斯坦的数据增强概率回归 [英] Probit regression with data augmentation in stan
问题描述
我正在尝试使用stan进行数据增强的概率模型.这是我们得到结果y
或0/1的地方,它告诉我们潜在变量ystar
的符号.到目前为止,这就是我所拥有的,但是我不确定如何在model
部分中添加有关y
的信息.有什么想法吗?
I am attempting to do a probit model with data augmentation using stan. This is where we have outcomes y
either 0/1 that tell us the sign of the latent variable ystar
. This is what I have so far, but I'm not sure how to add information in the model
section about y
. Any thoughts?
data {
int<lower=0> N; // number of obs
int<lower=0> K; // number of predictors
int<lower=0,upper=1> y[N]; // outcomes
matrix[N, K] x; // predictor variables
}
parameters {
vector[K] beta; // beta coefficients
vector[N] ystar; // latent variable
}
model {
vector[N] mu;
beta ~ normal(0, 100);
mu <- x*beta;
ystar ~ normal(mu, 1);
}
推荐答案
您可以
data {
int<lower=0> N; // number of obs
int<lower=0> K; // number of predictors
vector<lower=-1,upper=1> sign; // y = 0 -> -1, y = 1 -> 1
matrix[N, K] x; // predictor variables
}
parameters {
vector[K] beta; // beta coefficients
vector<lower=0>[N] abs_ystar; // latent variable
}
model {
beta ~ normal(0, 100);
// ignore the warning about a Jacobian from the parser
sign .* abs_ystar ~ normal(x * beta, 1);
}
You could do
data {
int<lower=0> N; // number of obs
int<lower=0> K; // number of predictors
vector<lower=-1,upper=1> sign; // y = 0 -> -1, y = 1 -> 1
matrix[N, K] x; // predictor variables
}
parameters {
vector[K] beta; // beta coefficients
vector<lower=0>[N] abs_ystar; // latent variable
}
model {
beta ~ normal(0, 100);
// ignore the warning about a Jacobian from the parser
sign .* abs_ystar ~ normal(x * beta, 1);
}
也就是说,没有理由在Stan中为二进制概率模型进行数据增强,除非某些结果丢失或有所遗漏.这样做更直接(并将参数空间减小为K而不是K + N)
data {
int<lower=0> N; // number of obs
int<lower=0> K; // number of predictors
int<lower=0,upper=1> y[N]; // outcomes
matrix[N, K] x; // predictor variables
}
parameters {
vector[K] beta; // beta coefficients
}
model {
vector[N] mu;
beta ~ normal(0, 100);
mu <- x*beta;
for (n in 1:N) mu[n] <- Phi(mu[n]);
y ~ bernoulli(mu);
}
如果您真正关心潜在的实用程序,则可以通过generated quantities
块中的拒绝采样来生成它,如下所示
generated quantities {
vector[N] ystar;
{
vector[N] mu;
mu <- x * beta;
for (n in 1:N) {
real draw;
draw <- not_a_number();
if (sign[n] == 1) while(!(draw > 0)) draw <- normal_rng(mu[n], 1);
else while(!(draw < 0)) draw <- normal_rng(mu[n], 1);
ystar[n] <- draw;
}
}
}
That said, there is no reason to do data augmentation in Stan for a binary probit model, unless some of the outcomes were missing or something. It is more straightforward (and reduces the parameter space to K instead of K + N) to do
data {
int<lower=0> N; // number of obs
int<lower=0> K; // number of predictors
int<lower=0,upper=1> y[N]; // outcomes
matrix[N, K] x; // predictor variables
}
parameters {
vector[K] beta; // beta coefficients
}
model {
vector[N] mu;
beta ~ normal(0, 100);
mu <- x*beta;
for (n in 1:N) mu[n] <- Phi(mu[n]);
y ~ bernoulli(mu);
}
If you really care about the latent utility, you could generate it via rejection sampling in the generated quantities
block, like this
generated quantities {
vector[N] ystar;
{
vector[N] mu;
mu <- x * beta;
for (n in 1:N) {
real draw;
draw <- not_a_number();
if (sign[n] == 1) while(!(draw > 0)) draw <- normal_rng(mu[n], 1);
else while(!(draw < 0)) draw <- normal_rng(mu[n], 1);
ystar[n] <- draw;
}
}
}
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