Abstract
Neural Networks (NNs) had wide interest due to empirical achievements on a wide range of learning issues. NNs are highly expressive models that can learn complex function approximations from input/output, with a particular ability to train them on massive data sets with stochastic optimization. The Bayesian approach to NNs can potentially avoid some of the problems of stochastic optimization. The use of Bayesian learning is well suited to the problem of real estate appraisals, in fact, Bayesian inference techniques are very interesting in order to deal with a small and noisy sample in the field of probabilistic inference carried out with neural model. For this purpose it has here been experimented a NNs model with Bayesian learning. The output distribution has been calculated operating a numerical integration on the weights space with the help of Markov Chain Hybrid Monte Carlo Method.
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Notes
- 1.
In the econometric appraisal of real estate prices the stochastic component is representative of the whole effect of the many disturbance factors which cannot be expressed through variables endogenous to the model. Such factors can be explained by: (1) the practical impossibility to assess the regression model with reference to the different samples drawn by the population of known-price sales; (2) the impossibility to quantify the influence of some important market factors (role of broker, contractual power in particular conditions, etc.) or institutional factors, for example, expectations related to fiscal charging, etc.; (3) the impossibility, once acknowledged the influence of a factor, to obtain quantitative information about it; (4) the need to use a number of explicative variables not too high also in relation to low sample size; (5) the chance that real estate sample is influenced by errors in the survey, as well as by random elements in the answers of interviewed (buyers, sellers and brokers).
- 2.
In real estate valuations each error is representative of the difference between the real estate value obtained from the model and the corresponding price drawn on the market.
- 3.
In the general case in which one does not hypothesize the independence of the observation, the problem becomes that of determining the conditional density of the target data p(t│x) conditioned on the input data.
- 4.
The sum of quadratic-error function remains valid even if a parametric approach is used, and the interpolating function will have a pre-defined algebraical form and will be expressed as a function of the parameters characterizing the functional form itself. The individuation of the parameters set which make the coat function minimum is, as known, almost always linked to the use of the square minimum algorithm, while in the case of neural network the algorithm mostly used is the back-propagation.
- 5.
All the deductive-axiomatic apparatus of the appraisal theory is based upon the hypothesis that the real estate sample is randomly drawn by a normal population referred to similar real estate sales with known price, and that with increasing of the sample dimension its distribution can be approximated to the normal [9]. Hardly ever these statistical hypothesis are mirrored in practice and this is because [33]: (a) the behaviour of the subjects which deal with the sampling hardly ever follows criteria which are based on random logic; (b) the atypicalness of real estate goods systematically invalidates the normal postulation of the population.
- 6.
The maximum likelihood approach represent a particular approximation of which we only consider the most probable weights vector which corresponds to the distribution peak.
- 7.
- 8.
The problem of a very good balance between bias and variance in order to avoid overfitting phenomena by the training data suggests the choice of a function smooth network [2].
- 9.
It is easy verify that, with α and β fixed, increasing the training set dimension the first is a representative term of the maximum likelihood, whereas the second is a representative term of the prior knowledge that decreases so that larger is the set and better the only term of maximum likelihood approximates the most probable weights distribution. Conversely for small data sets the role of the prior term in the determination of the most probable solution is essential [37].
- 10.
The elaboration have been carried out with Netlab free software (Aston University, Birmingham).
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Del Giudice, V., De Paola, P., Forte, F. (2017). Bayesian Neural Network Models in the Appraisal of Real Estate Properties. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2017. ICCSA 2017. Lecture Notes in Computer Science(), vol 10406. Springer, Cham. https://doi.org/10.1007/978-3-319-62398-6_34
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