# Belief Dynamics in Missing-Person Identification

#### Franco L. Marsico

#### 2026-08-26

## 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:

Quantiles of C_W+ under H_p for three pedigrees on the Argentine 15-STR database.
Pedigree | 50% | 75% | 90% | 95% | 99%
--- | --- | --- | --- | --- | ---
linearPed(2) (2nd degree) | 0.130 | 0.152 | 0.174 | 0.192 | 0.225
linearPed(3) (3rd degree) | 0.140 | 0.163 | 0.192 | 0.211 | 0.254
cousinPed(1) (3rd degree) | 0.141 | 0.163 | 0.192 | 0.213 | 0.258

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.

### Calibration and reporting helpers

The two steps of the workflow above — calibrating the pedigree-specific cutoff under \(H_p\) and producing a per-case reportable statement — are exposed as the high-level helpers `calibrate_concentration_cutoff()` and `fragility_report()`. They internally invoke `sim_lr_genetic()` and `concentration_index_positive()`, so a complete pipeline from a `pedtools::ped` object to a printable sentence is just two function calls:

    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

The returned `statement` field is a single character string of the form “Combined weight of evidence W = w bans, C_W+ = c, … a successful challenge of the top marker would leave (1-c)w bans of residual support”, suitable to paste verbatim into the laboratory’s case report.

## 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.