictive can have different coordinate values.

#### predictions_constant_data#

Model constants used to get the predictions samples. Its variables should have a counterpart in constant_data. However, variables in predictions_constant_data and their counterpart in constant_data can have different coordinate values.

## Planned features#

The InferenceData structure is still evolving, with some feature being currently developed. This section aims to describe the roadmap of the specification.

### Sampler parameters#

Parameters of the sampling algorithm and sampling backend to be used for analysis reproducibility.

## Examples#

In order to clarify the definitions above, an example of InferenceData generation for a 1D linear regression is available in several programming languages and probabilistic programming frameworks. This particular inference task has been chosen because it is widely well known while still being useful and it also allows to populate all the fields in the InferenceData object.