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Computing the required sample size
Published in Mark Stamp, Introduction to Machine Learning with Applications in Information Security, 2023
The three criteria (ALC, ACC, and WOC) described above are now used to compute the required sample size for estimating the population mean, assuming that the data come from a normal distribution. As we are uncertain about the population standard deviation σ (S*(z) in Equation (12.7) is only a prior point estimate of σ), a prior distribution is assigned to this parameter. It is convenient to assign a gamma distribution as a prior distribution to the reciprocal of the population variance, referred to as the precision parameter λ=1/σ2. More precisely, a prior bivariate normal-gamma distribution is assigned to the population mean and the precision parameter3. With this prior distribution, the posterior distribution of the population mean is fully defined, i.e., both the type of distribution and its parameters are known. The prior distribution is so-called conjugate with the normal distribution.
Applications in Finance
Published in Sylvia Frühwirth-Schnatter, Gilles Celeux, Christian P. Robert, Handbook of Mixture Analysis, 2019
John M. Maheu, Azam Shamsi Zamenjani
The latent log volatility ht follows a parametric, stationary, first-order autoregressive (AR) process defined with the AR parameter δ, but the rest of the model is nonparametric inasmuch no assumption is made about the underlying distribution of return innovations. Note that, assuming independence, ht ⊥ yt, Jensen & Maheu (2010) remove any leverage effect (Jacquier et al., 2004). Equations (17.24) and (17.25) assume the mixture’s probabilities and parameters follow the DP prior. The base distribution, H0, is a conjugate conditional normal-gamma distribution. The Sethuraman (1994) representation for this semi-parametric model is () yt∼∑g=1∞ηgN(μg,λg−2exp(ht)),
Log-Concave Functions
Published in Prem K. Kythe, Elements of Concave Analysis and Applications, 2018
(iv) The product of two log-concave functions is log-concave, which means that the joint densities formed by multiplying two probability densities (e.g., normal-gamma distribution, which always has a shape parameter ≥1 $ \ge 1 $ ) are log-concave. This property is extremely useful in general-purpose Gibbs sampling programs.
Real-time model for unit-level heating and cooling energy prediction in multi-family residential housing
Published in Journal of Building Performance Simulation, 2021
Sang Woo Ham, Panagiota Karava, Ilias Bilionis, James Braun
The real to nominal power ratio is obtained by calculating in each time step, and these values are expressed as a distribution instead of a single number. However, this requires a large amount of data to calculate all values for all combinations of the heat pump operation. Therefore, we approximate the distribution of as a normal distribution, and then the distribution is automatically updated with incoming data by using a Normal-Gamma distribution and its conjugate priors (Appendix C). Since the current posterior of is parameterized as a Normal-Gamma distribution, we can update the posterior of with new data without all the historic data.
Enhanced Lasso Regularization-Based Self-Adaptive Feature Selection Algorithm for the High-Dimensional Uncertainty Quantification of TREAT Transient Test Modeling
Published in Nuclear Technology, 2020
Haining Zhou, Volkan Seker, Thomas Downar
The reason for the observation can be intuitively explained by Fig. 21, which plots the distribution shapes of each of the input parameters. Figure 21 shows that using normal/gamma distribution does not provide a significant change in the overall distribution shape for the parameters studied. However, using uniform distribution effectively perturbs the possible ranges of the input parameters. Among the three parameters, fuel hydrogen content is known to have the most effect on the model while boron content has the least impact. If only gamma distribution and normal distribution are compared, according to Fig. 21 the distribution shape change causes the most effect on the hydrogen content. Therefore, we expect that in the gamma-distributed data set, the influence brought by the hydrogen content perturbation is enhanced and consequently undermines the impact of the boron content parameter. Figure 22 verifies the expectation.
Determination of Bayesian optimal warranty length under Type-II unified hybrid censoring scheme
Published in Quality Technology & Quantitative Management, 2022
Tanmay Sen, Ritwik Bhattacharya, Biswabrata Pradhan, Yogesh Mani Tripathi
where, , and are the observed values of , and , respectively. It is assumed that the joint prior distribution of follows a normal-gamma distribution with probability density function