Normal Likelihood and Conjugate Prior Sample Clauses

Normal Likelihood and Conjugate Prior. As explained in the background, after a logarithm transformation, the Armitage-Doll mul- tistage carcinogenesis model can be written as the format of log-linear model: log(r) = α + (s − 1) log(t). To simplify the expression, the above linear model can be expressed as y = Xβ + ϵ, where y is an n × 1 vector of observations, X is an n × p design matrix and β is a p × 1 vector of parameters and ϵ an n × 1 vector of random errors. In this example, y = log(r), X = (1, log(t)), and β = (α, s − 1)r. In most scenarios, the error terms ϵ are assumed have a N (0, σ2) distribution and the model is described as the normal linear model where y|β, σ2 ∼ N (Xβ, σ2 I) and I is an n × n identity matrix. The likelihood takes the format of f (y|β, σ2) = (2πσ2)—n/2 exp{−(y − Xβ)r(y − Xβ)/(2σ2)}. The Normal-inverse-gamma (NIG) prior is then employed to define parameters β and σ2 in the form of f (β, σ2) ∝ (σ2)—(d+p+2)/2 exp{−[(β − m)rV —1(β − m) + a]/(2σ2)}, where a, d ∈ R. The prior distribution can be written as (β, σ2) ∼ NIG(a, d, m, V ) and the normal-inverse gamma prior can be equivalently derived from σ2 ∼ IG(a, d) and β|σ2 ∼ N (m, σ2 V ). We can also obtain weak prior information about (β, σ2) within the conjugate family by letting prior variances go to infinity [X’Xxxxx, 1994]. If parameters a and d in the inverse gamma prior for variance σ2 are taken to be 0, we can derive an improper prior for σ2 as f (σ2) ∝ σ—2 based on the probability density function for inverse gamma distribution. In addition, if we let V —1 → 0, we can simplify the joint prior f (β, σ2) ∝ σ—(p+2). The posterior can be derived after combining the likelihood and the prior. By the conjugate structure, the posterior is still with normal-inverse-gamma f (β, σ2|y) ∝ f (y|β, σ2)f (β, σ2) ∝ (σ2)—(d+n+p+2)/2 exp{−Q/(2σ2)} where Q = (y − Xβ)r(y − Xβ) + (β − m)rV —1(β − m) + a = (β − m∗)r(V ∗)—1(β − m∗) + a∗, and the posterior distribution can be written as (β, σ2|y) ∼ NIG(a∗, d∗, m∗, V ∗) where V ∗ = (V —1 + XrX)—1, m∗ = (V —1 + XrX)—1(V —1m + Xry), a∗ = a + mrV —1m + yry − (m∗)r(V ∗)—1m∗, d∗ = d + n. If XrX is non-singular, the posterior estimate of β which is E(β|y) = m∗ can be written as m∗ = (V —1 + XrX)—1(V —1m + XrXβˆ) = (I − A)m + Aβˆ, where A = (V —1 + XrX)—1XrX. In the log-normal Armitage-Doll model, we have logti n (XrX)—1 = ,. Σ Σ logti Σ —1   Σ (logti)2 , . and Xry = . X xxxxx . ,  logti logri then we can calculate the classical estimate βˆ = (XrX)—1(Xry) and the posterior mean of β is a w...