128 Chapter 6. Comparing Homogeneous and Inhomogeneous Time Markov Chains for Modelling Deterioration in Sewer Pipe Networks Initially, parameters ⇀ are sampled from the prior P(⇀|M). By applying Bayes’ theorem, the posterior distribution P(⇀|y, M) is expressed as: P(⇀|y, M)= f(y|⇀, M)P(⇀|M) P(y|M) , where the posterior P(⇀|y, M) updates beliefs about the parameters after observing data. The marginal likelihood P(y|M) is given by: P(y|M)= f(y|⇀, M)P(⇀|M)d⇀, reflecting how well model M, across all parameter values, explains the observed data. Since the computation of P(y|M) is complex, it is assumed that the posterior is proportional to the product of the likelihood and the prior. P(⇀|y, M) ∋f(y|⇀, M)P(⇀|M) (6.3) For our optimisation problem, we first derive the following relations: Sk(t; ⇀, M)= k m=1 pm(t; ⇀, M), f(y|⇀, M)=→ dSk(t; ⇀, M) dt , where Sk(t; ⇀, M) is the survival functions, notice that Sk=F(t; ⇀, M)=1. Then the log-likelihood function (⇁) is defined by: ⇁ = t→y k→S nk,t log→ dSk(t; ⇀, M) dt (6.4) Here, ⇁ ↓(→△,0], nk,t denotes the number of pipes of age t found in states that transitioned fromk, denoted as k. E.g., if k =1, then k =↔2, F↗ (see Figure 6.1). The acceptance distribution Aof the M-H algorithm is given by: A(xt, xt+1)=min1, f(y|⇀t+1, M)P(⇀t+1|M) f(y|⇀t, M)P(⇀t|M) >U(0,1) (6.5) Here, xt and xt+1 are the current and proposed points in the parameter space, and ⇀t and ⇀t+1 the corresponding sets of hyper-parameters. The prior P(⇀t+1|M) is a uniform distribution U(,, ¯,), where , and ¯, define the range for each hyperparameter in ⇀.
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