Incorporating known detection probabilities into mark-recapture models

Modeling
,
1

Hi all,

I have a mark-recapture model that borrows heavily from code here. The model estimates detection probabilities and survival for sample units. The data I currently have has four time steps (for example, 1101 or 1001 or 1111).

I want to borrow the detection probabilities so that I can look at the same data set for just two time steps (e.g. 10 or 11), but I am not sure how to do this. I don’t think I need to stay with my current mark recapture model, but rather go with a simpler binomial model with a detection probability included. I am just not sure how to do this.

Any help on getting started would be appreciated. Here is the code I have for modeling temporal effects with no covariates

// This models is derived from section 12.3 of "Stan Modeling Language
// User's Guide and Reference Manual"

functions {
  int first_capture(int[] y_i) {
    for (k in 1:size(y_i))
      if (y_i[k])
        return k;
    return 0;
  }

  int last_capture(int[] y_i) {
    for (k_rev in 0:(size(y_i) - 1)) {
      // Compoud declaration was enabled in Stan 2.13
      int k = size(y_i) - k_rev;
      //      int k;
      //      k = size(y_i) - k_rev;
      if (y_i[k])
        return k;
    }
    return 0;
  }

  matrix prob_uncaptured(int nind, int n_occasions,
                         matrix p, matrix phi) {
    matrix[nind, n_occasions] chi;

    for (i in 1:nind) {
      chi[i, n_occasions] = 1.0;
      for (t in 1:(n_occasions - 1)) {
        // Compoud declaration was enabled in Stan 2.13
        int t_curr = n_occasions - t;
        int t_next = t_curr + 1;
        /*
        int t_curr;
        int t_next;

        t_curr = n_occasions - t;
        t_next = t_curr + 1;
        */
        chi[i, t_curr] = (1 - phi[i, t_curr])
                        + phi[i, t_curr] * (1 - p[i, t_next - 1]) * chi[i, t_next];
      }
    }
    return chi;
  }
}

data {
  int<lower=0> nind;            // Number of individuals
  int<lower=2> n_occasions;     // Number of capture occasions
  int<lower=0,upper=1> y[nind, n_occasions];    // Capture-history
}

transformed data {
  int n_occ_minus_1;
  // Compoud declaration is enabled in Stan 2.13
  //  int n_occ_minus_1 = n_occasions - 1;
  int<lower=0,upper=n_occasions> first[nind];
  int<lower=0,upper=n_occasions> last[nind];

  n_occ_minus_1 = n_occasions - 1;
  for (i in 1:nind)
    first[i] = first_capture(y[i]);
  for (i in 1:nind)
    last[i] = last_capture(y[i]);
}

parameters {
  vector<lower=0,upper=1>[n_occ_minus_1] alpha;  // Mean survival
  vector<lower=0,upper=1>[n_occ_minus_1] beta;   // Mean recapture
}

transformed parameters {
  matrix<lower=0,upper=1>[nind, n_occ_minus_1] phi;
  matrix<lower=0,upper=1>[nind, n_occ_minus_1] p;
  matrix<lower=0,upper=1>[nind, n_occasions] chi;

  // Constraints
  for (i in 1:nind) {
    for (t in 1:(first[i] - 1)) {
      phi[i, t] = 0;
      p[i, t] = 0;
    }
    for (t in first[i]:n_occ_minus_1) {
      phi[i, t] = alpha[t];
      p[i, t] = beta[t];
    }
  }

  chi = prob_uncaptured(nind, n_occasions, p, phi);
}

model {
  // Priors
  // Uniform priors are implicitly defined.
  //  alpha ~ uniform(0, 1);
  //  beta ~ uniform(0, 1);

  // Likelihood
  for (i in 1:nind) {
    if (first[i] > 0) {
      for (t in (first[i] + 1):last[i]) {
        1 ~ bernoulli(phi[i, t - 1]);
        y[i, t] ~ bernoulli(p[i, t - 1]);
      }
      1 ~ bernoulli(chi[i, last[i]]);
    }
  }
}
2

Hi @Wade! I’d like to help out but I’m not totally sure I understand. Can you clarify what you mean by “known detection probabilities” and “borrow the detection probabilities” here?

Are you trying to take the posterior for detection probabilities from the model with 4 time points, and subsequently use it as “known” or data in another model with 2 time points?

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