Bankruptcy Prediction through Soft Computing based Deep Learning Technique by Arindam Chaudhuri & Soumya K. Ghosh

Author:Arindam Chaudhuri & Soumya K. Ghosh
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

(b)Hyperprior which forms the distribution of the a priori distribution. They come in picture when conjugate priors are used. This becomes abstract and is taken away from the actual problem.

To verify abovementioned components toward rough Bayesian hierarchical structure, consider random variable Y with parameters α and 1 as the mean and variance, respectively, such that Y ∣ α ~ N(α, 1). The parameter α has prior distribution which is represented through normal distribution such that Y ∣ ϑ ~ N(ϑ, 1). The parameter ϑ is modeled through standard normal distribution N(0, 1). The hyperparameter distribution is N(0, 1) and shows the hyperprior distribution. The distribution of Y changes when another parameter is added such that Y ∣ α , ϑ ~ N(ϑ, 1). When another stage is present, then ϑ follows normal distribution with mean δ and variance ρ such that ϑ ~ N(δ, ρ). Here δ and ρ are the hyperparameters and they are distributed as hyperprior distributions.

Consider y j as an observation and α j as the parameter which governs through the data generation process for y j . It is assumed that the parameters α 1 ,  …  …  , α j are generated when exchange is performed from a common population having the distribution through the hyperparameter ω. The parameters α and ω represent the random variables. The hierarchical rough Bayesian model is drawn through the stages as highlighted below. This is a three-stage model which can be extended to n-stage model.

(a) Stage I: y j ∣α j  , ω ∼ Prob(y j | α j , ω.)

(b) Stage II: α j ∣ω ∼ Prob(α j | ω)

(c) Stage III: ω ∼ Prob(ω)

The likelihood as represented in the Stage I is Prob(y j | α j , ω) with Prob(α j , ω) as the a priori distribution. The likelihood depends on ω only through α j . The a priori distribution from Stage I is:

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