Confidence sequences and uniform boundaries

# Confidence sequences and uniform boundaries

This library supports calculation of uniform boundaries, confidence sequences, and always-valid p-values. These constructs are useful in sequential A/B testing, best-arm identification, and other sequential statistical procedures. The library is written in C++ and Python with a full Python interface and partial R interface. The main references are

This library is in early-stage development and should not be considered stable. Automated tests run on Python 3.7, 3.8, 3.9 and 3.10 on the latest Ubuntu and macOS.

The C++ implementation requires a compiler with C++14 to build the package, as well as the Boost C++ headers.

In the Python package, functions are split across modules by topic, as detailed below. In the R package, all functions mentioned below are exported in a single namespace.

## Installing the python package

Run `pip install confseq` at the command line.

## Installing the R package

Run the following in the R console:

```install.packages('devtools')
devtools::install_github('gostevehoward/confseq/r_package')
```

## Demos

### Estimating average treatment effect in a randomized trial

`demo/ate_demo.py` illustrates how to compute a confidence sequence for average treatment effect in a randomized trial with bounded potential outcomes, along with an always-valid p-value sequence. The method is based on Corollary 2 of the paper and uses the gamma-exponential mixture boundary. This demo requires `numpy` and `pandas`.

### Quantile confidence sequences

`demo/quantiles.py` illustrates how to use some of the included boundaries to construct confidence sequences for quantiles based on a stream of i.i.d. samples. The file includes a function to estimate a single, fixed quantile, as well as a function to estimate all quantiles simultaneously, with error control uniform over quantiles and time.

## Uniform boundaries

The `confseq.boundaries` Python module implements several uniform boundaries from the confidence sequences paper.

• There are four mixture boundaries. These are implemented by the functions `<TYPE>_log_mixture()` and `<TYPE>_mixture_bound()`, where `<TYPE>` is one of `normal` (Propositions 4 and 5), `gamma_exponential` (Proposition 8), `gamma_poisson` (Proposition 9), or `beta_binomial` (Propositions 6 and 7).

• `<TYPE>_log_mixture(s, v, ...)` returns the logarithm of the mixture supermartingale when called with S_t, the martingale, and V_t, the intrinsic time process. The reciprocal of the exponential of this value is an always-valid p-value. These functions are denoted log(m(s,v)) in the paper.
• `<TYPE>_mixture_bound(v, alpha, ...)` returns the uniform boundary with crossing probability at most alpha, evaluated at intrinsic time v.

Each function takes another required argument `v_opt` and an optional argument `alpha_opt=0.05`. These arguments are used to set the tuning parameter for each mixture, denoted by rho or r in the paper, optimizing the uniform boundary with crossing probability `alpha_opt` for intrinsic time `v_opt`. Such tuning is discussed in section 3.5 of the paper.

The gamma-exponential and gamma-Poisson mixtures also require a scale parameter `c`. The beta-binomial mixture requires range parameters `g` and `h`. Finally, the `normal_*` and `beta_binomial_*` functions accept an optional boolean parameter `is_one_sided` which is `True` by default. If `False`, the two-sided variants of these mixtures are used (Propositions 4 and 6).

• The polynomial stitching boundary (see Theorem 1 and the subsequent example) is implemented by `poly_stitching_bound`. Besides `v` and `alpha`, this function requires the tuning parameter `v_min` as well as optional parameters `c`, `s`, and `eta`, all documented in the paper.

• This module also includes a `bernoulli_confidence_interval` function which computes confidence sequences for the mean of any distribution with bounded support by making use of the sub-Bernoulli condition. Observations must be scaled so that the support is within the unit interval [0, 1].

All functions accept NumPy arrays in Python or vectors in R and perform vectorized operations.

## Quantile bounds

The `confseq.quantiles` Python module implements two quantile-uniform confidence sequences from the quantile paper.

• `empirical_process_lil_bound` is based on Theorem 2, and can be used to construct iterated-logarithm-rate confidence sequences for quantiles in which the confidence radius (in quantile space) is constant for all quantiles. This can also be used run the sequential Kolmogorov-Smirnov test described in section 7.2.
• `double_stitching_bound` is based on Theorem 3, and can be used to construct confidence sequences for quantiles in which the confidence radius (in quantile space) varies, getting smaller for extreme quantiles close to zero and one.

Finally, `quantile_ab_p_value` implements the two-sided sequential test of the hypothesis that two populations have equal values for some quantile, based on Theorem 5. The theorem covers tests of null hypothesis other than equality, as well as one-sided tests, but these are not yet implemented.

## C++ library

The main underlying implementation is in a single-file, header-only C++ library in `src/confseq/uniform_boundaries.h`. The top of the file defines a simplified interface mirroring the Python interface described above. Below that is an object-oriented interface useful for more involved work. The `confseq.boundaries` Python module is a wrapper generated by pybind11. The R package uses Rcpp.

Some implementations (such as betting-based or without-replacement confidence sequences) are only available in Python at the moment. Specifically, these include the implementations of

• `src/confseq/betting.py`
• `src/confseq/betting_strategies.py`
• `src/confseq/conjmix_bounded.py`, and
• `src/confseq/cs_plots.py`.

If you would like to help create an R interface for these methods, it would be appreciated!

## Unit tests

### C++

```make -C /path/to/confseq/test runtests
```

### Python (with random tests)

```pytest --ignore=test/googletest-1.8.1/
```

### Python (without random tests)

```pytest -m "not random" --ignore=test/googletest-1.8.1/
```

## Citing this software

Howard, S. R., Waudby-Smith, I. and Ramdas, A. (2019-), ConfSeq: software for confidence sequences and uniform boundaries, https://github.com/gostevehoward/confseq [Online; accessed <today>].

```@Misc{,
author = {Steven R. Howard, Ian Waudby-Smith, and Aaditya Ramdas},
title = {{ConfSeq}: software for confidence sequences and uniform boundaries},
year = {2021--},
url = "https://github.com/gostevehoward/confseq",
note = {[Online; accessed <today>]}
}
```

## Project details

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