Boundary kernels (internal API) — boundary

Low-level state machines behind the confidence sequences used by update_comparison(). Observations must already be rescaled to [0, 1]. These functions are exported for reuse by sister packages; ordinary users should call comparison_design() and update_comparison() instead.

Usage

boundary_init(
  name = seqbench_boundaries(),
  alpha = 0.05,
  c = 0.5,
  theta = 0.5,
  grid = 1001L,
  refine = TRUE,
  sd_max01 = NULL
)

boundary_update(state, x)

boundary_interval(state, thresholds = NULL)

Arguments

name: One of seqbench_boundaries().
alpha: Miscoverage level in (0, 1).
c: Truncation constant for the empirical-Bernstein and betting bets, in (0, 1). Waudby-Smith & Ramdas recommend 1/2 or 3/4.
theta: Hedging weight on the "long" capital in the betting boundary, in (0, 1). Default 1/2.
grid: Number of grid points on [0, 1] for the betting boundary.
refine: Logical; refine the betting interval endpoints by bisection on the exact capital (default TRUE). Without refinement the endpoints are the outer boundaries of the grid cells that contain the true endpoints, which is conservative and keeps the coverage guarantee.
sd_max01: For "bernstein_declared": declared upper bound on the conditional standard deviation of the observations, on the [0, 1] scale (at most 1/2). The boundary is a predictable-plug-in Bennett confidence sequence: for X in [0, 1] with conditional mean mu and conditional variance at most sd_max01^2, exp(lambda (X - mu) - sd_max01^2 (e^lambda - 1 - lambda)) is a supermartingale for every predictable lambda >= 0 (Bennett's inequality), and the same holds for -(X - mu). Coverage is guaranteed only if the declared bound is true.
state: A boundary state returned by boundary_init() or boundary_update().
x: A single number in [0, 1].
thresholds: Optional numeric vector of decision thresholds on the [0, 1] scale. When given to boundary_interval() for the betting boundary, an endpoint is refined only if its grid cell contains one of them, which is the only case in which refinement can change a decision; this keeps the per-step cost linear in the grid size instead of linear in the number of observations. Decisions and stopping times are identical to full refinement; the reported running intersection can be up to one grid cell wider per side. NULL (default) refines both endpoints.

Value

boundary_init() and boundary_update() return a state object of class seqbench_boundary. boundary_interval() returns a named numeric vector c(estimate, lower, upper) on the [0, 1] scale. For the betting boundary, NA endpoints mean that the exact confidence set is empty (a possible event, of probability at most alpha under the assumptions); a nonempty set narrower than one grid cell is located by searching the exact capital, never reported as empty.

Examples

st <- boundary_init("hoeffding", alpha = 0.05)
for (x in c(0.2, 0.3, 0.25)) st <- boundary_update(st, x)
boundary_interval(st)
#> estimate    lower    upper 
#>     0.25     0.00     1.00