
This is a basic description file to inform how to run experiments corresponding to those
in the paper. 

THE CURRENT VERSION IS NOT MEANT TO BE APPLICABLE TO DATASETS WITH MORE THAN A DOZEN VARIABLES.

The function 'app_witness_synthetic_default_run3' provides a starting example. It generates
four series of 100 synthetic problems as described in the paper (solvable/"hard", solvable/"easy",
not solvable/"hard", not solvable/"easy") and runs WPP under six different values of
relaxation parameters. Each relaxation parameter vector also contains 6 pieces of information,

|eta_x0^star - eta_x1^star| <= epsilons[1]

|eta_x0^star - P(Y = 1 | X = x, W = 0)| <= epsilons[2]

|eta_x1^star - P(Y = 1 | X = x, W = 1)| <= epsilons[3]
|delta_w0^star - P(X = 1 | W = 0)| <= epsilons[4]
|delta_w1^star - P(X = 1 | W = 1)| <= epsilons[4]
epsilons[5] * P(U) <= P(U | W = w) <= epsilons[6] * P(U),

where the notation is introduced in the paper. In 'app_witness_synthetic_default_run3',
we have epsilons[5] == epsilons[6] == 1, while the other four vary jointly in the
grid (0.05, 0.10, 0.15, 0.20, 0.25, 0.30). Runs of WPP are stored in lists such as
'result_solvable_hard', with evaluation metrics stored in the respective
'result_stats_solvable_hard'. Function 'app_plot_summary_witness' can provide some
visual depiction of e.g. ''result_stats_solvable_hard'. That is,

> app_plot_summary_witness(result_stats_solvable_hard[[4])]

will plot some graphs for the case epsilons[1] == ... == epsilons[4] == 0.20.

Simpler versions of ''app_witness_synthetic_default_run3', functions 
'app_witness_synthetic_default_run1' and 'app_witness_synthetic_default_run2'
variations of the analysis done by 'app_witness_synthetic_default_run3'.

Finally, 'app_witness_demo.R' is an analysis of a real-word dataset. Contact
us in case you are interested in this data. This file can be used as a template
for an application to other real cases.

THE CURRENT VERSION IS NOT MEANT TO BE APPLICABLE TO DATASETS WITH MORE THAN A DOZEN VARIABLES.

-Ricardo Silva and Robin Evans,
 October, 2014


