
library(CausalQueries)
library(knitr)
library(ggplot2)
library(rstan)
library(bayesplot)
rstan_options(refresh = 0)

Accessing the posterior

When you update a model using CausalQueries, CausalQueries generates and updates a stan model and saves the posterior distribution over parameters in the model.

The basic usage is:

data <- data.frame(X = rep(c(0:1), 10), Y = rep(c(0:1), 10))

model <- make_model("X -> Y") |> 
  update_model(data)

The posterior over parameters can be accessed thus:

inspect(model, "posterior_distribution")
#> 
#> posterior_distribution
#> Summary statistics of model parameters posterior distributions:
#> 
#>   Distributions matrix dimensions are 
#>   4000 rows (draws) by 6 cols (parameters)
#> 
#>      mean   sd
#> X.0  0.50 0.10
#> X.1  0.50 0.10
#> Y.00 0.08 0.07
#> Y.10 0.04 0.04
#> Y.01 0.80 0.11
#> Y.11 0.08 0.08

When querying a model you can request use of the posterior distribution with the using argument:

model |> 
  query_model(
    query = "Y[X=1] > Y[X=0]",
    using = c("priors", "posteriors")) |>
  kable(digits = 2)

query	given	using	case_level	mean	sd	cred.low	cred.high
Y[X=1] > Y[X=0]	-	priors	FALSE	0.25	0.19	0.01	0.71
Y[X=1] > Y[X=0]	-	posteriors	FALSE	0.80	0.11	0.55	0.96

Summary of stan performance

You can access a summary of the parameter values and convergence information as produced by stan thus:

inspect(model, "stan_summary")
#> 
#> stan_summary
#> Stan model summary:
#> 
#> Inference for Stan model: simplexes.
#> 4 chains, each with iter=2000; warmup=1000; thin=1; 
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#> 
#>              mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
#> X.0          0.50    0.00 0.10   0.30   0.43   0.50   0.57   0.70  2479    1
#> X.1          0.50    0.00 0.10   0.30   0.43   0.50   0.57   0.70  2479    1
#> Y.00         0.08    0.00 0.07   0.00   0.03   0.06   0.11   0.27  2197    1
#> Y.10         0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.14  4898    1
#> Y.01         0.80    0.00 0.11   0.55   0.74   0.82   0.88   0.96  4042    1
#> Y.11         0.08    0.00 0.08   0.00   0.02   0.06   0.11   0.27  4581    1
#> X0.Y00       0.04    0.00 0.04   0.00   0.01   0.03   0.05   0.14  2158    1
#> X1.Y00       0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.14  2278    1
#> X0.Y10       0.02    0.00 0.02   0.00   0.01   0.01   0.03   0.07  4630    1
#> X1.Y10       0.02    0.00 0.02   0.00   0.01   0.01   0.03   0.07  4638    1
#> X0.Y01       0.40    0.00 0.10   0.22   0.33   0.40   0.46   0.61  2694    1
#> X1.Y01       0.40    0.00 0.10   0.22   0.33   0.40   0.47   0.59  2860    1
#> X0.Y11       0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  4301    1
#> X1.Y11       0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.14  4295    1
#> lp__       -14.55    0.04 1.49 -18.17 -15.37 -14.22 -13.46 -12.64  1475    1
#> 
#> Samples were drawn using NUTS(diag_e) at Wed Nov 20 11:38:56 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at 
#> convergence, Rhat=1).

This summary provides information on the distribution of parameters as well as convergence diagnostics, summarized in the Rhat column. In the printout above the first six rows show the distribution of the model parameters; the next eight rows show the distribution over transformed parameters, here the causal types. The last row shows the unnormalized log density on Stan’s unconstrained space which, as described in Stan documentation is intended to diagnose sampling efficiency and evaluate approximations.

See stan documentation for further details.

Advanced diagnostics

If you are interested in advanced diagnostics of performance you can save and access the raw stan output.

model <- make_model("X -> Y") |> 
  update_model(data, keep_fit = TRUE)

Note that the summary for this raw output shows the labels used in the generic stan model: lambda for the vector of parameters, corresponding to the parameters in the parameters dataframe (inspect(model, "parameters_df")), and , if saved, a vector types for the causal types (see inspect(model, "causal_types")) and w for the event probabilities (inspect(model, "prior_event_probabilities")).

model |> inspect("stanfit")
#> 
#> stanfit
#> Stan model summary:
#> Inference for Stan model: simplexes.
#> 4 chains, each with iter=2000; warmup=1000; thin=1; 
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#> 
#>              mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
#> lambdas[1]   0.50    0.00 0.10   0.29   0.43   0.50   0.57   0.70  1836    1
#> lambdas[2]   0.50    0.00 0.10   0.30   0.43   0.50   0.57   0.71  1836    1
#> lambdas[3]   0.08    0.00 0.08   0.00   0.02   0.06   0.12   0.29  2147    1
#> lambdas[4]   0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  4020    1
#> lambdas[5]   0.80    0.00 0.11   0.52   0.74   0.82   0.88   0.95  3820    1
#> lambdas[6]   0.08    0.00 0.08   0.00   0.02   0.06   0.11   0.28  4298    1
#> types[1]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  2175    1
#> types[2]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  2120    1
#> types[3]     0.02    0.00 0.02   0.00   0.01   0.01   0.03   0.08  3734    1
#> types[4]     0.02    0.00 0.02   0.00   0.01   0.01   0.03   0.08  3938    1
#> types[5]     0.40    0.00 0.10   0.21   0.33   0.40   0.46   0.60  2105    1
#> types[6]     0.40    0.00 0.10   0.21   0.33   0.40   0.47   0.61  2243    1
#> types[7]     0.04    0.00 0.04   0.00   0.01   0.03   0.05   0.15  4060    1
#> types[8]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  3944    1
#> lp__       -14.68    0.05 1.60 -18.60 -15.50 -14.35 -13.47 -12.67  1179    1
#> 
#> Samples were drawn using NUTS(diag_e) at Wed Nov 20 11:38:57 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at 
#> convergence, Rhat=1).

You can then use diagnostic packages such as bayesplot.

model |> inspect("stanfit") |>
  bayesplot::mcmc_pairs(pars = c("lambdas[3]", "lambdas[4]", "lambdas[5]", "lambdas[6]"))
#> 
#> stanfit
#> Stan model summary:
#> Inference for Stan model: simplexes.
#> 4 chains, each with iter=2000; warmup=1000; thin=1; 
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#> 
#>              mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
#> lambdas[1]   0.50    0.00 0.10   0.29   0.43   0.50   0.57   0.70  1836    1
#> lambdas[2]   0.50    0.00 0.10   0.30   0.43   0.50   0.57   0.71  1836    1
#> lambdas[3]   0.08    0.00 0.08   0.00   0.02   0.06   0.12   0.29  2147    1
#> lambdas[4]   0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  4020    1
#> lambdas[5]   0.80    0.00 0.11   0.52   0.74   0.82   0.88   0.95  3820    1
#> lambdas[6]   0.08    0.00 0.08   0.00   0.02   0.06   0.11   0.28  4298    1
#> types[1]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  2175    1
#> types[2]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  2120    1
#> types[3]     0.02    0.00 0.02   0.00   0.01   0.01   0.03   0.08  3734    1
#> types[4]     0.02    0.00 0.02   0.00   0.01   0.01   0.03   0.08  3938    1
#> types[5]     0.40    0.00 0.10   0.21   0.33   0.40   0.46   0.60  2105    1
#> types[6]     0.40    0.00 0.10   0.21   0.33   0.40   0.47   0.61  2243    1
#> types[7]     0.04    0.00 0.04   0.00   0.01   0.03   0.05   0.15  4060    1
#> types[8]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  3944    1
#> lp__       -14.68    0.05 1.60 -18.60 -15.50 -14.35 -13.47 -12.67  1179    1
#> 
#> Samples were drawn using NUTS(diag_e) at Wed Nov 20 11:38:57 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at 
#> convergence, Rhat=1).

np <- model |> inspect("stanfit") |> bayesplot::nuts_params()
#> 
#> stanfit
#> Stan model summary:
#> Inference for Stan model: simplexes.
#> 4 chains, each with iter=2000; warmup=1000; thin=1; 
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#> 
#>              mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
#> lambdas[1]   0.50    0.00 0.10   0.29   0.43   0.50   0.57   0.70  1836    1
#> lambdas[2]   0.50    0.00 0.10   0.30   0.43   0.50   0.57   0.71  1836    1
#> lambdas[3]   0.08    0.00 0.08   0.00   0.02   0.06   0.12   0.29  2147    1
#> lambdas[4]   0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  4020    1
#> lambdas[5]   0.80    0.00 0.11   0.52   0.74   0.82   0.88   0.95  3820    1
#> lambdas[6]   0.08    0.00 0.08   0.00   0.02   0.06   0.11   0.28  4298    1
#> types[1]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  2175    1
#> types[2]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  2120    1
#> types[3]     0.02    0.00 0.02   0.00   0.01   0.01   0.03   0.08  3734    1
#> types[4]     0.02    0.00 0.02   0.00   0.01   0.01   0.03   0.08  3938    1
#> types[5]     0.40    0.00 0.10   0.21   0.33   0.40   0.46   0.60  2105    1
#> types[6]     0.40    0.00 0.10   0.21   0.33   0.40   0.47   0.61  2243    1
#> types[7]     0.04    0.00 0.04   0.00   0.01   0.03   0.05   0.15  4060    1
#> types[8]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  3944    1
#> lp__       -14.68    0.05 1.60 -18.60 -15.50 -14.35 -13.47 -12.67  1179    1
#> 
#> Samples were drawn using NUTS(diag_e) at Wed Nov 20 11:38:57 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at 
#> convergence, Rhat=1).
head(np) |> kable()

Chain	Iteration	Parameter	Value
1	1	accept_stat__	0.8906345
1	2	accept_stat__	0.8402453
1	3	accept_stat__	0.9390884
1	4	accept_stat__	0.9943855
1	5	accept_stat__	0.9663906
1	6	accept_stat__	0.8208385


model |> 
  inspect("stanfit") |>
  bayesplot::mcmc_trace(pars = "lambdas[5]", np = np) 
#> 
#> stanfit
#> Stan model summary:
#> Inference for Stan model: simplexes.
#> 4 chains, each with iter=2000; warmup=1000; thin=1; 
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#> 
#>              mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
#> lambdas[1]   0.50    0.00 0.10   0.29   0.43   0.50   0.57   0.70  1836    1
#> lambdas[2]   0.50    0.00 0.10   0.30   0.43   0.50   0.57   0.71  1836    1
#> lambdas[3]   0.08    0.00 0.08   0.00   0.02   0.06   0.12   0.29  2147    1
#> lambdas[4]   0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  4020    1
#> lambdas[5]   0.80    0.00 0.11   0.52   0.74   0.82   0.88   0.95  3820    1
#> lambdas[6]   0.08    0.00 0.08   0.00   0.02   0.06   0.11   0.28  4298    1
#> types[1]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  2175    1
#> types[2]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  2120    1
#> types[3]     0.02    0.00 0.02   0.00   0.01   0.01   0.03   0.08  3734    1
#> types[4]     0.02    0.00 0.02   0.00   0.01   0.01   0.03   0.08  3938    1
#> types[5]     0.40    0.00 0.10   0.21   0.33   0.40   0.46   0.60  2105    1
#> types[6]     0.40    0.00 0.10   0.21   0.33   0.40   0.47   0.61  2243    1
#> types[7]     0.04    0.00 0.04   0.00   0.01   0.03   0.05   0.15  4060    1
#> types[8]     0.04    0.00 0.04   0.00   0.01   0.03   0.06   0.15  3944    1
#> lp__       -14.68    0.05 1.60 -18.60 -15.50 -14.35 -13.47 -12.67  1179    1
#> 
#> Samples were drawn using NUTS(diag_e) at Wed Nov 20 11:38:57 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at 
#> convergence, Rhat=1).
#> No divergences to plot.

Inspecting posteriors

Accessing the posterior

Summary of stan performance

Advanced diagnostics