DP posterior survival and its Kaplan-Meier limit

With F ~ DP(alpha G0) and right-censored data the posterior mean survival interpolates between prior and data, and as alpha tends to zero it converges to the KAPLAN-MEIER estimator. That limit is the point of the section: it shows the DP prior smuggles in no information when alpha is small, and it gives the classical frequentist estimator a Bayesian reading. The discrepancy is computed rather than asserted, so shrinking alpha must shrink it.

Usage

Ghosalsurvdpkm(x, event = NULL, alpha = 1, g0_rate = NULL)

Arguments

x

Observed times, non-negative.

event

Event indicators 0/1; all events when NULL.

alpha

DP concentration, positive.

g0_rate

Rate of the exponential base measure; from the data when NULL.

Value

List with times, survival_dp, survival_km, max_abs_diff_to_km, alpha, limit_note, n_events, n, method.

Details

Formula: S_KM(t) = prod_{t_j <= t} (1 - d_j / R_j); S_DP = w S_KM + (1 - w) exp(-rate t), w = n / (alpha + n).

References

Ghosal & van der Vaart (2017), Fundamentals of Nonparametric Bayesian Inference, CUP, section 13.2; Susarla & Van Ryzin (1976); Kaplan & Meier (1958).

Examples

V <- c(1, 2, 3, 4, 5, 6, 7, 8)
Ghosalsurvdpkm(V)
#> $times
#> [1] 1 2 3 4 5 6 7 8
#> 
#> $survival_dp
#> [1] 0.86674860 0.73790893 0.61260190 0.49012359 0.36991033 0.25151079 0.13456357
#> [8] 0.01877926
#> 
#> $survival_km
#> [1] 0.875 0.750 0.625 0.500 0.375 0.250 0.125 0.000
#> 
#> $max_abs_diff_to_km
#> [1] 0.01877926
#> 
#> $alpha
#> [1] 1
#> 
#> $limit_note
#> [1] "alpha -> 0 gives Kaplan-Meier exactly; alpha -> infinity gives the base measure"
#> 
#> $n_events
#> [1] 8
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "DP posterior survival (Sec. 13.2); Kaplan-Meier is the alpha -> 0 limit"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"