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Sep 28, 2015, 9:31:47 PM9/28/15

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Hi,

I've been playing around with a mixed Gaussian model with a latent variable and have tried to run this model in stan. Here is a brief description of the model.

Two Gaussians are mixed with the weigh parameter \q, and the latent variable \z turns on and off each one of the Gaussians. \z follows the Bernoulli with \q. Simply, the model is

f(y_i| q, theta1, theta2) = q*Normal1(theta1) + (1-q)*Normal2(theta2)

and

f(y_i| q, z_i, theta1, theta2) = Normal1(theta1)^z_i * Normal2(theta2)^(1-z_i)

finally,

z_i ~ Bernoulli(q).

Using stan, I want to estimate theta1, theta2, q and z, among which z is a vector of 0,1 and gives how many times each one of Gaussians is chosen in the simulation. (As Swupnil kindly pointed out) This model boils down to finding the posterior: p(y | z, theta1, theta2) p(z | q) p(q) p(theta1) p(theta2) because the likelihood of y is not anymore dependent on q when it is conditioned on z.

Here is my code, which does not work at all..

##normal

latent<-

"

data{

int<lower=0> N;

real y[N];

}

parameters{

real signal_mu;

real<lower=0,upper=10> signal_sig;

real noise_mu;

real<lower=0,upper=100> noise_sig;

real<lower=0,upper=1> q;

int<lower=0,upper=1> z[N];

}

model{

real ps[N];

for(i in 1:N){

ps[i] <- z[i]*normal_log(y[i],signal_mu,signal_sig) +

(1-z[i])*normal_log(y[i],noise_mu,noise_sig);

}

increment_log_prob(log_sum_exp(ps));

signal_sig~uniform(0,10);

noise_sig~uniform(0,100);

z~bernoulli(q);

}

"

N<-length(data_all$bias)

y<-data_all$bias

library(rstan)

fit<-stan(model_code = "latent",data=c("N","y"),iter=100,chain=2)

The following error sign keeps coming out.

integer parameters or transformed parameters are not allowed; found declared type int, parameter name=z

Problem with declaration.

Then I checked out the manual and noticed that stan does not support sampling discrete parameters and I have to marginalize out those parameters. And I tried to derive the likelihood function marginalized on z

p(y | theta1, theta2, q) = sum_z (p(z,y | theta1, theta2, q)

= sum_z (p(z | q)*p(y | theta1, theta2, z, q)

= sum_z (p(z | q)*p(y | theta1, theta2, z)

= sum_z Bernoulli (z_i, q) * Normal1(theta1)^z_i * Normal2(theta2)^(1-z_i)

But I do not know how to code this in stan mainly because I still feel confused with the latent variable z, especially where and how to declare it. Could you kindly tell me how to make my code work? Any advice would be greatly appreciated!

Jangho

Sep 29, 2015, 5:06:16 PM9/29/15

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That's not right. Check out the mixtures chapter of the manual

and work through what log_sum_exp is doing.

The correct form for a mixture is this:

for (n in 1:N) {

real lp1;

real lp2;

lp1 <- bernoulli_log(1, q) + normal_log(y[i], mu[1], sigma[1]);

lp2 <- bernoulli_log(0, q) + normal_log(y[i], mu[2], sigma[2]);

increment_log_prob(log_sum_exp(lp1,lp2));

}

or

for (n in 1:N)

increment_log_prob(log_mix(q,

normal_log(y[i], mu[1], sigma[1]),

normal_log(y[i], mu[2], sigma[2])));

- Bob

>

> for(i in 1:N){

> ps[1] <- z[i]*normal_log(y[i],signal_mu,signal_sig); +

> --

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and work through what log_sum_exp is doing.

The correct form for a mixture is this:

for (n in 1:N) {

real lp1;

real lp2;

lp1 <- bernoulli_log(1, q) + normal_log(y[i], mu[1], sigma[1]);

lp2 <- bernoulli_log(0, q) + normal_log(y[i], mu[2], sigma[2]);

increment_log_prob(log_sum_exp(lp1,lp2));

}

or

for (n in 1:N)

increment_log_prob(log_mix(q,

normal_log(y[i], mu[1], sigma[1]),

normal_log(y[i], mu[2], sigma[2])));

- Bob

>

> for(i in 1:N){

> ps[1] <- z[i]*normal_log(y[i],signal_mu,signal_sig); +

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Sep 30, 2015, 1:11:21 PM9/30/15

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Thanks Bob for your reply. Actually, I did try the second form before and it worked fine. The problem is that I want to know not only the mixture weight \q but also the latent variable \z for each data point (in each iteration). In the above mixture model, one of two Gaussians is selected according to the z_i and I want to see whether 1 or 0 has been assigned to each data point. The primary reason I am doing this is to remove outliers (noise points) in my data. If a certain data point has many 0s, then it means this point comes from the noise distribution and I can remove it based on some criterion such as 95% of zeros in the iterations. (In the actual model, the signal distribution is the exponential normal, and the noise distribution is the normal with a large standard deviation.)

I wanted to code this up in stan but it is not working as you can see above. It would be appreciated, if you could give some advice!

JH

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Sep 30, 2015, 2:34:06 PM9/30/15

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There is no “true latent variable”. Because you don’t measure the latent variable

directly you’ll always be uncertain about it and hence have some mixture of 0 and 1,

which is what q tells you directly. If you want to remove outliers you can now do

something well posed, like construct a loss function and choose the option that

minimizes expected loss. Or just take z = 1 if q > 0.5 and z = 0 otherwise.

Oct 1, 2015, 5:34:38 PM10/1/15

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Thank you so much for your reply. Reading your email, I just realized that the q in my simulation should have been estimated for each data point. I ran it again and I was able to use the method you suggested. It worked perfectly. Thanks again. I really appreciate it.

JH

Nov 1, 2016, 2:22:02 PM11/1/16

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Hi all,

I'm trying to properly code a marginalized latent discrete parameter and I'm having trouble. I know that I am writing the code improperly, but I'm not sure what is wrong and wondering if someone can please help. The model is as follows:

**I simulated data using the following code:**

set.seed(135)

n <- 1000

mu <- 0.1 # background infection rate

rho <- 0.90 # detection probability

phi <- 0.95 # 1-probabiity of false detection (probbaility not detected when not present)

pi <- 0.2

covariates <- replicate(3, rnorm(n, 0, 1))

colnames(covariates) <- c('X1', 'X2', 'X3')

X <- cbind(covariates)

coefs <- c(1, -0.5, 0.3)

resp <- X %*% coefs + mu

inv.logit <- function(x){

exp(x)/(exp(x)+1)

}

psi <- inv.logit(resp)

z <- rbinom(n, 1, psi*rho + (1-psi)*(1-phi))

y <- rbinom(n, 1, z*pi)

**Here is the stan model: **

# specify model in stan

mod7 <- '

data{

int<lower=1> N;

int y[N];

int<lower=1> K;

matrix[N,K] X;

}

parameters{

vector[K] beta;

real<lower=0,upper=1> mu;

real<lower=0,upper=1> rho;

real<lower=0,upper=1> phi;

real<lower=0,upper=1> pi;

}

transformed parameters{

vector[N] Xbeta;

vector[N] psi;

vector[N] omega;

Xbeta <- X*beta;

for(n in 1:N){

psi[n] <- inv_logit(mu + Xbeta[n]);

}

}

model{

// priors - note strong priors just to see if I can get model to fit

beta ~ normal(0,1);

rho ~ beta(31.5, 3.5);

phi ~ beta(10, 1.47);

mu ~ beta(3.5, 31.5);

for(n in 1:N){

real z[2];

if (y[n] > 0){

increment_log_prob(bernoulli_log(y[n], psi[n]*rho + (1-psi[n])*(1-phi)));

z[1] <- bernoulli_log(1, psi[n]*rho);// true presence

z[2] <- bernoulli_log(0, (1-psi[n])*(1-phi)); // false positive

increment_log_prob(log_sum_exp(z));

} else {

z[1] <- bernoulli_log(1, psi[n]*(1-rho)); // false negative

z[2] <- bernoulli_log(0, (1-psi[n])*phi); // true negative

increment_log_prob(log_sum_exp(z));

}

increment_log_prob(bernoulli_log(y[n], z[1]*pi));

}

}'

mod7c <- stan_model(model_code=mod7)

mod7.fit <- sampling(mod7c, dat, iter=1000, chains=4, thin=1)

**I get several errors including:**

I know that I'm improperly marginalizing the parameter, but I don't know how to fix it. I've played with it a lot, but I haven't been successful. I'd appreciate any help you all can provide.

Thank you,

Mikey

I'm trying to properly code a marginalized latent discrete parameter and I'm having trouble. I know that I am writing the code improperly, but I'm not sure what is wrong and wondering if someone can please help. The model is as follows:

Process model:

logit(psi[i]) = mu + **X*****Beta**

Observation model:

z[i] ~ Bernoulli(psi[i]*rho + (1-psi[i])*(1-phi))

Data model:

y[i] ~ Bernoulli(z[i]*pi)

set.seed(135)

n <- 1000

mu <- 0.1 # background infection rate

rho <- 0.90 # detection probability

phi <- 0.95 # 1-probabiity of false detection (probbaility not detected when not present)

pi <- 0.2

covariates <- replicate(3, rnorm(n, 0, 1))

colnames(covariates) <- c('X1', 'X2', 'X3')

X <- cbind(covariates)

coefs <- c(1, -0.5, 0.3)

resp <- X %*% coefs + mu

inv.logit <- function(x){

exp(x)/(exp(x)+1)

}

psi <- inv.logit(resp)

z <- rbinom(n, 1, psi*rho + (1-psi)*(1-phi))

y <- rbinom(n, 1, z*pi)

# specify model in stan

mod7 <- '

data{

int<lower=1> N;

int y[N];

int<lower=1> K;

matrix[N,K] X;

}

parameters{

vector[K] beta;

real<lower=0,upper=1> mu;

real<lower=0,upper=1> rho;

real<lower=0,upper=1> phi;

real<lower=0,upper=1> pi;

}

transformed parameters{

vector[N] Xbeta;

vector[N] psi;

vector[N] omega;

Xbeta <- X*beta;

for(n in 1:N){

psi[n] <- inv_logit(mu + Xbeta[n]);

}

}

model{

// priors - note strong priors just to see if I can get model to fit

beta ~ normal(0,1);

rho ~ beta(31.5, 3.5);

phi ~ beta(10, 1.47);

mu ~ beta(3.5, 31.5);

for(n in 1:N){

real z[2];

if (y[n] > 0){

increment_log_prob(bernoulli_log(y[n], psi[n]*rho + (1-psi[n])*(1-phi)));

z[1] <- bernoulli_log(1, psi[n]*rho);// true presence

z[2] <- bernoulli_log(0, (1-psi[n])*(1-phi)); // false positive

increment_log_prob(log_sum_exp(z));

} else {

z[1] <- bernoulli_log(1, psi[n]*(1-rho)); // false negative

z[2] <- bernoulli_log(0, (1-psi[n])*phi); // true negative

increment_log_prob(log_sum_exp(z));

}

increment_log_prob(bernoulli_log(y[n], z[1]*pi));

}

}'

mod7c <- stan_model(model_code=mod7)

mod7.fit <- sampling(mod7c, dat, iter=1000, chains=4, thin=1)

"Exception thrown at line 64: stan::math::bernoulli_log: Probability parameter is -1.64453, but must be between (0, 1)"

I know that I'm improperly marginalizing the parameter, but I don't know how to fix it. I've played with it a lot, but I haven't been successful. I'd appreciate any help you all can provide.

Thank you,

Mikey

Nov 1, 2016, 6:27:47 PM11/1/16

to stan-...@googlegroups.com

> On Nov 1, 2016, at 2:22 PM, Mikey T <tab...@gmail.com> wrote:

>

> Hi all,

> I'm trying to properly code a marginalized latent discrete parameter and I'm having trouble. I know that I am writing the code improperly, but I'm not sure what is wrong and wondering if someone can please help. The model is as follows:

> Process model:

>

> logit(psi[i]) = mu + X*Beta

>

>

>

> Observation model:

>

> z[i] ~ Bernoulli(psi[i]*rho + (1-psi[i])*(1-phi))

>

>

>

> Data model:

>

> y[i] ~ Bernoulli(z[i]*pi)

crank to marginalize, then just code it up directly in Stan. First,

you have the joint distribution of the latent z and observed y based

on parameters

p(z, y | mu, X, beta, rho, psi, phi)

= PRODUCT_i p(z[i], y[i] | ...)

and then you can work on the inner term:

p(z[i], y[i] | ...)

= Bernoulli(z[i] | psi[i]*rho + (1-psi[i])*(1-phi))

* Bernoulli(y[i] | z[i] * pi)

where

psi[i] = inv_logit(mu + X * Beta); // sure that's not X[i] * Beta?

and then marginalize out the z[i]:

= SUM_{k in 0:1}

Bernoulli(k | psi[i]*rho + (1-psi[i])*(1-phi))

* Bernoulli(y[i] | k * pi)

Then put it on the log scale:

log(p(z[i], y[i] | ...)

= log(SUM_{k in 0:1}

Bernoulli(k | psi[i]*rho + (1-psi[i])*(1-phi))

* Bernoulli(y[i] | k * pi)

= log_sum_exp_{k in 0:1}

log Bernoulli(k | psi[i]*rho + (1-psi[i])*(1-phi))

+ log Bernoulli(y[i] | k * pi)

and you can just code that up directly using log_sum_exp

in Stan:

psi = inv_logit

log_sum_exp(bernoulli_lpmf(0 | psi[i] * rho + (1-psi[i]) * (1-phi))

+ bernoulli_lpmf(y[i] | 0),

bernoulli_lpmf(1 | psi[i] * rho + (1-psi[i]) * (1-phi))

+ bernoulli_lpmf(y[i] | pi));

You can also look at the zero-inflation and hurdle models

in the manual, which are similar.

- Bob

Nov 2, 2016, 10:15:06 AM11/2/16

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Hi Bob,

Thanks a lot for your help with this. I really appreciate your clear and thorough explanation. I also appreciate all of your work on the software and manuals.

Thank you,

Mikey

Thanks a lot for your help with this. I really appreciate your clear and thorough explanation. I also appreciate all of your work on the software and manuals.

Thank you,

Mikey

Nov 2, 2016, 3:48:30 PM11/2/16

to stan-...@googlegroups.com

> On Nov 2, 2016, at 10:15 AM, Mikey T <tab...@gmail.com> wrote:

>

> Hi Bob,

> Thanks a lot for your help with this. I really appreciate your clear and thorough explanation.

branch this on y[i] == 0 vs. y[i] == 1,

because the first term is -infinity when y[i] == 1.

real psi;

real alpha;

psi = inv_logit(mu + X[i] * Beta);

alpha = psi * rho + (1 - psi) * (1 - phi);

if (y[i] == 1)

target += bernoulli_lpmf(1 | alpha) + bernoulli_lpmf(1 | pi);

else

target += log_sum_exp(bernoulli_lpmf(0 | alpha),

bernoulli_lpmf(1 | alpha)

+ bernoulli_lpmf(0 | pi));

It'll avoid NaN values in derivatives (not sure if they'll pop

up with a -infinity term and then an exp).

It'd be more arithmetically stable if you stayed on the logit scale

for alpha and used bernoulli_logit_lpmf.

You'll want to double-check the algebra, too.

- Bob

Nov 7, 2016, 11:42:57 AM11/7/16

to Stan users mailing list

Hi Bob,

Thanks again for your help. I have run both of these suggestions and tried tinkering with the code slightly, but I am not able to get the chains to converge. If I put informed priors on the parameters, some of the chains correctly estimate while others are estimating a completely different area of parameter space. However, the model works well if I remove the last two components of the logit function [i.e., (1-psi[i])*(1-phi)].

Also, I'm wondering if the code you recommended last should be modified so that the probability of each condition is explicitly expressed in the conditional. I'm pasting this code here, but it did not work either.

// separate out detection components?

for(n in 1:N){

if(y[n] ==1){

target += log_sum_exp(bernoulli_lpmf(1|psi[n]*rho),

bernoulli_lpmf(0|(1-psi[n])*(1-phi))

+ bernoulli_lpmf(1|pi));

} else{

target += log_sum_exp(bernoulli_lpmf(0|(1-psi[n])*phi),

bernoulli_lpmf(1|psi[n]*(1-rho))

+ bernoulli_lpmf(0|pi));

}

}

I appreciate your help.

I have attached all R code below in case this helps.

Thank you,

Mikey

# r code to simulate data

Thanks again for your help. I have run both of these suggestions and tried tinkering with the code slightly, but I am not able to get the chains to converge. If I put informed priors on the parameters, some of the chains correctly estimate while others are estimating a completely different area of parameter space. However, the model works well if I remove the last two components of the logit function [i.e., (1-psi[i])*(1-phi)].

Also, I'm wondering if the code you recommended last should be modified so that the probability of each condition is explicitly expressed in the conditional. I'm pasting this code here, but it did not work either.

// separate out detection components?

for(n in 1:N){

if(y[n] ==1){

target += log_sum_exp(bernoulli_lpmf(1|psi[n]*rho),

bernoulli_lpmf(0|(1-psi[n])*(1-phi))

+ bernoulli_lpmf(1|pi));

} else{

target += log_sum_exp(bernoulli_lpmf(0|(1-psi[n])*phi),

bernoulli_lpmf(1|psi[n]*(1-rho))

+ bernoulli_lpmf(0|pi));

}

}

I appreciate your help.

I have attached all R code below in case this helps.

Thank you,

Mikey

# r code to simulate data

set.seed(135)

n <- 1000

mu <- 0.1 # background infection rate

rho <- 0.90 # detection probability

phi <- 0.95 # 1-probabiity of false detection (probbaility not detected when not present)

pi <- 0.2

covariates <- replicate(3, rnorm(n, 0, .5))

colnames(covariates) <- c('X1', 'X2', 'X3')

X <- cbind(covariates)

coefs <- c(1, -0.5, 0.3)

resp <- X %*% coefs + mu

inv.logit <- function(x){

exp(x)/(exp(x)+1)

}

psi <- inv.logit(resp)

#z <- rbinom(n, 1, psi*rho) # ignoring false detection here. model works when I do this

z <- rbinom(n, 1, psi*rho + (1-psi)*(1-phi))

y <- rbinom(n, 1, z*pi)

# specify model in stan

model <- '

data{

int<lower=1> N;

int y[N];

int<lower=1> K;

matrix[N,K] X;

}

parameters{

vector[K] beta;

real<lower=0,upper=1> mu;

real<lower=0,upper=1> rho;

real<lower=0,upper=1> phi;

real<lower=0,upper=1> pi;

}

transformed parameters{

vector[N] Xbeta;

vector[N] psi;

vector[N] omega;

Xbeta = X*beta;

for(n in 1:N){

psi[n] = inv_logit(mu + Xbeta[n]);

omega[n] = psi[n]*rho +(1-psi[n])*(1-phi);

//omega[n] = psi[n]*rho;

//omega[n] = inv_logit(psi[n]*rho +(1-psi[n])*(1-phi));

}

}

model{

// priors

beta ~ normal(0,1);

//beta[1] ~ normal(1,1);

//beta[2] ~ normal(-.5, 1);

//beta[3] ~ normal(0.3, 1);

rho ~ beta(3.92, 1.32); //beta(31.5, 3.5); //

phi ~ beta(3.46, 1.13); // beta(17.1, 0.9); //

mu ~ beta(1.32, 3.92); //beta(3.5, 31.5); //

pi ~ beta(3.28, 10.12); //beta(12.6, 50.4); //

// separate out detection components

//for(n in 1:N){

// if(y[n] ==1){

// target += log_sum_exp(bernoulli_lpmf(1|psi[n]*rho),

// bernoulli_lpmf(0|(1-psi[n])*(1-phi))

// + bernoulli_lpmf(1|pi));

// } else{

// target += log_sum_exp(bernoulli_lpmf(0|(1-psi[n])*phi),

// bernoulli_lpmf(1|psi[n]*(1-rho))

// + bernoulli_lpmf(0|pi));

// }

//}

// separating out conditional

//for(n in 1:N){

//if(y[n] == 1){

//target += bernoulli_lpmf(y[n]|omega[n]) + bernoulli_lpmf(y[n]|pi);

//} else{

//target += log_sum_exp(bernoulli_lpmf(0|omega[n]),

// bernoulli_lpmf(1|omega[n])

// + bernoulli_lpmf(y[n]|pi));

//}

//}

// one option

for(n in 1:N){

target += log_sum_exp(bernoulli_lpmf(0|omega[n]) +

bernoulli_lpmf(y[n]|0),

bernoulli_lpmf(1|omega[n])

+ bernoulli_lpmf(y[n]|pi));

}

}'

for(n in 1:N){

psi[n] = inv_logit(mu + Xbeta[n]);

omega[n] = psi[n]*rho +(1-psi[n])*(1-phi);

//omega[n] = psi[n]*rho;

//omega[n] = inv_logit(psi[n]*rho +(1-psi[n])*(1-phi));

}

}

model{

// priors

beta ~ normal(0,1);

//beta[1] ~ normal(1,1);

//beta[2] ~ normal(-.5, 1);

//beta[3] ~ normal(0.3, 1);

rho ~ beta(3.92, 1.32); //beta(31.5, 3.5); //

phi ~ beta(3.46, 1.13); // beta(17.1, 0.9); //

mu ~ beta(1.32, 3.92); //beta(3.5, 31.5); //

pi ~ beta(3.28, 10.12); //beta(12.6, 50.4); //

// separate out detection components

//for(n in 1:N){

// if(y[n] ==1){

// target += log_sum_exp(bernoulli_lpmf(1|psi[n]*rho),

// bernoulli_lpmf(0|(1-psi[n])*(1-phi))

// + bernoulli_lpmf(1|pi));

// } else{

// target += log_sum_exp(bernoulli_lpmf(0|(1-psi[n])*phi),

// bernoulli_lpmf(1|psi[n]*(1-rho))

// + bernoulli_lpmf(0|pi));

// }

//}

// separating out conditional

//for(n in 1:N){

//if(y[n] == 1){

//target += bernoulli_lpmf(y[n]|omega[n]) + bernoulli_lpmf(y[n]|pi);

//} else{

//target += log_sum_exp(bernoulli_lpmf(0|omega[n]),

// bernoulli_lpmf(1|omega[n])

// + bernoulli_lpmf(y[n]|pi));

//}

//}

// one option

for(n in 1:N){

target += log_sum_exp(bernoulli_lpmf(0|omega[n]) +

bernoulli_lpmf(y[n]|0),

bernoulli_lpmf(1|omega[n])

+ bernoulli_lpmf(y[n]|pi));

}

}'

Feb 16, 2017, 1:52:54 PM2/16/17

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Hi all,

I changed my model structure and I'm once again having trouble with the latent discrete parameter (Z). I think I marginalized it correctly, but I am having trouble getting the chains to converge. Here is the new model structure:

y_i ~ bernoulli(Z_i * rho + (1-Z_i)*(1-phi)

Z_i ~ bernoulli(psi_i)

logit(psi_i) = X_i***B**

I marginalized the latent parameter (Z), so that the code looks like this (in the model block):

for(n in 1:N){

target += log_sum_exp(bernoulli_lpmf(1|psi[n]) + bernoulli_lpmf(y[n]|rho),

bernoulli_lpmf(0|psi[n]) + bernoulli_lpmf(y[n]|(1-phi)));

}

The full code including code for simulation is below. (I have the same problem when I don't use those hyperpriors and just estimate rho and phi with the priors beta(1,1).)

Thank you in advance for your help,

Mikey

#- simulate some data

set.seed(135)

n <- 1000

I changed my model structure and I'm once again having trouble with the latent discrete parameter (Z). I think I marginalized it correctly, but I am having trouble getting the chains to converge. Here is the new model structure:

y_i ~ bernoulli(Z_i * rho + (1-Z_i)*(1-phi)

Z_i ~ bernoulli(psi_i)

logit(psi_i) = X_i*

for(n in 1:N){

target += log_sum_exp(bernoulli_lpmf(1|psi[n]) + bernoulli_lpmf(y[n]|rho),

bernoulli_lpmf(0|psi[n]) + bernoulli_lpmf(y[n]|(1-phi)));

}

The full code including code for simulation is below. (I have the same problem when I don't use those hyperpriors and just estimate rho and phi with the priors beta(1,1).)

Thank you in advance for your help,

Mikey

#- simulate some data

set.seed(135)

n <- 1000

rho <- 0.90 # detection probability

phi <- 0.95 # 1-probabiity of false detection (probbaility not detected when not present)

pi <- 0.2

covariates <- replicate(3, rnorm(n, 0, .5))

colnames(covariates) <- c('X1', 'X2', 'X3')

X <- cbind(rep(1, length(covariates[,1])),covariates)

coefs <- c(1, -1, -0.5, 0.3)

coefs <- c(1, -1, -0.5, 0.3)

resp <- X %*% coefs

inv.logit <- function(x){

exp(x)/(exp(x)+1)

}

psi <- inv.logit(resp)

exp(x)/(exp(x)+1)

}

psi <- inv.logit(resp)

z <- rbinom(n, 1, psi)

y <- rbinom(n, 1, z*rho +(1-z)*(1-phi))

y <- rbinom(n, 1, z*rho +(1-z)*(1-phi))

# specify model in stan

mod <-'

data{

int<lower=1> N;

int y[N];

int<lower=1> K;

matrix[N,K] X;

}

parameters{

vector[K] beta;

real<lower=0,upper=1> rho;

real<lower=0,upper=1> phi;

real<lower=0,upper=1> phi;

real<lower=0> sigma[K];

//int<lower=0> lp2[N];

real<lower=0,upper=1> mu_rho;

real<lower=0> sigma_rho;

real<lower=0,upper=1> mu_phi;

real<lower=0> sigma_phi;

//int<lower=0> lp2[N];

real<lower=0,upper=1> mu_rho;

real<lower=0> sigma_rho;

real<lower=0,upper=1> mu_phi;

real<lower=0> sigma_phi;

}

transformed parameters{

vector[N] Xbeta;

vector[N] psi;

Xbeta = X*beta;

for(n in 1:N){

for(n in 1:N){

psi[n] = inv_logit(Xbeta[n]);

}

}

model{

// hyperpriors

mu_rho ~ beta(42.573, 5.619222);

sigma_rho ~ gamma(10.63844, 0.2);

mu_phi ~ beta(42.573, 5.619222);

sigma_phi ~ gamma(10.63844, 0.2);

// priors

rho ~ beta(mu_rho*sigma_rho, (1-mu_rho)*sigma_rho);

phi ~ beta(mu_phi*sigma_phi, (1-mu_phi)*sigma_phi);

//sigma ~ cauchy(0,2.5);

//beta ~ normal(0,sigma);

beta ~ normal(0,1);

//**** Here is the marginalized parameter

for(n in 1:N){

target += log_sum_exp(bernoulli_lpmf(1|psi[n]) + bernoulli_lpmf(y[n]|rho),

bernoulli_lpmf(0|psi[n]) + bernoulli_lpmf(y[n]|(1-phi)));

}

}

generated quantities{

real<lower=0> mean_psi;

vector[N] log_lik;

real<lower=0> pi[N];

real<lower=0> mean_pi;

mean_psi = sum(psi)/N;

// calculating log likelihood to get waic

for(n in 1:N){

log_lik[n] = bernoulli_lpmf(y[n]|psi[n]);

pi[n] = psi[n]*(1-rho) + psi[n]*phi;

}

mean_pi = mean(pi);

}'

modc <- stan_model(model_code=mod)

}

}

model{

// hyperpriors

mu_rho ~ beta(42.573, 5.619222);

sigma_rho ~ gamma(10.63844, 0.2);

mu_phi ~ beta(42.573, 5.619222);

sigma_phi ~ gamma(10.63844, 0.2);

// priors

rho ~ beta(mu_rho*sigma_rho, (1-mu_rho)*sigma_rho);

phi ~ beta(mu_phi*sigma_phi, (1-mu_phi)*sigma_phi);

//sigma ~ cauchy(0,2.5);

//beta ~ normal(0,sigma);

beta ~ normal(0,1);

//**** Here is the marginalized parameter

for(n in 1:N){

target += log_sum_exp(bernoulli_lpmf(1|psi[n]) + bernoulli_lpmf(y[n]|rho),

bernoulli_lpmf(0|psi[n]) + bernoulli_lpmf(y[n]|(1-phi)));

}

}

generated quantities{

real<lower=0> mean_psi;

vector[N] log_lik;

real<lower=0> pi[N];

real<lower=0> mean_pi;

mean_psi = sum(psi)/N;

// calculating log likelihood to get waic

for(n in 1:N){

log_lik[n] = bernoulli_lpmf(y[n]|psi[n]);

pi[n] = psi[n]*(1-rho) + psi[n]*phi;

}

mean_pi = mean(pi);

}'

modc <- stan_model(model_code=mod)

Feb 16, 2017, 2:36:13 PM2/16/17

to stan-...@googlegroups.com

> On Feb 16, 2017, at 1:52 PM, Mikey T <tab...@gmail.com> wrote:

>

> Hi all,

> I changed my model structure and I'm once again having trouble with the latent discrete parameter (Z). I think I marginalized it correctly, but I am having trouble getting the chains to converge. Here is the new model structure:

>

> y_i ~ bernoulli(Z_i * rho + (1-Z_i)*(1-phi)

>

> Z_i ~ bernoulli(psi_i)

>

> logit(psi_i) = X_i*B

>

> I marginalized the latent parameter (Z), so that the code looks like this (in the model block):

>

> for(n in 1:N){

> target += log_sum_exp(bernoulli_lpmf(1|psi[n]) + bernoulli_lpmf(y[n]|rho),

> bernoulli_lpmf(0|psi[n]) + bernoulli_lpmf(y[n]|(1-phi)));

> }

psi = inv_logit(X * B);

(It's just shorter and more efficient written as above assuming X is

a matrix.)

It's more efficient to just the bernoulli_logit directly on X[n] * B (saving

the X[n] * B calculation to be reused).

If you want to do the inv_logit yourself, then you can just use

target += log_mix(psi[n], bernoulli_lpmf(y[n] | rho),

bernoulli_lpmf(y[n] | 1 - phi));

Your priors for the sigmas might be causing problems

>

> sigma_rho ~ gamma(10.63844, 0.2);

>

> sigma_phi ~ gamma(10.63844, 0.2);

because they're not on the unit scale (mean 0, variance 1) but

instead on a roughly (mean 11, variance 200) scale.

But it's not extreme enough it should cause the fitting to fail.

What actually happens when you run it?

You also don't need a beta(1, 1) prior for a probability --- that's

just uniform, and uniform is the default. But again, that's just

for efficiency.

- Bob

Feb 16, 2017, 3:07:27 PM2/16/17

to Stan users mailing list

Thank you for your quick response. When I run the model, the mean estimates for all of the parameters are good, but there are really large credible intervals. Looking at the traceplots (attached), the chains don't appear to be converging properly. Also, the R_hats and n_eff are off what they should be.

Just to be clear, are you suggesting that I replace:

Just to be clear, are you suggesting that I replace:

target += log_sum_exp(bernoulli_lpmf(1|psi[n]) + bernoulli_lpmf(y[n]|rho),

bernoulli_lpmf(0|psi[n]) + bernoulli_lpmf(y[n]|(1-phi)));

with:

target += log_mix(psi[n], bernoulli_lpmf(y[n] | rho),

bernoulli_lpmf(y[n] | 1 - phi));

I'm not sure if these are identical? In the attached traceplot I did make this substitution and I used gamma(1,0.001) for the priors on sigmas.

Feb 16, 2017, 3:27:19 PM2/16/17

to Stan users mailing list

I changed the priors for sigmas to gamma(2,0.0001), as I think this is more appropriate. As I mentioned, the estimates are getting close to proper values, but the n_eff is really low and Rhat is higher than it should be. Also, I get the error message that there are many divergent pairs. I attached the traceplot from using this prior.

Thank you for your help!

Thank you for your help!

Feb 22, 2017, 1:23:55 PM2/22/17

to stan-...@googlegroups.com

Noooo! You don't want those super-fat gamma priors.

Andrew Gelman has written a bunch of papers on this and goes over

it in BDA.

If you expect really big values for sigma, you might want to

rescale the problem.

- Bob

> <trace.pdf>

Andrew Gelman has written a bunch of papers on this and goes over

it in BDA.

If you expect really big values for sigma, you might want to

rescale the problem.

- Bob

May 18, 2017, 3:53:30 PM5/18/17

to Stan users mailing list

Hi all,

I'm working on another model now that has latent discrete parameters (this time they are unbounded, which makes it more complicated). I'll present with just the most simple model structure. It is an open population mixture model where I'm looking N (true population size) as a function of Survival (S) and gains (G), with n_it observations. I get the error message: "Rejecting initial value: Error evaluating log probability at initial value". I tried specifying initial values for omega and gamma, but maybe I am missing something.

I'm working on another model now that has latent discrete parameters (this time they are unbounded, which makes it more complicated). I'll present with just the most simple model structure. It is an open population mixture model where I'm looking N (true population size) as a function of Survival (S) and gains (G), with n_it observations. I get the error message: "Rejecting initial value: Error evaluating log probability at initial value". I tried specifying initial values for omega and gamma, but maybe I am missing something.

S_it ~ Binomial(N_it-1, omega)

G_it ~ Binomial(N_it-1, gamma)

N_it = S_it + G_it

n_it ~ Poisson(N_it)

Here is my Stan code:

'

data{

int<lower=0> N_cam; // number cameras (i)

int<lower=0> N_t; // number of occasions (t)

int<lower=0> obs[N_cam, N_t]; // observations (nit)

int<lower=1> K; // upper bound of population size could allow this to change at each time?

}

transformed data{

int<lower=0> max_obs[N_cam];

for(i in 1:N_cam){

max_obs[i] = max(obs[i]);

}

}

parameters{

real<lower=0,upper=1> gamma; // arrival rate

real<lower=0,upper=1> omega; // survival probability

//real<lower=0,upper=1> pi; // probability of detecting same individual more than once

real<lower=0> sigma;

}

transformed parameters{

}

model{

// specify priors

gamma ~ beta(1,1);

omega ~ beta(1,1);

for(i in 1:N_cam){

for(t in 2:N_t){

int S[i,t];

int G[i,t];

int N;

vector[K-max_obs[i] + 1] lp;

// try specifying values at timestep 1

S[i,1] = 10;

G[i,1] = 2;

// this is the part of the code that I know is incorrect, but Im not sure how to do it properly

for(j in 1:(K-max_obs[i] +1)){ // or for (j in 1:K)

lp[j] = binomial_lpmf(S[i,t]|(G[i,(t-1)] + S[i,(t-1)]), omega)

+ binomial_lpmf(G[i,t]|(G[i,(t-1)] + S[i,(t-1)]), gamma);

target += log_sum_exp(lp);

N = S[i,t] + G[i,t];

sigma ~ gamma(2,1);

//obs[i,t] ~ neg_binomial(-pow(mu,2)/(mu-sigma), mu/(mu-sigma));

obs[i,t] ~ poisson(N);

}

}

}

}',

# And here is the code to simulate data

mean_pop <- 20

ncam <- 10

Nt <- 5 # might need to make this specific to each camera

gamma <- 0.1 # arrival rate

omega <- 0.8 # survival rate

pi <- 0.2 # pr detecting more than once

S <- matrix(NA, ncam, Nt)

G <- matrix(NA, ncam, Nt)

N_tot <- matrix(NA, ncam, Nt)

N1 <- rpois(ncam, mean_pop)

N_tot[,1] <- N1

for(i in 1:ncam){

for(t in 2:Nt){

S[i,t] <- rbinom(1,N_tot[,t-1],(omega*(1-pi)))

G[i,t] <- rbinom(1,N_tot[,t-1], gamma)

N_tot[i,t] <- rpois(1,S[i,t] + G[i,t])

}

}

Thank you in advance for your help,

Mikey

May 18, 2017, 3:53:57 PM5/18/17

to Stan users mailing list

Thank you in advance for your help,

Mikey

On Wednesday, February 22, 2017 at 11:23:55 AM UTC-7, Bob Carpenter wrote:

May 18, 2017, 3:56:13 PM5/18/17

to Stan users mailing list

Sorry, Google cut off the actual model structure:

On Thursday, May 18, 2017 at 1:53:57 PM UTC-6, Mikey T wrote:

Hi all,

I'm working on another model now that has latent discrete parameters (this time they are unbounded, which makes it more complicated). I'll present with just the most simple model structure. It is an open population mixture model where I'm looking N (true population size) as a function of Survival (S) and gains (G), with n_it observations. I get the error message: "Rejecting initial value: Error evaluating log probability at initial value". I tried specifying initial values for omega and gamma, but maybe I am missing something.

<w:LsdException Locked="false" SemiHidden="true" UnhideWhenUsed

May 18, 2017, 4:24:44 PM5/18/17

to stan-...@googlegroups.com

Stan can't handle latent discrete unbounded parameters unless you have an analytic marginalization.

The variables you used in your math didn't match your code, so I couldn't follow where you had

the latent discrete parameter.

If you can't get it to initialize, it means you don't have a finite log density at initial values.

That's usually because you've given something an illegal argument somewhere. If you're not getting

error messages, I think that's a bug in 2.15 RStan that shut off the messages and we'll be putting

out a patch release soon. Sorry about that.

- Bob

The variables you used in your math didn't match your code, so I couldn't follow where you had

the latent discrete parameter.

If you can't get it to initialize, it means you don't have a finite log density at initial values.

That's usually because you've given something an illegal argument somewhere. If you're not getting

error messages, I think that's a bug in 2.15 RStan that shut off the messages and we'll be putting

out a patch release soon. Sorry about that.

- Bob

May 19, 2017, 11:00:28 AM5/19/17

to Stan users mailing list

Thank you Bob. I updated the math so that the symbols match the code. I used a maximum bound on population size (K) so that N (the latent discrete parameter) is bounded. The trouble is that S and G are also discrete and their sum makes up N.

S_it ~ Binomial(N_it-1, omega)

G_it ~ Binomial(N_it-1, gamma)

N_it = S_it + G_it

obs_it ~ Poisson(N_it)

//sigma ~ gamma(2,1);

//obs[i,t] ~
neg_binomial(-pow(mu,2)/(mu-sigma), mu/(mu-sigma));

obs[i,t] ~ poisson(N);

}

}

}

}'

# And here is the R code to simulate data

mean_pop <- 20

ncam <- 10

Nt <- 5 # might need to make this specific to each camera

gamma <- 0.1 # arrival rate

omega <- 0.8 # survival rate

pi <- 0.2 # pr detecting more than once

S <- matrix(NA, ncam, Nt)

G <- matrix(NA, ncam, Nt)

N_tot <- matrix(NA, ncam, Nt)

N1 <- rpois(ncam, mean_pop)

N_tot[,1] <- N1

for(i in 1:ncam){

for(t in 2:Nt){

S[i,t] <- rbinom(1,N_tot[,t-1],(omega*(1-pi)))

G[i,t] <- rbinom(1,N_tot[,t-1], gamma)

N_tot[i,t] <- rpois(1,S[i,t] + G[i,t])

}

}

data <- list(N_cam=ncam, N_t=Nt, obs=N_tot, K=1000)

Thank you for any help you can provide,Mikey

May 20, 2017, 3:16:45 PM5/20/17

to stan-...@googlegroups.com

> On May 19, 2017, at 11:00 AM, Mikey T <tab...@gmail.com> wrote:

>

> Thank you Bob. I updated the math so that the symbols match the code. I used a maximum bound on population size (K) so that N (the latent discrete parameter) is bounded. The trouble is that S and G are also discrete and their sum makes up N.

>

> S_it ~ Binomial(N_it-1, omega)

>

> G_it ~ Binomial(N_it-1, gamma)

>

> N_it = S_it + G_it

>

> obs_it ~ Poisson(N_it)

this is just a not-quite-TeX not-quite code way to write obs[i,t]

But then I don't see any indexed N in your code, just an N_t.

This gets really confusing when N_ sometimes is used for naming

and sometimes for subscripting. Just remember it's *a lot* harder

for someone to read math than it is to write it.

Is N_it-1 supposed to be N[i, t-1]. And only obs[i,t] is observed

among N, S, and G?

If so, you can't code this model in Stan---it's combinatorially

intractable and we don't let you code discrete parameters.

- Bob

May 22, 2017, 8:02:41 AM5/22/17

to Stan users mailing list

Hi Bob,

Yes-you are correct about the improper way in which I wrote out the math (I'm not sure why I wrote it this way). I understand and thank you for your help. I'll work on building the model differently.

Thank you,

Mikey

Yes-you are correct about the improper way in which I wrote out the math (I'm not sure why I wrote it this way). I understand and thank you for your help. I'll work on building the model differently.

Thank you,

Mikey

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