Formula and Auto prior from brms

[jandraor]

For discussion on 15/08/2022:

• How will users supply priors?
• How will users supply the measurement model (likelihood function/sampling statement)?

[hyunjimoon]

From https://hyunjimoon.github.io/SBC/articles/brms.html,

Auto-prior class and formula

priors_func <- prior(normal(0,1), class = "b") +
  prior(normal(5,1), class = "Intercept") +
  prior(normal(0,5), class = "sigma") +
  prior(normal(0,0.75), class = "sd")

backend_func <- SBC_backend_brms(y ~ x + (1 | group),  
                            prior = priors_func, chains = 1,
                            template_data = datasets_func$generated[[1]])

1. what are the benefits of using class = "b", "sigma", "sd"? (more detailed level than for stan block parsing)
2. str "normal(0,1)" VS str "normal" real[] (0,1)
3. Example to persuade using formula is useful for hierarchical ODE

From https://magesblog.com/post/2021-02-08-fitting-multivariate-ode-models-with-brms/

Auto-parameterization

  prior(normal(0,5), class = "sigma") +
  prior(normal(0,0.75), class = "sd")

for good modeling practice (grid search from hypercube)

7. How useful is standardizing as below (specific references?) Can we automate?

for efficient computation

8. Hierarchical auto-parameterization from Mike's

• Hier-ADC here

and

data {

int<lower=0> N; // Number of observations vector[N] y; // Observations real<lower=0> sigma; // Measurement variability

// Number of individual contexts in hierarchy int<lower=0> K; // Individual context from which each observation is generated int<lower=1, upper=K> indiv_idx[N];

int<lower=0, upper=K> K_ncp; // Number of noncentered individuals int<lower=1, upper=K> ncp_idx[K_ncp]; // Index of noncentered individuals

int<lower=0, upper=K> K_cp; int<lower=1, upper=K> cp_idx[K_cp];

// Number of centered individuals // Index of noncentered individuals

}

parameters { real mu; real<lower=0> tau; vector[K_ncp] theta_ncp; vector[K_cp] theta_cp; }

// Population location // Population scale // Non-centered individual parameters // Ccentered individual parameters

transformed parameters { // Recentered individual parameters vector[K] theta; theta[ncp_idx] = mu + tau * theta_ncp; theta[cp_idx] = theta_cp; }

model {

mu ~ normal(0, 5); tau ~ normal(0, 5);

// Prior model // Prior model

theta_ncp ~ normal(0, 1); // Non-centered hierarchical model theta_cp ~ normal(mu, tau); // Centered hierarchical model

y ~ normal(theta[indiv_idx], sigma); // Observational model

}

[hyunjimoon]

[hyunjimoon]

3. How useful is standardizing (specific references?) + automation?
4. Hierarchical auto-parameterization

data {

int<lower=0> N; // Number of observations vector[N] y; // Observations real<lower=0> sigma; // Measurement variability

// Number of individual contexts in hierarchy int<lower=0> K; // Individual context from which each observation is generated int<lower=1, upper=K> indiv_idx[N];

int<lower=0, upper=K> K_ncp; // Number of noncentered individuals int<lower=1, upper=K> ncp_idx[K_ncp]; // Index of noncentered individuals

int<lower=0, upper=K> K_cp; int<lower=1, upper=K> cp_idx[K_cp];

// Number of centered individuals // Index of noncentered individuals

}

parameters { real mu; real<lower=0> tau; vector[K_ncp] theta_ncp; vector[K_cp] theta_cp; }

// Population location // Population scale // Non-centered individual parameters // Ccentered individual parameters

transformed parameters { // Recentered individual parameters vector[K] theta; theta[ncp_idx] = mu + tau * theta_ncp; theta[cp_idx] = theta_cp; }

model {

mu ~ normal(0, 5); tau ~ normal(0, 5);

// Prior model // Prior model

theta_ncp ~ normal(0, 1); // Non-centered hierarchical model theta_cp ~ normal(mu, tau); // Centered hierarchical model

y ~ normal(theta[indiv_idx], sigma); // Observational model

}