Binary Treatment (T)

Binary Treatment CausalGPSLC with Covariates (X) and Latent Confounders (U)

No Covariates (no X), Binary Treatment CausalGPSLC

No Covariates (no X), Continuous CausalGPSLC

No latent confounders (no U), Binary Treatment CausalGPSLC

No covariates (no X), no latent confounders (no U) for Binary Treatment CausalGPSLC

No Covariates (no X), No Latent Confounders (no U), Continuous CausalGPSLC

No Latent Confounders (no U), Continuous CausalGPSLC

Continous CausalGPSLC, with Latent Confounders (U) and Covariates (X)

Confounders (U)

Latent confounders that CausalGPSLC performs inference over.

Either 1D or 2D matrices of Float64 values.

Continuous Treatment (T)

Intervention (doT)

ReshapeableMatrix

Matrix that can be reshaped.

SupportedRBFData

Viable inputs to the rbfKernelLog function that are nested lists.

SupportedRBFLengthscale

Viable inputs to the rbfKernelLog function as kernel lengthscales.

SupportedRBFMatrix

Viable inputs to the rbfKernelLog function in linear algebra datatypes.

SupportedRBFVector

Viable inputs to the rbfKernelLog function in linear algebra datatypes.

Treatment (T)

Is made up of BinaryTreatment which is an alias for Vector{Bool} and ContinuousTreatment which is an alias for Vector{Float64}. These types support other vector types to afford compatibility with internal libraries.

Gen function to generate binary treatment (T) from real value

Sample binary treatment from prior (T)

Sample Binary T from confounders (U)

Sample Binary T from confounders (U) and covariates (X)

Sample Binary T from covariates (X)

Gen function to generate lengthscale parameter for GP

Gen function to generate noise from inv_gamma

Sample continuous treatment from prior (T)

Sample Continuous T from confounders (U)

Sample T from confounders (U) and covariates (X)

Sample Binary T from covariates (X)

Gen function to generate scale parameter for GP

Gen function to generate latent confounders (U) from mvnormal distribution

Sample U

Gen function to generate covariates (n,X_k) from mvnormal distribution

Sample covariates from prior (X)

Sample X from U

Sample Y from only treatment (T)

Sample Y from confounders (U) and treatment (T)

Sample Y from confounders (U), covariates (X), and treatment (T)

Sample Y from covariates (X) and treatment (T)

Treatment to outcome lengthscale

Latent confounders to treatment and outcome lengthscale

Latent confounders to treatment and outcome lengthscale when nX is known

Covariates to treatment and outcome lengthscale

paramProposal(trace, variance, addr)

Like a Gaussian drift, we match the moments of our proposal with the previous noise sample with a fixed variance. See https://arxiv.org/pdf/1605.01019.pdf.

Sample noise from prior for treatment (T)

Generate noise terms from noise prior

Sample noise for prior from confounders (U)

Sample noise from prior for covariates (X)

Sample noise from prior for outcome (Y)

Sample kernel scale from prior for treatment (T)

Generate kernel scales from prior for covariates (X)

Sample kernel scale from prior for outcome (Y)

SigmaU

structured prior for U.

Covariates (X) Observed confounders and covariates.

Matrix{Float64} is the only valid structure for covariates

GPSLCObject

This is the struct in CausalGPSLC.jl that contains the data, hyperparamters, prior parameters, and posterior samples. It provides the primary interfaces to abstract the internals of CausalGPSLC away from the higher-order functions like sampleITE, sampleSATE, and predictCounterfactualEffects.

Returned by gpslc

Constructor for GPSLCObject that samples from the posterior before constructing the GPSLCObject.

GPSLCObject(hyperparams, priorparams, SigmaU, obj, X, T, Y)
GPSLCObject(hyperparams, priorparams, SigmaU, obj, nothing, T, Y)
GPSLCObject(hyperparams, priorparams, nothing, nothing, X, T, Y)
GPSLCObject(hyperparams, priorparams, nothing, nothing, nothing, T, Y)

Full Model or model with no observed Covariates

No Confounders

No Confounders, No Covariates

HyperParameters

Define the high-level attributes of the inference procedure. More information on each of the attributes can be found in getHyperParameters.

Object Labels for instances (obj)

Optional for CausalGPSLC, but per publication it improves performance.

Outcome (Y)

The outcome for the series of Gaussian Process predictions is a Vector{Float64}. Currently only continuous values are supported as outcomes for input data.

PriorParameters

A dictionary of shapes and scales for various Inverse Gamma distributions used as priors for kernel parameters and other parameters. More information on each of the attributes can be found in getPriorParameters.

SupportedCovarianceMatrix

Viable inputs to the processCov function.

ITEDistributions(g, doT)

Collect MeanITEs and CovITEs from the posterior with conditionalITE.

ITEsamples(MeanITEs, CovITEs, nSamplesPerMixture)

Individual Treatment Effect Samples

Returns nMixtures * nSamplesPerMixture outcome (Y) samples for each individual [nMixtures * nSamplesPerMixture, n] where nMixtures is the number of posterior samples (nOuter)

Binary Treatment Full Model

Binary Treatment with No Confounders

Continuous Treatment Full Model

Continuous Treatment No Confounders

nESInner is not used to sample anything via elliptical slice sampling It is required here for compatibility with the binary treatment version which uses ES to learn hyperparameters for the support of binary variables which are not usually supported by Gaussian processes.

Binary Treatment No Covariates

Binary Treatment No Confounders No Covariates

Continuous Treatment No Covariates

Continuous Treatment No Confounders No Covariates

SATEDistributions

Collect SATE Mean and Variance corresponding to each posterior sample.

SATEsamples(MeanSATEs, VarSATEs, nSamplesPerMixture)

Collect Sample Average Treatment Effect corresponding to each posterior sample.

Returns a vector of nSamplesPerMixture samples for each posterior sample's SATE distribution parameters.

conditionalITE(g, psindex, doT)

Wrapper for conditionalITE that extracts parameters from g::GPSLCObject at posterior sample psindex and applies intervention doT.

conditionalITE(nothing, nothing, tyLS, yNoise, yScale, nothing, nothing, T, Y, doT)
conditionalITE(nothing, xyLS, tyLS, yNoise, yScale, nothing, X, T, Y, doT)
conditionalITE(uyLS, nothing, tyLS, yNoise, yScale, U, nothing, T, Y, doT)
conditionalITE(uyLS, xyLS, tyLS, yNoise, yScale, U, X, T, Y, doT)

Conditional Individual Treatment Estimation

conditionalITE takes in parameters (presumably from posterior inference) as well as the observed and inferred data to produce individual treatment effects.

Params:

• uyLS: (optional) Kernel lengthscale for latent confounders to outcome
• xyLS: (optional) Kernel lengthscale for covariates to outcome
• tyLS: Kernel lengthscale for treatment to outcome
• yNoise: Gaussian noise for outcome prediction
• yScale: Gaussian scale for outcome prediction
• U: (optional) Latent confounders
• X: (optional) Covariates
• T: Treatment
• Y: Outcome
• doT: Treatment intervention

Returns:

• MeanITE::Vector{Float64}: The mean value for the n individual treatment effects
• CovITE::Matrix{Float64}: The covariance matrix for the n individual treatment effects.

conditionalSATE

Conditional Sample Average Treatment Effect

Expit is the inverse of the logit function, mapping a Real to [0,1]

extractParameters(g, posteriorSampleIdx)

Get inferred parameters from g's posteriorSampleIdxth posterior sample. Parameters are uyLS, xyLS, tyLS, yNoise, yScale, U, some of which are allowed to be Nothing

generateSigmaU(nIndividualsArray)
generateSigmaU(nIndividualsArray, eps)
generateSigmaU(nIndividualsArray, eps, cov)

Generate block matrix for U given object counts

SigmaU is shorthand for the object structure of the latent confounder

getAddresses(choicemap)

Debugging tool to print all available address keys in choicemap

getHyperParameters()

Returns default values for hyperparameters

• nU = 1: Number of latent confounding variables assumed to be influencing all the instances that belong to one object. Inference will be performed over these values.
• nOuter = 20: Number of posterior samples to draw.
• nMHInner = 5: Number of internal Metropolis-Hastings updates to make per posterior sample.
• nESInner = 5: Number of elliptical-slice sampling updates to make per posterior for latent confounders and binary treatment.
• nBurnIn = 5: Number of posterior samples to discard when making predictions and estimates.
• stepSize = 1: How frequently to use posterior samples (1 being every one after burnIn, higher being every stepSizeth).
• predictionCovarianceNoise=1e-10: Predicting with Gaussian processes requires use of covariance matrices that are Symmetric Positive Definite, and this covariance noise on the diagonal ensures these operations can be performed in a stable and consistent way.

getN(g)

Number of individuals in dataset.

getNU(g)

Number of latent confounders to perform inference over (hyperparameter).

getNX(g)

Number of covariates (and observed confounders) in dataset.

getNumPosteriorSamples(g)

Number of posterior samples that will be used based on hyperparameters.

(total posterior samples - burn in) / step size = nBurnIn:stepSize:nOuter

getPriorParameters()

These are standard values for scale and shape of Inverse Gamma priors over kernel parameters, confounder structure covariance noise, and confounder Gaussian prior covariance.

• uNoiseShape::Float64=4.0: shape of the InvGamma prior over the noise of U
• uNoiseScale::Float64=4.0: scale of the InvGamma prior over the noise of U
• xNoiseShape::Float64=4.0: shape of the InvGamma prior over the noise of X
• xNoiseScale::Float64=4.0: scale of the InvGamma prior over the noise of X
• tNoiseShape::Float64=4.0: shape of the InvGamma prior over the noise of T
• tNoiseScale::Float64=4.0: scale of the InvGamma prior over the noise of T
• yNoiseShape::Float64=4.0: shape of the InvGamma prior over the noise of Y
• yNoiseScale::Float64=4.0: scale of the InvGamma prior over the noise of Y
• xScaleShape::Float64=4.0: shape of the InvGamma prior over kernel scale of X
• xScaleScale::Float64=4.0: scale of the InvGamma prior over kernel scale of X
• tScaleShape::Float64=4.0: shape of the InvGamma prior over kernel scale of T
• tScaleScale::Float64=4.0: scale of the InvGamma prior over kernel scale of T
• yScaleShape::Float64=4.0: shape of the InvGamma prior over kernel scale of Y
• yScaleScale::Float64=4.0: scale of the InvGamma prior over kernel scale of Y
• uxLSShape::Float64=4.0: shape of the InvGamma prior over kernel lengthscale of U and X
• uxLSScale::Float64=4.0: scale of the InvGamma prior over kernel lengthscale of U and X
• utLSShape::Float64=4.0: shape of the InvGamma prior over kernel lengthscale of U and T
• utLSScale::Float64=4.0: scale of the InvGamma prior over kernel lengthscale of U and T
• xtLSShape::Float64=4.0: shape of the InvGamma prior over kernel lengthscale of X and T
• xtLSScale::Float64=4.0: scale of the InvGamma prior over kernel lengthscale of X and T
• uyLSShape::Float64=4.0: shape of the InvGamma prior over kernel lengthscale of U and Y
• uyLSScale::Float64=4.0: scale of the InvGamma prior over kernel lengthscale of U and Y
• xyLSShape::Float64=4.0: shape of the InvGamma prior over kernel lengthscale of X and Y
• xyLSScale::Float64=4.0: scale of the InvGamma prior over kernel lengthscale of X and Y
• tyLSShape::Float64=4.0: shape of the InvGamma prior over kernel lengthscale of T and Y
• tyLSScale::Float64=4.0: scale of the InvGamma prior over kernel lengthscale of T and Y
• sigmaUNoise::Float64=1.0e-13: noise added to matrix to make covariance stable and invertible
• sigmaUCov::Float64=1.0: assumed covariance over structured confounders
• drift::Float64=0.5: as in the paper, Metropolis Hastings Gaussian Drift

getProposalAddress(name, i, j)

Optimizes paramProposal by providing compact way to access trace address symbols

gpslc(filename * ".csv")
gpslc(filename * ".csv"; hyperparams=hyperparams, priorparams=priorparams))

gpslc(DataFrame(X1=...,X2=...,T=...,Y=...,obj=...))
gpslc(DataFrame(X1=...,X2=...,T=...,Y=...,obj=...); hyperparams=hyperparams, priorparams=priorparams)

Run posterior inference on the input data.

Datatypes of DataFrame or CSV must follow these standards:

• T (Boolean/Float64)
• Y (Float64)
• X1...XN (Float64...Float64)
• obj (Any)

Optional parameters

• hyperparams::HyperParameters=getHyperParameters(): Hyper parameters primarily define the high level amount of inference to perform.
• priorparams::PriorParameters=getPriorParameters(): Prior parameters define the high level priors to draw from when constructing kernel functions and latent confounder structure.

Returns a GPSLCObject which stores the hyperparameters, prior parameters, data, and posterior samples.

No Confounders No Covariates Continuous/Binary

No Confounders Continuous/Binary

likelihoodDistribution

A utility that uses multiple dispatch to take in one set of parameters for CausalGPSLC and the observed data, plus an intervention doT, and outputs the necessary matrices to compute MeanITE and CovITE, as well as other predictions, like directly predicting $Y_cf$.

No Covariates Continuous/Binary

loadData("path/to/filename.csv")

Returns DataFrame of CSV at path.

loadGPSLCObject(filename)
loadGPSLCObject("path/to/filename")
loadGPSLCObject("path/to/filename.gpslc")

This function will load and return the GPSLCObject contained in <filename>.gpslc.

Note: the extension .gpslc is optional and will be added if it is not included.

Logit maps a [0,1] onto the Real values

predictCounterfactualOutcomes(g, nSamplesPerMixture)
predictCounterfactualOutcomes(g, nSamplesPerMixture; fidelity=100)
predictCounterfactualOutcomes(g, nSamplesPerMixture; fidelity=100, minDoT=0, maxDoT=5)

Params

• g::GPSLCObject: The GPSLCObject that inference has already been computed for.
• nSamplesPerMixture::Int64: The number of outcome samples to

draw from each set of inferred posterior parameters.

• fidelity::Int64: How many intervention values to use to cover the domain of treatment values. Higher means more samples.
• minDoT::Float64=min(g.T...): The lowest interventional treatment to use.Defaults to the data g.T's lowest treatment value.
• maxDoT::Float64=max(g.T...): The highest interventional treatment to use. Defaults to the data g.T's highest treatment value.

julia> ite, doT = predictCounterfactualEffects(g, 30; fidelity=100)

Returns

• ite::Matrix{Float64}: An array of size [d, n, numPosteriorSamples * nSamplesPerMixture] where d is the number of interventional values defined by fidelity and the range of treatments in g.T - doTrange::Vector{Float64}: The list values of doT used, in order that matches the rows of ite.

prepareData(df, confounderEps, confounderCov)
prepareData("path/to/filename.csv", confounderEps, confounderCov)

Prepare Data Creates the latent confounding structure from the object labels in the data. Parses matrices for the observed covariates, treatments, and outcomes.

Returns: X, T, Y, SigmaU

Convert covariance matrix back from log-space, scale and add noise (if passed)

2D rbfKernelLog

Radial Basis Function Kernel applied element-wise to two vectors X1 and X2 passed

Params:

• X1: First array of values
• X2: Second array of values
• LS: Lengthscale array

Output normalized by LS squared

removeAdjacent(vector)

Return vector where each element is distinct from the previous one

sampleITE(g, doT)
sampleITE(g, doT; samplesPerPosterior=10)

Estimate Individual Treatment Effect with CausalGPSLC model

Params:

• g::GPSLCObject: Contains data and hyperparameters
• doT: The requested intervention (e.g. set all treatments to 1.0)
• samplesPerPosterior: How many ITE samples to draw per posterior sample in g.

Returns:

ITEsamples: n x m matrix where n is the number of individuals, and m is the number of samples.

samplePosterior(hyperparameters, priorparameters, SigmaU, X, T, Y)

Draw samples from the posterior given the observed data. Params:

• hyperparams::HyperParameters
• priorparams::PriorParameters
• SigmaU::ConfounderStructure
• X::Covariates
• T::Treatment
• Y::Outcome

Posterior samples are returned as a Vector of Gen choicemaps.

sampleSATE(g, doT)
sampleSATE(g, doT; samplesPerPosterior=10)

Estimate Sample Average Treatment Effect with CausalGPSLC model

Using sampleITE, samples can be drawn for the sample average treatment effect

Params:

• g::GPSLCObject: Contains data and hyperparameters
• doT: The requested intervention (e.g. set all treatments to 1.0)
• samplesPerPosterior: How many samples to draw per posterior sample in g.

Returns:

SATEsamples: n x m matrix where n is the number of individuals, and m is the number of samples.

saveGPSLCObject(g, filename)
saveGPSLCObject(g, "path/to/filename")
saveGPSLCObject(g, "path/to/filename.gpslc")

This function will save the GPSLCObject g to the file <filename>.gpslc. This GPSLCObject, including the posterior samples contained within it can be retrieved with the loadGPSLCObject function.

Note: The extension .gpslc is optional and will be added if it is not included.

summarizeEstimates(samples)
summarizeEstimates(samples; savetofile="ite_samples.csv")

Summarize Predicted Estimates (Counterfactual Outcomes or Individual Treatment Effects)

Create dataframe of mean, lower and upper quantiles of the samples from sampleITE or predictCounterfactualEffects.

Params:

• samples: The n x m array of samples from sampleSATE or sampleITE
• savetofile::String: Optionally save the resultant DataFrame as CSV to the filename passed
• credible_interval::Float64: A real in [0,1] where 0.90 is the default for a 90% credible interval

Returns:

• df: Dataframe of Individual, Mean, LowerBound, and UpperBound values for the credible intervals around the sample.

toMatrix(X, n, m)

Convert a vector of vectors or similar to a 2D matrix. Only call if you know all subvectors are same length.

toTupleOfVectors(matrix)

Convert matrix to tuple of vectors.