Bayesian estimation of rainfall dispersion in Thailand using gamma distribution with excess zeros

The BAY-J and HPD-J intervals

The Jeffreys’ prior for δ in a binomial distribution is $p\left(\delta \right)\propto {\left(\delta \right)}^{-\frac{1}{2}}{\left(1-\delta \right)}^{\frac{1}{2}}$ (Bolstad & Curran, 2016). This leads to obtaining the marginal posterior distribution of δ as (20)${\displaystyle {\delta }_{jef}|x\sim Beta\left(n_{\left(0\right)}+\frac{1}{2},n_{\left(1\right)}+\frac{3}{2}\right).}$

Jeffreys’ prior for σ² in a lognormal distribution is p(σ²) ∝ σ⁻². Therefore, the marginal posterior distribution of σ² becomes (21)${\displaystyle {\sigma }_{jef}^2|x\sim IG\left(\frac{n_{\left(1\right)}}{2},\frac{\sum _{i=1}^n{\left(x_i-\mu \right)}^2}{2}\right).}$

The marginal posterior distribution of µis (22)${\displaystyle {\mu }_{jef}|{\sigma }^2,x\sim N\left(\bar{x},{\sigma }_{jef}^2/n_{\left(1\right)}\right).}$

We compute the mean and variance of gamma by using μ_jef|σ², x and ${\sigma }_{jef}^2|x$ as follows: (23)${\displaystyle M_{BAY-J}={\left\{\frac{{\mu }_{jef}}{2}+\sqrt{{\left(\frac{{\mu }_{jef}}{2}\right)}^2+{\sigma }_{jef}^2}\right\}}^3}$ (24)${\displaystyle V_{BAY-J}={\left\{\frac{{\mu }_{jef}+\sqrt{{\mu }_{jef}^2+4{\sigma }_{jef}^2}}{2\left(9^{-1/4}\right){\left({\sigma }_{jef}^2\right)}^{-1/4}}\right\}}^4.}$

So that (25)${\displaystyle {\widehat{\tau }}_{BAY-J}=\left(1-{\delta }_{jef}\right)\cdot V_{BAY-J}+{\delta }_{jef}\left(1-{\delta }_{jef}\right)\cdot M_{BAY-J}^2.}$

The confidence interval and HPD interval of τ based on the Jeffreys’ prior are obtained as (26)${\displaystyle C{I}_{BAY-J}=\left[{\hat{\tau }}_{BAY-J}\left(\alpha /2\right),{\hat{\tau }}_{BAY-J}\left(1-\alpha /2\right)\right].}$

The BAY-U and HPD-U intervals

The uniform prior for δ in a binomial distribution is p(δ) ∝ 1 (Bolstad & Curran, 2016). This leads to obtaining the marginal posterior distribution of δ as (27)${\displaystyle {\delta }_{unif}|x\sim Beta\left(n_{\left(0\right)}+1,n_{\left(1\right)}+1\right).}$

The uniform prior for σ² is σ² ∝ 1 (Kalkur & Rao, 2017). Subsequently, the marginal posterior distribution of σ² becomes (28)${\displaystyle {\sigma }_{unif}^2|x\sim IG\left(\frac{n_{\left(1\right)}-2}{2},\frac{\sum _{i=1}^n{\left(x_i-\mu \right)}^2}{2}\right).}$

The marginal posterior distribution of µas (29)${\displaystyle {\mu }_{unif}|{\sigma }^2,x\sim N\left(\bar{x},{\sigma }_{unif}^2/n_{\left(1\right)}\right).}$

We compute the mean and variance of a gamma distribution using μ_unif|σ², x and ${\sigma }_{unif}^2|x$ as follows: (30)${\displaystyle M_{BAY-U}={\left\{\frac{{\mu }_{unif}}{2}+\sqrt{{\left(\frac{{\mu }_{unif}}{2}\right)}^2+{\sigma }_{unif}^2}\right\}}^3}$ (31)${\displaystyle V_{BAY-U}=\left\{\frac{{\mu }_{unif}+\sqrt{{\mu }_{unif}^2+4{\sigma }_{unif}^2}}{2\left(9^{-1/4}\right){\left({\sigma }_{unif}^2\right)}^{-1/4}}\right\}{.}^4}$

So that (32)${\displaystyle {\widehat{\tau }}_{BAY-U}=\left(1-{\delta }_{unif}\right)\cdot V_{BAY-U}+{\delta }_{unif}\left(1-{\delta }_{unif}\right)\cdot M_{BAY-U}^2.}$

The confidence interval and HPD interval of τ based on the uniform prior are respectively obtained as (33)${\displaystyle C{I}_{BAY-U}=\left[{\hat{\tau }}_{BAY-U}\left(\alpha /2\right),{\hat{\tau }}_{BAY-U}\left(1-\alpha /2\right)\right].}$

The BAY-NGB and HPD-NGB intervals

Maneerat & Niwitpong (2021) defined the normal-gamma-beta prior as (34)${\displaystyle p\left(\tau \right)\propto {\lambda }^{-1}{\left[\left(\delta \right)\left(1-\delta \right)\right]}^{-1/2}}$ where λ = σ⁻², (μ, λ) follows a normal-gamma distribution and δ follows a beta distribution (Maneerat & Niwitpong, 2021). Thus, the marginal posterior distributions of δ, σ² and µrespectively become (35)${\displaystyle {\delta }_{NGB}|x\sim Beta\left(n_{\left(0\right)}+\frac{1}{2},n_{\left(1\right)}+\frac{1}{2}\right)}$ (36)${\displaystyle {\sigma }_{NGB}^2|x\sim IG\left(\frac{n_{\left(1\right)}-1}{2},\frac{\sum _{i=1}^{n_{\left(1\right)}}{\left(x_i-\mu \right)}^2}{2}\right)}$ (37)${\displaystyle {\mu }_{NGB}|x\sim t_{2\left(n_{\left(1\right)}-1\right)}\left(\bar{x},\frac{\sum _{i=1}^n{\left(x_i-\bar{x}\right)}^2}{n_{\left(1\right)}\left(n_{\left(1\right)}-1\right)}\right).}$ We compute the mean and variance of a gamma distribution by using μ_NGB|x and ${\sigma }_{NGB}^2|x$ as follows: (38)${\displaystyle M_{BAY-NGB}={\left\{\frac{{\mu }_{NGB}}{2}+\sqrt{{\left(\frac{{\mu }_{NGB}}{2}\right)}^2+{\sigma }_{NGB}^2}\right\}}^3}$ (39)${\displaystyle V_{BAY-NGB}={\left\{\frac{{\mu }_{NGB}+\sqrt{{\mu }_{NGB}^2+4{\sigma }_{NGB}^2}}{2\left(9^{-1/4}\right){\left({\sigma }_{NGB}^2\right)}^{-1/4}}\right\}}^4.}$

So that (40)${\displaystyle {\widehat{\tau }}_{BAY-NGB}=\left(1-{\delta }_{NGB}\right)\cdot V_{BAY-NGB}+{\delta }_{NGB}\left(1-{\delta }_{NGB}\right)\cdot M_{BAY-NGB}^2.}$

The confidence interval and HPD interval of τ based on the normal-gamma-beta prior are respectively obtained as (41)${\displaystyle C{I}_{BAY-NGB}=\left[{\hat{\tau }}_{BAY-NGB}\left(\alpha /2\right),{\hat{\tau }}_{BAY-NGB}\left(1-\alpha /2\right)\right].}$

 
_______________________________________________________________________________________________________ 
Algorithm 3 Bayesian interval___________________________________________________ 
 1:  Generate x from a gamma distribution with excess zeros, compute ^ δ , ^ μ , and 
     ^ σ2. 
  2:  Generate δ|x from Eqs. (20), (27) and (35). 
  3:  Generate σ2|x from Eqs. (21), (28) and (36). 
  4:  Given σ2|x generate μ|σ2,x. 
  5:  Compute mean and variance of gamma distribution from Eqs.  (23), (24), 
     (30), (31), (38) and (39). 
  6:  Compute ^ τ from Eqs. (25), (32) and (40). 
  7:  Compute the 95% confidence intervals and HPD for ^ τ from Eqs. (26), (33) 
     and (41). 
  8:  Repeat Steps 1–7 10,000 times to compute the CPs and ALs.__________________