Dealing with latent discrete parameters in Stan
6 likes9,983 views
Stan can handle latent discrete parameters in population ecology models by marginalizing them out. This allows estimating quantities like population size and transition probabilities. Stan formulations for capture-recapture and multistate models use matrix calculations in hidden Markov models. Stan shows faster computation and better sampling than OpenBUGS for these examples, though the models can be more complex to specify in Stan.
![Data
[,1] [,2] [,3]
[1,] 0 0 1
[2,] 1 1 1
[3,] 1 0 0
[4,] 1 0 1
[5,] 1 0 1
[6,] 1 0 1
:
[85,] 0 1 0
[86,] 0 1 1
[87,] 1 1 0
Individuals
Survey occasions](https://image.slidesharecdn.com/dealingwithlatentdiscreteparametersinstan-160604211959/85/Dealing-with-latent-discrete-parameters-in-Stan-10-320.jpg)
![Data augmentation
[,1] [,2] [,3]
[1,] 0 0 1
[2,] 1 1 1
[3,] 1 0 0
[4,] 1 0 1
:
[87,] 1 1 0
[88,] 0 0 0
:
[236,] 0 0 0
[237,] 0 0 0
Add 150
dummy records](https://image.slidesharecdn.com/dealingwithlatentdiscreteparametersinstan-160604211959/85/Dealing-with-latent-discrete-parameters-in-Stan-12-320.jpg)
![BUGS
model {
# Priors
omega ~ dunif(0, 1) # Inclusion probability
p ~ dunif(0, 1) # Detection probability
# Likelihood
for (i in 1:M){
z[i] ~ dbern(omega) # Inclusion indicators
for (j in 1:T) {
y[i, j] ~ dbern(p.eff[i, j])
p.eff[i, j] <- z[i] * p
}
}
# Derived quantities
N <- sum(z[])
}](https://image.slidesharecdn.com/dealingwithlatentdiscreteparametersinstan-160604211959/85/Dealing-with-latent-discrete-parameters-in-Stan-14-320.jpg)
![data {
int<lower=0> M; // Size of augmented data set
int<lower=0> T; // Number of sampling occasions
int<lower=0,upper=1> y[M, T]; // Capture-history matrix
}
transformed data {
int<lower=0> s[M]; // Totals in each row
int<lower=0> C; // Size of observed data set
C <- 0;
for (i in 1:M) {
s[i] <- sum(y[i]);
if (s[i] > 0)
C <- C + 1;
}
}
parameters {
real<lower=0,upper=1> omega; // Inclusion probability
real<lower=0,upper=1> p; // Detection probability
}](https://image.slidesharecdn.com/dealingwithlatentdiscreteparametersinstan-160604211959/85/Dealing-with-latent-discrete-parameters-in-Stan-16-320.jpg)
![model {
for (i in 1:M) {
real lp[2];
if (s[i] > 0) {
// Included
increment_log_prob(bernoulli_log(1, omega)
+ binomial_log(s[i], T, p));
} else {
// Included
lp[1] <- bernoulli_log(1, omega)
+ binomial_log(0, T, p);
// Not included
lp[2] <- bernoulli_log(0, omega);
increment_log_prob(log_sum_exp(lp));
}
}
}](https://image.slidesharecdn.com/dealingwithlatentdiscreteparametersinstan-160604211959/85/Dealing-with-latent-discrete-parameters-in-Stan-17-320.jpg)
![Data
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 1 3 3 2 3 1
[2,] 1 1 1 1 1 2
[3,] 1 1 1 1 3 3
[4,] 1 3 3 3 3 3
[5,] 1 3 3 3 3 3
[6,] 1 1 2 3 3 3
:
[798,] 3 3 3 3 2 3
[799,] 3 3 3 3 2 3
[800,] 3 3 3 3 2 3
Individuals
Survey occasion
Values
1: seen (captured) at site A, 2: seen (captured) at site B,
3: not seen (not captured)](https://image.slidesharecdn.com/dealingwithlatentdiscreteparametersinstan-160604211959/85/Dealing-with-latent-discrete-parameters-in-Stan-24-320.jpg)
![model {
:
# Define probabilities of state S(t+1) given S(t)
ps[1, 1] <- phiA * (1 - psiAB)
ps[1, 2] <- phiA * psiAB
ps[1, 3] <- 1 - phiA
ps[2, 1] <- phiB * psiBA
ps[2, 2] <- phiB * (1 - psiBA)
ps[2, 3] <- 1 - phiB
ps[3, 1] <- 0
ps[3, 2] <- 0
ps[3, 3] <- 1
# Define probabilities of O(t) given S(t)
po[1, 1] <- pA
po[1, 2] <- 0
po[1, 3] <- 1 - pA
po[2, 1] <- 0
po[2, 2] <- pB
po[2, 3] <- 1 - pB
po[3, 1] <- 0
po[3, 2] <- 0
po[3, 3] <- 1
:](https://image.slidesharecdn.com/dealingwithlatentdiscreteparametersinstan-160604211959/85/Dealing-with-latent-discrete-parameters-in-Stan-30-320.jpg)
![for (i in 1:nind) {
# Define latent state at first capture
z[i, f[i]] <- y[i, f[i]]
for (t in (f[i] + 1):n.occasions) {
# State process
z[i, t] ~ dcat(ps[z[i, t - 1], ])
# Observation process
y[i, t] ~ dcat(po[z[i, t], ])
}
}
f[]: array containing first capture occasion
ps[,]: state transition matrix
po[,]: observation matrix](https://image.slidesharecdn.com/dealingwithlatentdiscreteparametersinstan-160604211959/85/Dealing-with-latent-discrete-parameters-in-Stan-31-320.jpg)
![model {
real acc[3];
vector[3] gamma[n_occasions];
// See Stan Modeling Language User's Guide and Reference Manual
for (i in 1:nind) {
if (f[i] > 0) {
for (k in 1:3)
gamma[f[i], k] <- (k == y[i, f[i]]);
for (t in (f[i] + 1):n_occasions) {
for (k in 1:3) {
for (j in 1:3)
acc[j] <- gamma[t - 1, j] * ps[j, k]
* po[k, y[i, t]];
gamma[t, k] <- sum(acc);
}
}
increment_log_prob(log(sum(gamma[n_occasions])));
}
}
}](https://image.slidesharecdn.com/dealingwithlatentdiscreteparametersinstan-160604211959/85/Dealing-with-latent-discrete-parameters-in-Stan-34-320.jpg)
![Stan code
data {
int<lower=0> R; // Number of sites
int<lower=0> T; // Number of temporal replications
int<lower=0> y[R, T]; // Counts
int<lower=0> K; // Upper bounds of population size
}
model {
:
// Likelihood
for (i in 1:R) {
vector[K+1] lp;
for (n in max_y[i]:K) {
lp[n + 1] <- binomial_log(y[i], n, p)
+ poisson_log(n, lambda);
}
increment_log_prob(log_sum_exp(lp[(max_y[i] + 1):(K + 1)]));
}
:
}](https://image.slidesharecdn.com/dealingwithlatentdiscreteparametersinstan-160604211959/85/Dealing-with-latent-discrete-parameters-in-Stan-38-320.jpg)
Dealing with latent discrete parameters in Stan
- 1. Dealing with latent discrete parameters in Stan ITÔ, Hiroki 2016-06-04 Tokyo.Stan Michael Betancourt’s Stan Lecture
- 2. About me • An end user of statistical software • Researcher of forest ecology • Species composition of forests • Forest dynamics
- 3. Contents 1. Introduction • Population Ecology 2. Examples 1. Capture-recapture data and data augmentation 2. Multistate model
- 4. Bayesian Population Analysis using WinBUGS (BPA) • My colleagues and I translated “Bayesian Population Analysis using WinBUGS” by M. Kéry and M. Schaub into Japanese. • Many practical examples for population ecology
- 5. Stan translation of BPA models https://github.com/stan-dev/example-models/tree/master/BPA
- 6. Population ecology • A subfield of ecology • Studies of population • Changes in size (number of individuals) of animals, plants and other organisms • Estimation of population size, growth rate, extinction probability, etc.
- 7. Population ecology • Latent discrete parameters are often used. • Unobserved status • Present or absent • Dead or alive • But Stan does not support discrete parameters. • Marginalizing out is required to deal with discrete parameters.
- 8. Example 1 Capture-recapture data and data augmentation (in Chapter 6 of BPA)
- 9. Capture-recapture data Capture Release Recapture 🐞 ① 🐞 ② 🐞 ③ 🐞 ④ 🐞 ⑤ 🐞 ⑥ 🐞 ⑦ 🐞 ⑧ 🐞 ⑨ 🐞 ① 🐞 ③ 🐞 ④ 🐞 ⑧ 🐞 ⑩ 🐞 ⑪ 🐞 ⑫ 🐞 ⑬ Assume closed population: fixed size, no recruitment, no death, no immigration nor emigration
- 10. Data [,1] [,2] [,3] [1,] 0 0 1 [2,] 1 1 1 [3,] 1 0 0 [4,] 1 0 1 [5,] 1 0 1 [6,] 1 0 1 : [85,] 0 1 0 [86,] 0 1 1 [87,] 1 1 0 Individuals Survey occasions
- 11. Estimation • Population size (total number of individuals including unobserved) • Detection (capture) probability
- 12. Data augmentation [,1] [,2] [,3] [1,] 0 0 1 [2,] 1 1 1 [3,] 1 0 0 [4,] 1 0 1 : [87,] 1 1 0 [88,] 0 0 0 : [236,] 0 0 0 [237,] 0 0 0 Add 150 dummy records
- 13. Model
- 14. BUGS model { # Priors omega ~ dunif(0, 1) # Inclusion probability p ~ dunif(0, 1) # Detection probability # Likelihood for (i in 1:M){ z[i] ~ dbern(omega) # Inclusion indicators for (j in 1:T) { y[i, j] ~ dbern(p.eff[i, j]) p.eff[i, j] <- z[i] * p } } # Derived quantities N <- sum(z[]) }
- 15. Stan
- 16. data { int<lower=0> M; // Size of augmented data set int<lower=0> T; // Number of sampling occasions int<lower=0,upper=1> y[M, T]; // Capture-history matrix } transformed data { int<lower=0> s[M]; // Totals in each row int<lower=0> C; // Size of observed data set C <- 0; for (i in 1:M) { s[i] <- sum(y[i]); if (s[i] > 0) C <- C + 1; } } parameters { real<lower=0,upper=1> omega; // Inclusion probability real<lower=0,upper=1> p; // Detection probability }
- 17. model { for (i in 1:M) { real lp[2]; if (s[i] > 0) { // Included increment_log_prob(bernoulli_log(1, omega) + binomial_log(s[i], T, p)); } else { // Included lp[1] <- bernoulli_log(1, omega) + binomial_log(0, T, p); // Not included lp[2] <- bernoulli_log(0, omega); increment_log_prob(log_sum_exp(lp)); } } }
- 18. generated quantities { int<lower=C> N; N <- C + binomial_rng(M, omega * pow(1 - p, T)); }
- 19. Results
- 20. Computing times OpenBUGS Stan Number of chains 3 4 Burn-in or warmup + iterations / chain 500 + 1000 500 + 1000 Computing time (sec) 5.9 5.8 Effective sample size of p 320 1867 Eff. sample size / time (sec-1) 54.6 319.5 Environment: 2.8 GHz Xeon W3530, Ubuntu 14.04, No parallell computing. Compilation time is not included in Stan. Times were measured using system.time(). The values are mean of 3 measurements.
- 22. Example 2 Multistate model (in chapter 9 of BPA)
- 23. Multistate model 🐞 ① 🐞 ② 🐞 ③ 🐞 ④ 🐞 ⑤ 🐞⑥ 🐞⑦ 🐞⑧ 🐞⑨ Site A Site B Capture Release Recapture 🐞① 🐞 ② 🐞 ⑤ 🐞 ⑥ 🐞⑦ 🐞⑫ 🐞⑪ Site A Site B 🐞 ⑩
- 24. Data [,1] [,2] [,3] [,4] [,5] [,6] [1,] 1 3 3 2 3 1 [2,] 1 1 1 1 1 2 [3,] 1 1 1 1 3 3 [4,] 1 3 3 3 3 3 [5,] 1 3 3 3 3 3 [6,] 1 1 2 3 3 3 : [798,] 3 3 3 3 2 3 [799,] 3 3 3 3 2 3 [800,] 3 3 3 3 2 3 Individuals Survey occasion Values 1: seen (captured) at site A, 2: seen (captured) at site B, 3: not seen (not captured)
- 25. Estimation • Survival probability (at site A and B) • Movement probability (from site A to B and B to A) • Detection (capture) probability (at site A and B)
- 26. State transition State at time t+1 Site A Site B Dead Site A φA(1-ψAB) φAψAB 1-φA State at time t Site B φBψBA φB(1-ψBA) 1-φB Dead 0 0 1 φA: survival probability at site A, φB: survival probability at site B, ψAB: movement probability from A to B, ψBA: movement probability from B to A
- 27. Observation Observation at time t Site A Site B Not seen Site A pA 0 1-pA State at time t Site B 0 pB 1-pB Dead 0 0 1 pA: detection probability at site A, pB: detection probability at site B
- 28. Model
- 29. BUGS
- 30. model { : # Define probabilities of state S(t+1) given S(t) ps[1, 1] <- phiA * (1 - psiAB) ps[1, 2] <- phiA * psiAB ps[1, 3] <- 1 - phiA ps[2, 1] <- phiB * psiBA ps[2, 2] <- phiB * (1 - psiBA) ps[2, 3] <- 1 - phiB ps[3, 1] <- 0 ps[3, 2] <- 0 ps[3, 3] <- 1 # Define probabilities of O(t) given S(t) po[1, 1] <- pA po[1, 2] <- 0 po[1, 3] <- 1 - pA po[2, 1] <- 0 po[2, 2] <- pB po[2, 3] <- 1 - pB po[3, 1] <- 0 po[3, 2] <- 0 po[3, 3] <- 1 :
- 31. for (i in 1:nind) { # Define latent state at first capture z[i, f[i]] <- y[i, f[i]] for (t in (f[i] + 1):n.occasions) { # State process z[i, t] ~ dcat(ps[z[i, t - 1], ]) # Observation process y[i, t] ~ dcat(po[z[i, t], ]) } } f[]: array containing first capture occasion ps[,]: state transition matrix po[,]: observation matrix
- 32. Stan Treating as a Hidden Markov model
- 33. Site B Site B Site A DeadSite A Hidden Markov model State Observation Seen at site A Dead Not seen Seen at site B Seen at site A Not seen Not seen
- 34. model { real acc[3]; vector[3] gamma[n_occasions]; // See Stan Modeling Language User's Guide and Reference Manual for (i in 1:nind) { if (f[i] > 0) { for (k in 1:3) gamma[f[i], k] <- (k == y[i, f[i]]); for (t in (f[i] + 1):n_occasions) { for (k in 1:3) { for (j in 1:3) acc[j] <- gamma[t - 1, j] * ps[j, k] * po[k, y[i, t]]; gamma[t, k] <- sum(acc); } } increment_log_prob(log(sum(gamma[n_occasions]))); } } }
- 35. Results
- 36. Computing times OpenBUGS Stan Number of chains 3 4 Burn-in or warmup + iterations / chain 500 + 2000 500 + 1000 Computing time (sec) 327.9 193.5 Effective sample size of pA 140 649 Eff. sample size / time (sec-1) 0.4 3.4 Environment: 2.8 GHz Xeon W3530, Ubuntu 14.04, No parallell computing. Compilation time is not included in Stan. Times were measured using system.time(). The values are mean of 3 measurements.
- 37. Binomial-mixture model (in Chapter 12 of BPA)
- 38. Stan code data { int<lower=0> R; // Number of sites int<lower=0> T; // Number of temporal replications int<lower=0> y[R, T]; // Counts int<lower=0> K; // Upper bounds of population size } model { : // Likelihood for (i in 1:R) { vector[K+1] lp; for (n in max_y[i]:K) { lp[n + 1] <- binomial_log(y[i], n, p) + poisson_log(n, lambda); } increment_log_prob(log_sum_exp(lp[(max_y[i] + 1):(K + 1)])); } : }
- 39. Computing times OpenBUGS Stan (K=100) Number of chains 3 4 Burn-in or warmup + iterations / chain 200 + 1000 500 + 1000 Computing time (sec) 25.4 1007.8 Effective sample size of λ 3000 401 Eff. sample size / time (sec-1) 118.3 0.4 Environment: 2.8 GHz Xeon W3530, Ubuntu 14.04, No parallell computing. Compilation time is not included in Stan. Times were measured using system.time(). The values are mean of 3 measurements.
- 40. Summary • Stan can deal with models including discrete parameters by marginalizing. • Divide and sum up every cases • However, the formulation used are less straightforward.