Explicit sum vs. matrix multiplication

library(rstan)

## Loading required package: Rcpp
## Loading required package: inline
## 
## Attaching package: 'inline'
## 
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## 
## rstan (Version 2.7.0-1, packaged: 2015-07-17 18:12:01 UTC, GitRev: 05c3d0058b6a)
## For execution on a local, multicore CPU with excess RAM we recommend calling
## rstan_options(auto_write = TRUE)
## options(mc.cores = parallel::detectCores())

explicit_sum <- "
data {
  int<lower=0> N;

  vector[N] v1;
  vector[N] v2;
  vector[N] v3;
  vector[N] v4;

  vector[N] y;
}

parameters {
  real<lower=0.0001> sigma;
  real c[4];
}

transformed parameters {
  vector[N] theta;
  theta <- c[1] * v1 + c[2] * v2 + c[3] * v3 + c[4] * v4; 
}

model {
  c ~ normal(0, 10);
  sigma ~ cauchy(0, 25);

  y ~ normal(theta, sigma);
}
"

sum_via_matrix <- "
data {
  int<lower=0> N;
  
  matrix[4, N] V;

  row_vector[N] y;
}

parameters {
  real<lower=0.0001> sigma;
  row_vector[4] c;
}

transformed parameters {
  row_vector[N] theta;
  theta <- c * V; 
}

model {
  c ~ normal(0, 10);
  sigma ~ cauchy(0, 25);
  
  y ~ normal(theta, sigma);
}
"


set.seed(42)
N <- 10000
y <- rnorm(N)
v1 <- rnorm(N)
v2 <- rnorm(N)
v3 <- rnorm(N)
v4 <- rnorm(N)

V <- matrix( c(v1, v2, v3, v4), byrow = T, ncol=N)

fit1 <- stan(model_code = explicit_sum, seed = 42)

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fit2 <- stan(model_code = sum_via_matrix, seed = 42)

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c1_1 <- extract(fit1, "c[1]")$c
mean(c1_1)

## [1] -0.007307882

sd(c1_1)

## [1] 0.01011407

c1_2 <- extract(fit2, "c[1]")$c
mean(c1_2)

## [1] -0.007190169

sd(c1_2)

## [1] 0.01010169