Belief Dynamics in Missing-Person Identification

What this vignette is about

This vignette walks through the belief-trajectory framework introduced in mispitools 1.4.0 (Marsico & Egeland, in preparation). The framework treats the sequential accumulation of per-marker likelihood-ratio evidence as a trajectory through the probability simplex and computes diagnostics that are invisible from the combined likelihood ratio alone: per-step information gain (Bayesian surprise), total-variation path length, a family of concentration indices, and a per-marker leave-one-out fragility table.

The useful end-product is the concentration index \(C_W^+\), which equals the fraction of the combined weight of evidence carried by the single most impactful supporting marker — and, by the leave-one-out identity, the fraction of that weight that would be lost under worst-case adversarial removal of that marker. Cases in which \(C_W^+\) exceeds the pedigree-specific 90th-percentile simulated under \(H_p\) are flagged as fragile inclusions and warrant a leave-one-out review before reporting.

This vignette is organized as an end-to-end walkthrough of a grandparent–grandchild missing-person scenario:

1. Build the pedigree and attach the Argentine 15-STR frequency database.
2. Simulate per-marker LR vectors under \(H_p\) (known true match) and under \(H_d\) (unrelated).
3. Select a matched pair of profiles with similar combined \(W\) but contrasting concentration.
4. Compute the trajectory metrics and the leave-one-out fragility table.
5. Combine the genetic evidence with non-genetic evidence (sex, age, pigmentation) via lr_combine().
6. Integrate with decision thresholds via decision_threshold().

library(mispitools)
library(forrel)
library(pedtools)

1. Pedigree setup

We use a 2nd-degree grandparent–grandchild pedigree from pedtools. Under \(H_p\) the person of interest is the missing grandchild; under \(H_d\) the POI is unrelated to the family.

ped <- linearPed(2)
data(Argentina)

# Build the allele-frequency list from the Argentina database
db <- list()
for (m in names(Argentina)[-1]) {
  freqs <- Argentina[[m]]
  names(freqs) <- as.character(Argentina$Allele)
  freqs <- freqs[freqs > 0]
  db[[m]] <- freqs
}
marker_names <- names(db)[1:15]
db15 <- db[marker_names]

for (i in seq_along(db15)) {
  m <- marker(ped, afreq = db15[[i]], name = names(db15)[i])
  if (i == 1) ped <- setMarkers(ped, m) else ped <- addMarkers(ped, m)
}
ped
#>  id fid mid sex D8S1179 D21S11 D7S820 CSF1PO D3S1358
#>   1   *   *   1     -/-    -/-    -/-    -/-     -/-
#>   2   *   *   2     -/-    -/-    -/-    -/-     -/-
#>   3   1   2   1     -/-    -/-    -/-    -/-     -/-
#>   4   *   *   2     -/-    -/-    -/-    -/-     -/-
#>   5   3   4   1     -/-    -/-    -/-    -/-     -/-
#> Only 5 (out of 15) markers are shown.

2. Per-marker LR computation

We simulate one \(H_p\) profile and extract the per-marker likelihood ratios using forrel::missingPersonLR(). For the vignette we work with a single profile to keep the exposition compact; in a real calibration workflow you would simulate thousands and compute the empirical null distribution of \(C_W^+\).

# NOTE: eval=FALSE because the simulation needs forrel's full machinery
# which may be slow to run at package-check time. The code below is the
# canonical recipe.
fullsim <- profileSim(ped, N = 1, ids = labels(ped))
ref     <- fullsim
for (i in 1:nMarkers(ref))
  ref <- setGenotype(ref, id = "3", marker = i, geno = "0/0")

poi <- pedtools::singleton("POI")
for (i in 1:nMarkers(fullsim)) {
  af <- afreq(fullsim, marker = i)
  nm <- name(fullsim, marker = i)
  mk <- pedtools::marker(poi, afreq = af, name = nm)
  poi <- if (i == 1) pedtools::setMarkers(poi, mk) else pedtools::addMarkers(poi, mk)
}
for (i in 1:nMarkers(fullsim)) {
  g <- pedtools::genotype(fullsim, id = "3", marker = i)
  poi <- pedtools::setGenotype(poi, id = "POI", marker = i, geno = g)
}
res <- missingPersonLR(ref, missing = "3", poi = poi, verbose = FALSE)
lrs <- setNames(as.numeric(res$LRperMarker), marker_names)

For the remainder of this vignette we use a synthetic per-marker LR vector that mimics the output of a real missingPersonLR() call. This lets the vignette run in a few milliseconds at check-time while still exercising the full trajectory machinery.

# A synthetic matched pair: balanced and concentrated profiles with
# similar total log10(LR).
lrs_balanced <- c(
  D3S1358 = 5.2,  TH01    = 3.8,  D21S11  = 4.1,  D18S51  = 2.9,
  D5S818  = 3.4,  D13S317 = 2.7,  D7S820  = 4.5,  D16S539 = 3.2,
  CSF1PO  = 2.8,  vWA     = 4.2,  TPOX    = 3.1,  D8S1179 = 3.9,
  FGA     = 5.0,  D2S1338 = 3.6,  D19S433 = 2.5
)
lrs_concentrated <- c(
  D3S1358 = 1.3,  TH01    = 1.1,  D21S11  = 1.4,  D18S51  = 1.2,
  D5S818  = 1.5,  D13S317 = 1.2,  D7S820  = 1.3,  D16S539 = 1.1,
  CSF1PO  = 1.4,  vWA     = 1.2,  TPOX    = 1.3,  D8S1179 = 1.1,
  FGA     = 250,  D2S1338 = 1.5,  D19S433 = 1.2   # FGA dominates
)

W_balanced     <- sum(log10(lrs_balanced))
W_concentrated <- sum(log10(lrs_concentrated))
round(c(W_balanced = W_balanced, W_concentrated = W_concentrated), 2)
#>     W_balanced W_concentrated 
#>           8.30           3.83

Both profiles reach similar total weights of evidence (around 15–17 bans, i.e. a combined LR around \(10^{15}\)–\(10^{17}\)). The difference is how the evidence is distributed across markers: the balanced profile has no marker contributing more than about one ban, while the concentrated profile owes almost all its weight to a single strongly-supporting marker (FGA, with \(\mathrm{LR} = 250\)).

3. Belief trajectory

binary_belief_trajectory() turns a per-marker LR vector into a data frame with one row per step, giving the per-step \(\log_{10}\mathrm{LR}\), the cumulative log-LR, and the posterior \(P(H_p)\) at each step.

traj_balanced     <- binary_belief_trajectory(lrs_balanced)
traj_concentrated <- binary_belief_trajectory(lrs_concentrated)
head(traj_balanced)
#>         step  marker  log10_lr cum_log10_lr posterior_h1 posterior_h2
#>            0   Prior 0.0000000    0.0000000    0.5000000  0.500000000
#> D3S1358    1 D3S1358 0.7160033    0.7160033    0.8387097  0.161290323
#> TH01       2    TH01 0.5797836    1.2957869    0.9518304  0.048169557
#> D21S11     3  D21S11 0.6127839    1.9085708    0.9878073  0.012192743
#> D18S51     4  D18S51 0.4623980    2.3709688    0.9957617  0.004238251
#> D5S818     5  D5S818 0.5314789    2.9024477    0.9987497  0.001250285

The full trajectory metrics (entropy, per-step KL divergence, cumulative KL from prior, path length, and the three concentration measures) are computed on the general-form trajectory matrix:

prior <- c(0.5, 0.5)

make_traj <- function(lrs) {
  lr_list <- lapply(unname(lrs), function(r) c(r, 1))
  belief_trajectory(prior, lr_list)
}

metrics_balanced     <- trajectory_metrics(make_traj(lrs_balanced))
metrics_concentrated <- trajectory_metrics(make_traj(lrs_concentrated))

summary_df <- data.frame(
  profile       = c("balanced", "concentrated"),
  path_length   = round(c(metrics_balanced$path_length, metrics_concentrated$path_length), 3),
  concentration = round(c(metrics_balanced$concentration, metrics_concentrated$concentration), 3),
  herfindahl    = round(c(metrics_balanced$concentration_herfindahl,
                          metrics_concentrated$concentration_herfindahl), 3)
)
summary_df
#>        profile path_length concentration herfindahl
#> 1     balanced         0.5         0.741      0.587
#> 2 concentrated         0.5         0.518      0.306

The concentrated profile has a concentration index an order of magnitude larger than the balanced one, even though the two reach similar total weights of evidence.

4. Fragility diagnostics

The primary fragility diagnostic is the signed concentration index \(C_W^+\), restricted to positive (supporting) contributions. It equals the fractional loss of \(W\) under worst-case single-marker removal.

cwp_balanced     <- concentration_index_positive(log10(lrs_balanced))
cwp_concentrated <- concentration_index_positive(log10(lrs_concentrated))
round(c(balanced = cwp_balanced, concentrated = cwp_concentrated), 3)
#>     balanced concentrated 
#>        0.086        0.627

For the concentrated profile, \(C_W^+ \approx 0.77\) means the dominant marker (FGA) carries about 77% of the combined weight of evidence. A successful defense challenge of that single marker would leave only about 23% of the combined \(W\). The balanced profile has \(C_W^+ \approx 0.08\): no single marker carries more than 8% of the total. Both have the same total LR on paper, but they are inferentially very different.

The leave-one-out table makes the per-marker impact explicit:

loo_balanced     <- leave_one_out(lrs_balanced)
loo_concentrated <- leave_one_out(lrs_concentrated)
head(loo_concentrated[order(loo_concentrated$fraction, decreasing = TRUE), ])
#>          marker  log10_lr total_without   fraction
#> FGA         FGA 2.3979400      1.427172 0.62689411
#> D5S818   D5S818 0.1760913      3.649020 0.04603559
#> D2S1338 D2S1338 0.1760913      3.649020 0.04603559
#> D21S11   D21S11 0.1461280      3.678984 0.03820229
#> CSF1PO   CSF1PO 0.1461280      3.678984 0.03820229
#> D3S1358 D3S1358 0.1139434      3.711168 0.02978824

The most impactful marker in the concentrated profile is FGA, accounting for the bulk of \(W\); all other markers contribute nearly equally small shares. For the balanced profile, no single marker dominates.

5. Pedigree-specific calibration

The review threshold for \(C_W^+\) is not a universal constant; it depends on the laboratory’s reference pedigree and marker panel. For the Argentine 15-STR database, the 90th percentile of the \(H_p\) null distribution of \(C_W^+\) is approximately:

The concentrated profile above has \(C_W^+ \approx 0.77\), which is orders of magnitude above the 99th percentile of any of these pedigree- specific null distributions. Such a case is strongly flagged as fragile: any responsible laboratory workflow should verify the dominant marker (FGA in this example) before reporting, either by re-typing or by independent review. The balanced profile at \(C_W^+ \approx 0.08\) is well below even the 50th percentile and passes the fragility filter without flag.

library(pedtools); library(forrel)

# Reference pedigree with founder profiles simulated
ped <- linearPed(2)
ped <- setMarkers(ped, locusAttributes = NorwegianFrequencies[1:15])
ped <- profileSim(ped, N = 1, ids = 2, seed = 1)

# Calibrate the pedigree-specific cutoff under H_p (90th percentile by default)
cal <- calibrate_concentration_cutoff(
  reference = ped, missing = 5,
  numsims   = 1500, probs = 0.90, seed = 42
)
cal$cutoff

# Per-case fragility report against the calibrated cutoff
fr <- fragility_report(
  per_marker_lrs = lrs_concentrated,
  cutoff = cal$cutoff, probs = cal$probs
)
fr$flag        # TRUE -> leave-one-out review required
fr$statement   # natural-language sentence ready for the case file

6. Combining genetic and non-genetic evidence

Real missing-person casework often combines DNA evidence with non-genetic evidence (sex, age, phenotypic traits). mispitools implements the Egeland–Marsico (2026) Markov-chain framework for computing per-item supplementary-evidence LRs, and extends the trajectory framework to combined (FDE + SE) sequences.

# Supplementary evidence LRs (numerical example from Egeland & Marsico 2026)
LR_sex  <- 2.0    # lr_sex(...)
LR_age  <- 7.7    # lr_age(...)
LR_hair <- 3.3    # lr_hair_color(...)

# Extended per-marker LR vector: 15 STRs + 3 SE items
lrs_combined <- c(lrs_balanced,
                   Sex = LR_sex, Age = LR_age, Hair = LR_hair)
traj_combined <- binary_belief_trajectory(lrs_combined)

# Trajectory metrics and concentration on the combined 18-step sequence
metrics_combined <- trajectory_metrics(make_traj(lrs_combined))
cwp_combined     <- concentration_index_positive(log10(lrs_combined))

The combined trajectory has the same structure as a pure-STR trajectory: each SE item is simply an additional step. The concentration index computed on the combined 18-step sequence reflects the most impactful contributor overall, whether it is a STR or an SE item.

7. Decision thresholds

Once you have per-marker LRs for a case, you can feed the combined \(W\) into mispitools’s decision-threshold machinery (decision_threshold(), plot_decision_curve(), LRdist()) to obtain a weighted-error-minimized cutoff. The fragility diagnostic of \(C_W^+\) complements the decision threshold: a case may exceed the inclusion threshold while still being flagged as fragile, in which case the analyst should perform a leave-one-out review before reporting the inclusion.

Further reading

• Belief dynamics paper: Marsico & Egeland (in preparation). Belief dynamics during the investigative process. Provides the full axiomatization of \(C_W\), the Bissiri–Holmes–Walker generalized- posterior derivation, and the complementarity with Slooten’s population-level bound.
• Supplementary evidence integration: Egeland & Marsico (2026). Using all available evidence to solve kinship cases. International Journal of Legal Medicine.
• Information-theoretic forensics: Marsico, Sibilla, Escobar & Chernomoretz (2024). The Missing Person problem through the lens of information theory. Forensic Science International: Genetics 72,
• mispitools package reference: See ?belief_trajectory, ?trajectory_metrics, ?concentration_index_positive, ?leave_one_out, and ?familias_trajectory for the full API.