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lumns whose entries are corresponding to a binary number h. H_N is harmonic number H_N=∑_(i=1)^N▒1/i, and α (moral rate) controls the number of features, associated to the each observation (each row). In particular, we consider the unlimited number of nonzero columns in the above equation (Griffiths & Ghahramani, 2006).
We use the culinary metaphor for defining the IBP process. This generator process is as follows: the first people (originator creating first pool) starts at the left of a buffet, and serves himself the Poisson(α) number of foods. The nth people simultaneously moves along the buffet, samples the previous served foods with probability m_k⁄n (based on their popularity), m_k is the number of peoples who sample the food k before the people nth. nth people also serves the new foods for himself with probability Poisson(α⁄n). Notice that even if total number of existing features are unlimited, actual number of used features K^+, are usually finite and drawn from the Poisson(α∑_(i=1)^N▒1/i) distribution. In this paper, we consider a new representation of the IBP process namely stick-breaking construction.

Stick-Breaking Construction
The stick-breaking construction is an alternative representation for IBP which imposes a specific ordering on the features (The & Gorur & Ghahramani, 2007). To generate a matrix Z using the stick-breaking construction, we start with assigning a parameter π_k∈(0, 1) to each columns of Z (which are the probability of having particular feature in an object). In other words, the probability of each feature is explicitly represented by a stick of unit length. Also, let Z be a N×K stochastic binary matrix whose each entry z_nk, represents possession the feature k by the object n. Set a Beta(α⁄K,1) prior over π_k, ∀k, where α is concentration parameter of the IBP. Given π_k, every z_nk in the column k is sampled as a Bernoulli random variable Bernoulli(π_k), where π_k, ∀k is generated through stick-breaking process as the follows; we first draw a sequence of independent random variables v_1, v_2, … each of them has beta distribution, Beta(α, 1). Next, we set π_1=v_1. for each dimension k, we set π_k=v_k π_(k-1)=∏_(i=1)^k▒v_i , which is a decreasing sequence of probabilities π_k. In the limit K→∞, π_k, ∀k follows the stick-breaking construction for representing the IBP.
In particular, given a finite set of data, the probability of seeing feature k is decreasing exponentially with k. Notice that π_k, ∀k has a markov property because π_(k+1) is conditionally independent of all of the π_k, ∀k=1,…,k-1, given π_k.
It is assumed that, using finite Bernoulli-Beta probability model p_K, the IBP is well approximated as presented below
π_k~Beta(α⁄K,1) Independently ∀k∈{1…K},
z_nk~Bernoulli(π_k ) Independently ∀k∈{1…K}, ∀n∈{1…N},
A_k0~Gamma(α, 1⁄α) Independently ∀k∈{1…K},
X_n0~Poisson(Z_n A) Independently ∀n∈{1…N},
Where K is the finite truncation level. Griffiths and Ghahramani showed that as K tends to infinity, this finite-dimensional approximation converges in distribution to the IBP. (Griffiths & Ghahramani, 2006).
Under finite-dimensional approximation, the joint probability of data X and hidden variables W={π, Z, A} is
p_K (W, X├|θ┤)=∏_(k=1)^K▒(p(π_k ├|α┤)p(A_k0 ├|α, 1⁄α┤)∏_(n=1)^N▒〖p(z_nk |π_k ┤)〗) ∏_(n=1)^N▒〖p(X_n0 |Z_i0 A┤)〗
We denote the parameter set by θ={α,1⁄α}.
According to the above specified theoretical framework and the securitization literature, let the K-dimensional latent variable vector Z_n0 be the K-dimensional vector e in the investor’s optimization problem, which previously was determined the probability distribution of the outcomes. Notice that Z_n0 is a random variable with parameters π_k, ∀k. These are the parameters of the prior probability distribution function for random variable z_nk, which affect every dimensions (every entry in the vector Z_n0) of the originator’s effort vector. Thus, these parameters determine the latent variable vector and as a result the realized observation vector X_n0.
Thus, using joint posterior probability of data and hidden variables we can continue. In this paper, we consider one pool of mortgages in the MBS, so, Z will be 1×K binary matrix (K-dimensional vector) and X will be a 1×12 matrix of observation which here is the number of defaults, arriving during the proposed interval (12-dimensional vector). Therefore we have
f(X,Y├|e┤)=p_K (W, X)=∏_(k=1)^K▒[p(π_k ├|α┤)p(A_k0 )p(z_1k ├|π_k ┤)] p(X_10 ├|Z_10 A┤) (4)
In this paper, in order to model the data related to the defaults of underlying loans, the finite Gamma-Poisson model with binary latent feature is considered. In this model, each vector X_n0 is represented by a 12-dimensional vector of loans’ defaults status. It is assumed that X_n0 have a Poisson distribution with mean λ=Z_n0 A and covariance matrix ∑_n▒〖=Z_n0 A〗, which Z_n0 is a binary K-dimensional vector and A is a K×12 matrix of weights with a Gamma distribution, having parameters λ and 1⁄λ.
Therefore, conditional probability distribution of X_10 is as follows
p(X_10 ├|Z_10, A┤)=∏_(d=1)^D▒〖p(x_1d ├|Z_10, A_0d ┤)=〗 ∏_(d=1)^D▒(e^(-Z_10 A_0d ) 1/(x_1d !) 〖(Z_10 A_0d)〗^(X_10 ) )
=∏_(d=1)^D▒〖e^(-(∑_k▒〖z_1k a_kd 〗) ) 1/(x_1d !) (∑_k▒〖z_1k a_kd 〗)^(x_1d ) 〗
=e^(-(∑_k▒∑_d▒〖z_1k a_kd 〗) )×(∏_(d=1)^D▒1/(x_1d )!)×(∏_(d=1)^D▒(∑_k▒〖z_1k a_kd 〗)^(x_1d ) ).
Thus, we will have
f(X,Y├|e┤)=∏_(k=1)^K▒[p(π_k ├|α┤)p(A_k0 )p(z_1k ├|π_k ┤)] 〖×e〗^(-(∑_k▒∑_d▒〖z_1k a_kd 〗) )×(∏_(d=1)^D▒1/(x_1d )!)×(∏_(d=1)^D▒(∑_k▒〖z_1k a_kd 〗)^(x_1d ) ) (5)
We use equation (5) and replace it into the (f_(e_k ) (X,Y├|e┤))/f(X,Y├|e┤) ratio in equation (2), obtained by resolving the cost minimization problem. By simplifying, a relation is derived from equation (2) as follows
1/(U^’ (s(X,Y)))=λ+(∑_(k=1)^K▒〖μ_k f_(e_k ) (X,Y├|e┤) 〗)/f(X,Y├|e┤) (6)
As a result, the second-best optimal compensation scheme for originator s(X,Y), is a function of observations x_1d, and inferred various dimensions of the effort, and intensity of influencing of each of these dimensions a_k1 on the credit performance of the pool of loans, and its shape is depend on risk-averse originator preferences.
For a general utility function U(.), the optimal scheme is solved as follows
s(X,Y)=Ψ(λ+(∑_(k=1)^K▒〖μ_k f_(e_k ) (X,Y├|e┤) 〗)/f(X,Y├|e┤) ),
Ψ=(1/(U^’ (.)))^(-1).
As is expected, incentive is decreasing with individual’s attitudes toward risk.
Another implication of the equation (2) is that the optimal contract is not necessary to take a linear form. The optimal shape of s(X,Y) is a function of the information contents of the observations X, and does not likely take a simple variation styles with (X,Y).

Results from Simulation
The convergence of Variational inference only guarantees to a local optima. Therefore, by using standard optimization techniques, one can avoid small minima. For each run, a random restart number is given, and the hyperparameters for feature and noise variance are adjusted such that the posterior is made smooth.
By using IBP model, we analysis modeling synthetic revenues of loan pools in MBS. In this model, the streams of cash flows are determined by the number of efforts (namely, numerous underwriting standards), performed by originator during the underwriting mortgage loans. This model assumes that the number of exerted efforts is unknown, and for each dimension of effort, the streams of cash flows generated by MBS are determined such that if for underwriting this loan(s), this dimension of effort is implemented. Latent variable z_nk, indicates that if for observation (loans pool) n, dimension k is exerted. Therefore, The goal is to discover both identities and the number of chosen efforts, from observing revenue streams generated by the loan pools underlying MBS security. For this purpose, in the paper, the finite Linear-Gaussian model with binary latent features is considered.

Synthetic Data
In this section, using synthetic data, variational approach for finite binary Linear-Gaussian model is run. The set of synthetic data includes A and Z matrices, randomly generated from truncated stick-breaking prior. In this paper, the synthetic data includes 20 observations of revenues generated from MBS securities which we denote by X. Each cash flow is a vector of dimension D=16, namely each observation X_n0 is a vector of dimension 16 (time dimension). Choosing the truncation level 4 for K, determines that each revenue observation X_n0, is generated from a4-dimensional subset of several hidden efforts (latent features). In other words, for each observation X_n0, at most 4 dimensions of underwriting standards are exerted by originator to screen loan applicants.
In MBS the values of these latent features (hidden efforts) are corresponds to the rows of weight matrix A. The rows of matrix A quantitatively determine the impact of intensity of each dimension of effort on the revenue stream of the underlying loan pools.
For each row of latent feature matrix, Z_n0, each entry was set to 1 with probability 0.5 and was set to 0 otherwise. Then, each observation, related to each different loan poolsX_n0, was generated by adding a white noise with variance σ_n^2=0.1000 to the linear combination of weight matrix based on binary feature matrix, Z_n0 A.
In this Variation Bayes procedure, we set the initial value of α equal to 1, and randomly draw Z from IBP(α), and set the initial value of σ_A^2 equal to 0.4364. Type of model was chosen LG. Sampling for optimization are ran for 1000 steps. The

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