Parameter

• Probability parameter \(p \in (0, 1)\)

Probability Mass Function

\[ \begin{aligned} \mathrm{P} [Y = y | p] &= \begin{cases} 1 - p & \text{ for } y = 0 \\ p & \text{ for } y = 1 \\ \end{cases} \\ \end{aligned} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= p \\ \mathrm{var}[Y] &= p (1 - p) \\ \end{aligned} \]

Score

\[ \nabla_{m} (y; p) = \begin{cases} \frac{1}{p - 1} & \text{ for } y = 0 \\ \frac{1}{p} & \text{ for } y = 1 \\ \end{cases} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{p, p} (p) &= \frac{1}{p (1 - p)} \\ \end{aligned} \]

Parameters

• Worth parameters \(w_i \in (0, \infty), i = 1, \ldots, n\)

Vector Notation

• Worth vector \(\boldsymbol{w}\) of length \(n\)

Probability Mass Function

\[ \begin{aligned} \mathrm{P} [\boldsymbol{Y} = \boldsymbol{y} | \boldsymbol{w}] &= \frac{1}{\sum_{i=1}^n w_i} \prod_{i=1}^n w_i^{y_i} \end{aligned} \]

Moments

\[ \begin{aligned} \mathrm{E}[\boldsymbol{Y}] &= \frac{1}{\sum_{i=1}^n w_i} \boldsymbol{w} \\ \mathrm{var}[\boldsymbol{Y}] &= \frac{1}{\sum_{i=1}^n w_i} \mathrm{diag} (\boldsymbol{w}) - \frac{1}{\left( \sum_{i=1}^n w_i \right)^2} \boldsymbol{w} \boldsymbol{w}^\intercal \\ \end{aligned} \]

Score

\[ \nabla_{\boldsymbol{w}} (\boldsymbol{y}; \boldsymbol{w}) = \boldsymbol{y} \oslash \boldsymbol{w} - \frac{1}{\sum_{i=1}^n w_i} \boldsymbol{1}_n \]

Fisher Information

\[ \mathcal{I}_{\boldsymbol{w}, \boldsymbol{w}} (\boldsymbol{w}) = \mathrm{diag} \left( \sum_{i=1}^n w_i \boldsymbol{1}_n \oslash \boldsymbol{w} \right) - \frac{1}{\left( \sum_{i=1}^n w_i \right)^2} \boldsymbol{1}_{n \times n} \]

Parameters

• Worth parameters \(w_i \in (0, \infty), i = 1, \ldots, n\)

Ranking Notation

• Worth parameters by rank \(w_{j^{\mathrm{th}}}, j = 1, \ldots, n\)

Probability Mass Function

\[ \mathrm{P} [\boldsymbol{Y} = \boldsymbol{y} | w_1, \ldots, w_n] = \prod_{j=1}^n \frac{w_{j^{\mathrm{th}}}}{\sum_{k=j}^n w_{k^{\mathrm{th}}}} \]

Score

\[ \nabla_{w_i} (\boldsymbol{y}; w_1, \ldots, w_n) = \frac{1}{w_i} - \sum_{j=1}^{y_i} \frac{1}{\sum_{k = j}^n w_{k^{\mathrm{th}}}} \]

Parameters

• Mean parameter \(m \in (0, \infty)\)
• Dispersion parameter \(s \in (0, \infty)\)

Probability Mass Function

\[ \mathrm{P} [Y = y | m, s] \approx \frac{1}{1 + \frac{1 - s}{12 s m} \left(1 + \frac{1}{s m} \right)} \sqrt{s} \frac{y^y}{y!} \left( \frac{m}{y} \right)^{s y} \exp(s y - s m - y) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &\approx m \\ \mathrm{var}[Y] &\approx \frac{m}{s} \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s) &\approx \frac{s}{m} (y - m) \\ \nabla_{s} (y; m, s) &\approx \frac{1}{2 s} - m \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s) &\approx \frac{s}{m} \\ \mathcal{I}_{m, s} (m, s) &\approx 0 \\ \mathcal{I}_{s, s} (m, s) &\approx \frac{1}{2 s^2} \\ \end{aligned} \]

Parameter

• Mean parameter \(m \in (0, \infty)\)

Probability Mass Function

\[ \mathrm{P} [Y = y | m] = \frac{1}{1 + m} \left( \frac{m}{1 + m} \right)^{y} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m \\ \mathrm{var}[Y] &= m (1 + m) \\ \end{aligned} \]

Score

\[ \nabla_{m} (y; m) = \frac{y - m}{m (1 + m) } \]

Fisher Information

\[ \mathcal{I}_{m, m} (m) = \frac{1}{m (1 + m)} \]

Parameter

• Probability parameter \(p \in (0, 1)\)

Probability Mass Function

\[ \mathrm{P} [Y = y | p] = p (1 - p)^{y} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= \frac{1 - p}{p} \\ \mathrm{var}[Y] &= \frac{1 - p}{p^2} \\ \end{aligned} \]

Score

\[ \nabla_{p} (y; p) = \frac{p y + p - 1}{p (p - 1)} \]

Fisher Information

\[ \mathcal{I}_{p, p} (p) = \frac{1}{p^2 (1 - p)} \]

Parameters

• Mean parameter \(m \in (0, \infty)\)
• Dispersion parameter \(s \in (0, \infty)\)

Probability Mass Function

\[ \mathrm{P} [Y = y | m, s] = \frac{\Gamma (y + s^{-1})}{\Gamma (y + 1) \Gamma (s^{-1})} \left( \frac{1}{1 + s m} \right)^{s^{-1}} \left( \frac{s m}{1 + s m} \right)^{y} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m \\ \mathrm{var}[Y] &= m (1 + s m) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s) &= \frac{y - m}{m (1 + s m) } \\ \nabla_{s} (y; m, s) &= \frac{ y - m}{s (1 + s m)} + \frac{1}{s^2} \left( \ln(1 + s m) + \psi_0 \left( \frac{1}{s} \right) - \psi_0 \left( y + \frac{1}{s} \right) \right) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s) &= \frac{1}{m (1 + s m)} \\ \mathcal{I}_{m, s} (m, s) &= 0 \\ \mathcal{I}_{s, s} (m, s) &\approx \frac{1}{s^4} \left( \ln(1 + s m) + \psi_0 \left( \frac{1}{s} \right) - \psi_0 \left( m + \frac{1}{s} \right) \right)^2 \\ \end{aligned} \]

Parameters

• Probability parameter \(p \in (0, 1)\)
• Size parameter \(r \in (0, \infty)\)

Probability Mass Function

\[ \mathrm{P} [Y = y | p, r] = \frac{\Gamma(y + r)}{\Gamma(y + 1) \Gamma(r)} (1 - p)^y p^r \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= \frac{r (1 - p)}{p} \\ \mathrm{var}[Y] &= \frac{r (1 - p)}{p^2} \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{p} (y; p, r) &= \frac{p r + p y - r}{p (p - 1)} \\ \nabla_{r} (y; p, r) &= \ln(p) - \psi_0(r) + \psi_0(y + r) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{p, p} (p, r) &= \frac{r}{p^2 (1 - p)} \\ \mathcal{I}_{p, r} (p, r) &= -\frac{1}{p} \\ \mathcal{I}_{r, r} (p, r) &\approx \left( \ln(p) - \psi_0(r) + \psi_0 \left( \frac{r}{p} \right) \right)^2 \\ \end{aligned} \]

Parameter

• Mean parameter \(m \in (0, \infty)\)

Probability Mass Function

\[ \mathrm{P} [Y = y | m] = \frac{m^y}{y!} \exp(-m) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m \\ \mathrm{var}[Y] &= m \\ \end{aligned} \]

Score

\[ \nabla_{m} (y; m) = \frac{y - m}{m} \]

Fisher Information

\[ \mathcal{I}_{m, m} (m) = \frac{1}{m} \]

Parameters

• Mean parameter \(m \in (0, \infty)\)
• Dispersion parameter \(s \in (0, \infty)\)
• Zero inflation parameter \(p \in (0, 1)\)

Probability Mass Function

\[ \begin{aligned} \mathrm{P} [Y = y | m, s, p] &= \begin{cases} p + (1 - p) \left( \frac{1}{1 + s m} \right)^{s^{-1}} & \text{ for } y = 0 \\ (1 - p) \frac{\Gamma (y + s^{-1})}{\Gamma (y + 1) \Gamma (s^{-1})} \left( \frac{1}{1 + s m} \right)^{s^{-1}} \left( \frac{s m}{1 + s m} \right)^{y} & \text{ for } y \geq 1 \\ \end{cases} \\ \end{aligned} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m (1 - p) \\ \mathrm{var}[Y] &= m(1 - p) (1 + p m + s m) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s, p) &= \begin{cases} \frac{p - 1}{(1 + s m) \left( 1 + p (1 + s m)^{s^{-1}} - p \right)} & \text{ for } y = 0 \\ \frac{y - m}{m (1 + s m) } & \text{ for } y \geq 1 \\ \end{cases} \\ \nabla_{s} (y; m, s, p) &= \begin{cases} \frac{(1 - p) \left( (1 + s m) \ln(1 + s m) -s m \right) }{ s^2 (1 + s m) \left( 1 + p (1 + s m)^{s^{-1}}- p \right) } & \text{ for } y = 0 \\ \frac{ s (y - m) + (1 + s m) \left( \ln(1 + s m) + \psi_0 \left( s^{-1} \right) - \psi_0 \left( y + s^{-1} \right) \right) }{s^2 (1 + s m)} & \text{ for } y \geq 1 \\ \end{cases} \\ \nabla_{p} (y; m, s, p) &= \begin{cases} \frac{(1 + s m)^{s^{-1}} - 1}{1 + p (1 + s m)^{s^{-1}}- p} & \text{ for } y = 0 \\ \frac{1}{p - 1} & \text{ for } y \geq 1 \\ \end{cases} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s, p) &= \frac{p(p - 1)}{(1 + s m)^2 \left( 1 + p (1 + s m)^{s^{-1}} - p \right)} + \frac{1 -p}{m(1 + s m)} \\ \mathcal{I}_{m, s} (m, s, p) &= \frac{\left( p - p^2 \right) \left( (1 + s m) \ln(1 + s m) - s m \right) }{s^2 (1 + s m)^2 \left( 1 + p (1 + s m)^{s^{-1}} -p \right)} \\ \mathcal{I}_{m, p} (m, s, p) &= \frac{-1}{ (1 + s m) \left( 1 + p (1 + s m)^{s^{-1}} - p \right) }\\ \mathcal{I}_{s, s} (m, s, p) &\approx \frac{1}{s^4} \left( \ln(1 + s m) + \psi_0 \left( s^{-1} \right) - \psi_0 \left( y + s^{-1} \right) \right)^2 \left( 1 - p - (1 - p) \left( 1 + s m \right)^{-s^{-1}} \right) \\ & \qquad + \frac{(1 - p)^2 \left( (1 + s m) \ln(1 + s m) - s m \right)^2} {s^4 (1 + s m)^{2 + s^{-1}} \left( 1 + p (1 + s m)^{s^{-1}} - p \right)} \\ \mathcal{I}_{s, p} (m, s, p) &= \frac{(1 + s m) \ln(1 + s m) - s m}{s^2 (1 + s m) \left( 1 + p (1 + s m)^{s^{-1}} - p \right)} \\ \mathcal{I}_{p, p} (m, s, p) &= \frac{1 - (1 + s m)^{s^{-1}}}{(p - 1) \left( 1 + p (1 + s m)^{s^{-1}} - p \right)} \end{aligned} \]

Parameters

• Mean parameter \(m \in (0, \infty)\)
• Zero inflation parameter \(p \in (0, 1)\)

Probability Mass Function

\[ \begin{aligned} \mathrm{P} [Y = y | m, p] &= \begin{cases} p + (1 - p) \exp(-m) & \text{ for } y = 0 \\ (1 - p) \frac{m^y}{y!} \exp(-m) & \text{ for } y \geq 1 \\ \end{cases} \\ \end{aligned} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m (1 - p) \\ \mathrm{var}[Y] &= m(1 - p) (1 + p m) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s, p) &= \begin{cases} \frac{p - 1}{p \exp(m) - p + 1} & \text{ for } y = 0 \\ \frac{y - m}{m} & \text{ for } y \geq 1 \\ \end{cases} \\ \nabla_{p} (y; m, s, p) &= \begin{cases} \frac{\exp(m) - 1}{p \exp(m) - p + 1} & \text{ for } y = 0 \\ \frac{1}{p - 1} & \text{ for } y \geq 1 \\ \end{cases} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s, p) &= \frac{p (p - 1)}{p \exp(m) - p + 1} - \frac{p - 1}{m} \\ \mathcal{I}_{m, p} (m, s, p) &= - \frac{1}{p \exp(m) - p + 1} \\ \mathcal{I}_{p, p} (m, s, p) &= \frac{\exp(m) - 1}{(1 - p) (p \exp(m) - p + 1)} \\ \end{aligned} \]

Parameters

• First rate parameter \(r_1 \in (0, \infty)\)
• Second rate parameter \(r_2 \in (0, \infty)\)

Probability Mass Function

\[ \begin{aligned} \mathrm{P} [Y = y | r_1, r_2] &= \exp(-r_1 - r_2) \left( \frac{r_1}{r_2} \right)^{\frac{y}{2}} I_y \left( 2 \sqrt{r_1 r_2} \right) \end{aligned} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= r_1 - r_2 \\ \mathrm{var}[Y] &= r_1 + r_2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{r_1} (y; r_1, r_2) &= \sqrt{\frac{r_2}{r_1}} \frac{I_{y-1} \left( 2 \sqrt{r_1 r_2} \right)}{I_y \left( 2 \sqrt{r_1 r_2} \right) } - 1 \\ \nabla_{r_2} (y; r_1, r_2) &= \sqrt{\frac{r_1}{r_2}} \frac{I_{y-1} \left( 2 \sqrt{r_1 r_2} \right)}{I_y \left( 2 \sqrt{r_1 r_2} \right) } -\frac{y}{r_2} - 1 \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{r_1, r_1} (r_1, r_2) &\approx \frac{r_2}{r_1} \left( \frac{I_{r_1 - r_2 - 1} \left(2 \sqrt{r_1 r_2} \right) }{I_{r_1 - r_2} \left(2 \sqrt{r_1 r_2} \right) } \right)^2 - 2 \sqrt{\frac{r_2}{r_1}} \frac{I_{r_1 - r_2 - 1} \left(2 \sqrt{r_1 r_2} \right) }{I_{r_1 - r_2} \left(2 \sqrt{r_1 r_2} \right) } + 1 \\ \mathcal{I}_{r_1, r_2} (r_1, r_2) &\approx \left( \frac{I_{r_1 - r_2 - 1} \left(2 \sqrt{r_1 r_2} \right) }{I_{r_1 - r_2} \left(2 \sqrt{r_1 r_2} \right) } \right)^2 - 2 \sqrt{\frac{r_1}{r_2}} \frac{I_{r_1 - r_2 - 1} \left(2 \sqrt{r_1 r_2} \right) }{I_{r_1 - r_2} \left(2 \sqrt{r_1 r_2} \right) } + \frac{r_1}{r_2} \\ \mathcal{I}_{r_2, r_2} (r_1, r_2) &\approx \frac{r_1}{r_2} \left( \frac{I_{r_1 - r_2 - 1} \left(2 \sqrt{r_1 r_2} \right) }{I_{r_1 - r_2} \left(2 \sqrt{r_1 r_2} \right) } \right)^2 - 2 \left( \frac{r_1}{r_2} \right)^{\frac{3}{2}} \frac{I_{r_1 - r_2 - 1} \left(2 \sqrt{r_1 r_2} \right) }{I_{r_1 - r_2} \left(2 \sqrt{r_1 r_2} \right) } + \left( \frac{r_1}{r_2} \right)^2 \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in \mathbb{R}\)
• Dispersion parameter \(s \in (0, \infty)\)

Probability Mass Function

\[ \mathrm{P} [Y = y | m, s] = \exp(-|m| - s) \left( \frac{|m| + m + s}{|m| - m + s} \right)^{\frac{y}{2}} I_y \left( \sqrt{s^2 + 2 |m| s} \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m \\ \mathrm{var}[Y] &= |m| + s \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s) &= \frac{y}{2|m| + s} + \frac{\mathrm{sgn}(m) s}{2 \sqrt{s^2 + 2 |m| s}} \frac{ I_{y-1} \left( \sqrt{s^2 + 2 |m| s} \right) + I_{y+1} \left( \sqrt{s^2 + 2 |m| s} \right) }{ I_y \left( \sqrt{s^2 + 2 |m| s} \right) } - \mathrm{sgn}(m) \\ \nabla_{s} (y; m, s) &= - \frac{m y}{s^2 + 2 |m| s} + \frac{|m| + s}{2 \sqrt{s^2 + 2 |m| s}} \frac{ I_{y-1} \left( \sqrt{s^2 + 2 |m| s} \right) + I_{y+1} \left( \sqrt{s^2 + 2 |m| s} \right) }{ I_y \left( \sqrt{s^2 + 2 |m| s} \right) } - 1 \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s) &\approx \frac{s^2}{4 \left( s^2 + 2|m|s \right)} \left( \frac{2 (|m| + s)}{\sqrt{s^2 + 2 |m| s}} - \frac{ I_{m-1} \left( \sqrt{s^2 + 2 |m| s} \right) + I_{m+1} \left( \sqrt{s^2 + 2 |m| s} \right) }{ I_m \left( \sqrt{s^2 + 2 |m| s} \right)} \right)^2 \\ \mathcal{I}_{m, s} (m, s) &\approx \frac{\mathrm{sgn}(m) (|m| + s) s}{4 \left( s^2 + 2|m|s \right)} \left( \frac{2 (|m| + s)}{\sqrt{s^2 + 2 |m| s}} - \frac{ I_{m-1} \left( \sqrt{s^2 + 2 |m| s} \right) + I_{m+1} \left( \sqrt{s^2 + 2 |m| s} \right) }{ I_m \left( \sqrt{s^2 + 2 |m| s} \right)} \right)^2 \\ \mathcal{I}_{s, s} (m, s) &\approx \frac{(|m| + s)^2}{4 \left( s^2 + 2|m|s \right)} \left( \frac{2 (|m| + s)}{\sqrt{s^2 + 2 |m| s}} - \frac{ I_{m-1} \left( \sqrt{s^2 + 2 |m| s} \right) + I_{m+1} \left( \sqrt{s^2 + 2 |m| s} \right) }{ I_m \left( \sqrt{s^2 + 2 |m| s} \right)} \right)^2 \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in \mathbb{R}\)
• Variance parameter \(s \in (|m|, \infty)\)

Probability Mass Function

\[ \mathrm{P} [Y = y | m, s] = \exp(-s) \left( \frac{s + m}{s - m} \right)^{\frac{y}{2}} I_y \left( \sqrt{s^2 - m^2} \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m \\ \mathrm{var}[Y] &= s \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s) &= \frac{s y}{s^2 - m^2} - \frac{m}{2 \sqrt{s^2 - m^2}} \frac{ I_{y-1} \left( \sqrt{s^2 - m^2} \right) + I_{y+1} \left( \sqrt{s^2 - m^2} \right) }{ I_y \left( \sqrt{s^2 - m^2} \right) } \\ \nabla_{s} (y; m, s) &= -\frac{m y}{s^2 - m^2} + \frac{s}{2 \sqrt{s^2 - m^2}} \frac{ I_{y-1} \left( \sqrt{s^2 - m^2} \right) + I_{y+1} \left( \sqrt{s^2 - m^2} \right) }{ I_y \left( \sqrt{s^2 - m^2} \right) } - 1\\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s) &\approx \frac{m^2}{4 \left( s^2 - m^2 \right)} \left( \frac{2 s}{\sqrt{s^2 - m^2}} - \frac{ I_{m-1} \left( \sqrt{s^2 - m^2} \right) + I_{m+1} \left( \sqrt{s^2 - m^2} \right) }{ I_m \left( \sqrt{s^2 - m^2} \right) } \right)^2 \\ \mathcal{I}_{m, s} (m, s) &\approx - \frac{m s}{4 \left( s^2 - m^2 \right)} \left( \frac{2 s}{\sqrt{s^2 - m^2}} - \frac{ I_{m-1} \left( \sqrt{s^2 - m^2} \right) + I_{m+1} \left( \sqrt{s^2 - m^2} \right) }{ I_m \left( \sqrt{s^2 - m^2} \right) } \right)^2 \\ \mathcal{I}_{s, s} (m, s) &\approx \frac{s^2}{4 \left( s^2 - m^2 \right)} \left( \frac{2 s}{\sqrt{s^2 - m^2}} - \frac{ I_{m-1} \left( \sqrt{s^2 - m^2} \right) + I_{m+1} \left( \sqrt{s^2 - m^2} \right) }{ I_m \left( \sqrt{s^2 - m^2} \right) } \right)^2 \\ \end{aligned} \]

Parameters

• First rate parameter \(r_1 \in (0, \infty)\)
• Second rate parameter \(r_2 \in (0, \infty)\)
• Inflation parameter \(p \in (0, 1)\)

Probability Mass Function

\[ \begin{aligned} \mathrm{P} [Y = y | r_1, r_2, p] &= \begin{cases} p + (1 - p) \exp(-r_1 - r_2) I_0 \left( 2 \sqrt{r_1 r_2} \right) & \text{ for } y = 0 \\ (1 - p) \exp(-r_1 - r_2) \left( \frac{r_1}{r_2} \right)^{\frac{y}{2}} I_y \left( 2 \sqrt{r_1 r_2} \right) & \text{ for } y \neq 0 \\ \end{cases} \\ \end{aligned} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= (1 - p) (r_1 - r_2) \\ \mathrm{var}[Y] &= (1 - p) \left( p \left( r_1 - r_2 \right)^2 + r_1 + r_2 \right) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{r_1} (y; r_1, r_2, p) &= \begin{cases} \frac{(p - 1) \left( \sqrt{r_1 r_2} I_0 \left( 2 \sqrt{r_1 r_2} \right) - r_2 I_1 \left( 2 \sqrt{r_1 r_2} \right) \right)}{\sqrt{r_1 r_2} \left( p \exp(r_1 + r_2) + (1 - p) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right)} & \text{ for } y = 0 \\ \sqrt{\frac{r_2}{r_1}} \frac{I_{y-1} \left( 2 \sqrt{r_1 r_2} \right)}{I_y \left( 2 \sqrt{r_1 r_2} \right) } - 1 & \text{ for } y \neq 0 \\ \end{cases} \\ \nabla_{r_2} (y; r_1, r_2, p) &= \begin{cases} \frac{(p - 1) \left( \sqrt{r_1 r_2} I_0 \left( 2 \sqrt{r_1 r_2} \right) - r_1 I_1 \left( 2 \sqrt{r_1 r_2} \right) \right)}{\sqrt{r_1 r_2} \left( p \exp(r_1 + r_2) + (1 - p) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right)} & \text{ for } y = 0 \\ \sqrt{\frac{r_1}{r_2}} \frac{I_{y-1} \left( 2 \sqrt{r_1 r_2} \right)}{I_y \left( 2 \sqrt{r_1 r_2} \right) } -\frac{y}{r_2} - 1 & \text{ for } y \neq 0 \\ \end{cases} \\ \nabla_{p} (y; r_1, r_2, p) &= \begin{cases} \frac{\exp(r_1 + r_2) - I_0 \left( 2 \sqrt{r_1 r_2} \right)}{p \exp(r_1 + r_2) + (1 - p) I_0 \left( 2 \sqrt{r_1 r_2} \right)} & \text{ for } y = 0 \\ \frac{1}{p - 1} & \text{ for } y \neq 0 \\ \end{cases} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{r_1, r_1} (r_1, r_2, p) &\approx (1 - p) \left( 1 - \exp(-r_1 - r_2) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right) \left( 1 - \sqrt{\frac{r_2}{r_1}} \frac{I_{r_1 - r_2 -1} \left( 2 \sqrt{r_1 r_2} \right)}{I_{r_1 - r_2} \left( 2 \sqrt{r_1 r_2} \right)} \right)^2 \\ & \qquad + \frac{(1 - p)^2 \exp(-r_1 - r_2) \left( \sqrt{r_1 r_2} I_0 \left( 2 \sqrt{r_1 r_2} \right) - r_2 I_1 \left( 2 \sqrt{r_1 r_2} \right) \right)^2}{r_1 r_2 \left( p \exp(r_1 + r_2) + (1 - p) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right)} \\ \mathcal{I}_{r_1, r_2} (r_1, r_2, p) &\approx (1 - p) \left( 1 - \exp(-r_1 - r_2) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right) \left( 1 - \sqrt{\frac{r_2}{r_1}} \frac{I_{r_1 - r_2-1} \left( 2 \sqrt{r_1 r_2} \right)}{I_{r_1 - r_2} \left( 2 \sqrt{r_1 r_2} \right)} \right) \\ & \qquad \times \left( \frac{r_1}{r_2} - \sqrt{\frac{r_1}{r_2}} \frac{I_{r_1 - r_2 - 1} \left( 2 \sqrt{r_1 r_2} \right)}{I_{r_1 - r_2} \left( 2 \sqrt{r_1 r_2} \right)} \right) \\ & \qquad + \frac{(1 - p)^2 \exp(-r_1 - r_2) \left( \sqrt{r_1 r_2} I_0 \left( 2 \sqrt{r_1 r_2} \right) - r_2 I_1 \left( 2 \sqrt{r_1 r_2} \right) \right)}{r_1 r_2 \left( p \exp(r_1 + r_2) + (1 - p) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right)} \\ & \qquad \times \left( \sqrt{r_1 r_2} I_0 \left( 2 \sqrt{r_1 r_2} \right) - r_1 I_1 \left( 2 \sqrt{r_1 r_2} \right) \right) \\ \mathcal{I}_{r_1, p} (r_1, r_2, p) &= \frac{(p - 1) \left( 1 - \exp(-r_1 - r_2 ) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right)}{\sqrt{r_1 r_2} \left( p \exp(r_1 + r_2) + (1 - p) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right)} \\ & \qquad \times \left( \sqrt{r_1 r_2} I_0 \left( 2 \sqrt{r_1 r_2} \right) - r_2 I_1 \left( 2 \sqrt{r_1 r_2} \right) \right) \\ \mathcal{I}_{r_2, r_2} (r_1, r_2, p) &\approx (1 - p) \left( 1 - \exp(-r_1 - r_2) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right) \left( \frac{r_1}{r_2} - \sqrt{\frac{r_1}{r_2}} \frac{I_{r_1 - r_2 - 1} \left( 2 \sqrt{r_1 r_2} \right)}{I_{r_1 - r_2} \left( 2 \sqrt{r_1 r_2} \right)} \right)^2 \\ & \qquad + \frac{(1 - p)^2 \exp(-r_1 - r_2) \left( \sqrt{r_1 r_2} I_0 \left( 2 \sqrt{r_1 r_2} \right) - r_1 I_1 \left( 2 \sqrt{r_1 r_2} \right) \right)^2}{r_1 r_2 \left( p \exp(r_1 + r_2) + (1 - p) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right)} \\ \mathcal{I}_{r_2, p} (r_1, r_2, p) &= \frac{(p - 1) \left( 1 - \exp(-r_1 - r_2 ) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right)}{\sqrt{r_1 r_2} \left( p \exp(r_1 + r_2) + (1 - p) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right)} \\ & \qquad \times \left( \sqrt{r_1 r_2} I_0 \left( 2 \sqrt{r_1 r_2} \right) - r_1 I_1 \left( 2 \sqrt{r_1 r_2} \right) \right) \\ \mathcal{I}_{p, p} (r_1, r_2, p) &= \frac{\exp(r_1 + r_2) - I_0 \left( 2 \sqrt{r_1 r_2} \right)}{(1 - p) \left( p \exp(r_1 + r_2) + (1 - p) I_0 \left( 2 \sqrt{r_1 r_2} \right) \right)} \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in \mathbb{R}\)
• Dispersion parameter \(s \in (0, \infty)\)
• Inflation parameter \(p \in (0, 1)\)

Probability Mass Function

\[ \begin{aligned} \mathrm{P} [Y = y | m, s, p] &= \begin{cases} p + (1 - p) \exp(-|m| - s) I_0 \left( \sqrt{s^2 + 2 |m| s} \right) & \text{ for } y = 0 \\ (1 - p) \exp(-|m| - s) \left( \frac{|m| + m + s}{|m| - m + s} \right)^{\frac{y}{2}} I_y \left( \sqrt{s^2 + 2 |m| s} \right) & \text{ for } y \neq 0 \\ \end{cases} \\ \end{aligned} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= (1 - p) m \\ \mathrm{var}[Y] &= (1 - p) \left( |m| + s + p m^2 \right) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s, p) &= \begin{cases} \frac{\mathrm{sgn}(m) (p - 1) \left( \sqrt{s^2 + 2 |m| s} I_0 \left( \sqrt{s^2 + 2 |m| s} \right) - s I_1 \left( \sqrt{s^2 + 2 |m| s} \right) \right)}{\sqrt{s^2 + 2 |m| s} \left( (1 - p) I_0 \left( \sqrt{s^2 + 2 |m| s} \right) + p \exp(|m| + s) \right)} & \text{ for } y = 0 \\ \frac{y}{2|m| + s} + \frac{\mathrm{sgn}(m) s}{2 \sqrt{s^2 + 2 |m| s}} \frac{ I_{y-1} \left( \sqrt{s^2 + 2 |m| s} \right) + I_{y+1} \left( \sqrt{s^2 + 2 |m| s} \right) }{ I_y \left( \sqrt{s^2 + 2 |m| s} \right) } - \mathrm{sgn}(m) & \text{ for } y \neq 0 \\ \end{cases} \\ \nabla_{s} (y; m, s, p) &= \begin{cases} \frac{ (p - 1) \left( \sqrt{s^2 + 2 |m| s} I_0 \left( \sqrt{s^2 + 2 |m| s} \right) - (|m| + s) I_1 \left( \sqrt{s^2 + 2 |m| s} \right) \right) }{\sqrt{s^2 + 2 |m| s} \left( (1 - p) I_0 \left( \sqrt{s^2 + 2 |m| s} \right) + p \exp(|m| + s) \right)} & \text{ for } y = 0 \\ - \frac{m y}{s^2 + 2 |m| s} + \frac{|m| + s}{2 \sqrt{s^2 + 2 |m| s}} \frac{ I_{y-1} \left( \sqrt{s^2 + 2 |m| s} \right) + I_{y+1} \left( \sqrt{s^2 + 2 |m| s} \right) }{ I_y \left( \sqrt{s^2 + 2 |m| s} \right) } - 1 & \text{ for } y \neq 0 \\ \end{cases} \\ \nabla_{p} (y; m, s, p) &= \begin{cases} \frac{\exp(|m| + s) - I_0 \left( \sqrt{s^2 + 2 |m| s} \right)}{p \exp(|m| + s) + (1 - p) I_0 \left( \sqrt{s^2 + 2 |m| s} \right)} & \text{ for } y = 0 \\ \frac{1}{p - 1} & \text{ for } y \neq 0 \\ \end{cases} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s, p) &\approx \frac{s^2 (1 - p) \left( 1 - \exp(-|m|-s) I_{0} \left( \sqrt{s^2 + 2 |m| s} \right) \right)}{4 s^2 + 8 |m| s} \\ & \qquad \times \left( \frac{2 (|m| + s)}{\sqrt{s^2 + 2 |m| s}} - \frac{ I_{m-1} \left( \sqrt{s^2 + 2 |m| s} \right) + I_{m+1} \left( \sqrt{s^2 + 2 |m| s} \right) }{ I_m \left( \sqrt{s^2 + 2 |m| s} \right)} \right)^2 \\ & \qquad + \frac{(1 - p)^2 \exp(-|m| - s) }{\left( s^2 + 2 |m| s \right) \left( p \exp(|m| + s) + (1 - p) I_0 \left( \sqrt{s^2 + 2 |m| s} \right) \right)} \\ & \qquad \times \left( \sqrt{s^2 + 2 |m| s} I_0 \left( \sqrt{s^2 + 2 |m| s} \right) - s I_1 \left( \sqrt{s^2 + 2 |m| s} \right) \right)^2 \\ \mathcal{I}_{m, s} (m, s, p) &\approx \frac{\mathrm{sgn}(m) s (1 - p) (|m| + s) \left( 1 - \exp(-|m|-s) I_{0} \left( \sqrt{s^2 + 2 |m| s} \right) \right)}{4 s^2 + 8 |m| s} \\ & \qquad \times \left( \frac{2 (|m| + s)}{\sqrt{s^2 + 2 |m| s}} - \frac{ I_{m-1} \left( \sqrt{s^2 + 2 |m| s} \right) + I_{m+1} \left( \sqrt{s^2 + 2 |m| s} \right) }{ I_m \left( \sqrt{s^2 + 2 |m| s} \right)} \right)^2 \\ & \qquad + \frac{\mathrm{sgn}(m) (1 - p)^2 \exp(-|m| - s)}{\left( s^2 + 2 |m| s \right) \left( p \exp(|m| + s) + (1 - p) I_0 \left( \sqrt{s^2 + 2 |m| s} \right) \right)} \\ & \qquad \times \left( \sqrt{s^2 + 2 |m| s} I_0 \left( \sqrt{s^2 + 2 |m| s} \right) - s I_1 \left( \sqrt{s^2 + 2 |m| s} \right) \right) \\ & \qquad \times \left( \sqrt{s^2 + 2 |m| s} I_0 \left( \sqrt{s^2 + 2 |m| s} \right) - (|m| + s) I_1 \left( \sqrt{s^2 + 2 |m| s} \right) \right) \\ \mathcal{I}_{m, p} (m, s, p) &= \frac{\mathrm{sgn}(m) (p - 1) \left( 1 - \exp(-|m| - s) I_0 \left( \sqrt{s^2 + 2 |m| s} \right) \right)}{\sqrt{s^2 + 2 |m| s} \left( p \exp(|m| + s) + (1 - p) I_0 \left( \sqrt{s^2 + 2 |m| s} \right) \right)} \\ & \qquad \times \left( \sqrt{s^2 + 2 |m| s} I_0 \left( \sqrt{s^2 + 2 |m| s} \right) - s I_1 \left( \sqrt{s^2 + 2 |m| s} \right) \right) \\ \mathcal{I}_{s, s} (m, s, p) &\approx \frac{(1 - p) (|m| + s)^2 \left( 1 - \exp(-|m|-s) I_{0} \left( \sqrt{s^2 + 2 |m| s} \right) \right)}{4 s^2 + 8 |m| s} \\ & \qquad \times \left( \frac{2 (|m| + s)}{\sqrt{s^2 + 2 |m| s}} - \frac{ I_{m-1} \left( \sqrt{s^2 + 2 |m| s} \right) + I_{m+1} \left( \sqrt{s^2 + 2 |m| s} \right) }{ I_m \left( \sqrt{s^2 + 2 |m| s} \right)} \right)^2 \\ & \qquad + \frac{(1 - p)^2 \exp(-|m| - s)}{\left( s^2 + 2 |m| s \right) \left( p \exp(|m| + s) + (1 - p) I_0 \left( \sqrt{s^2 + 2 |m| s} \right) \right)} \\ & \qquad \times \left( \sqrt{s^2 + 2 |m| s} I_0 \left( \sqrt{s^2 + 2 |m| s} \right) - (|m| + s) I_1 \left( \sqrt{s^2 + 2 |m| s} \right) \right)^2 \\ \mathcal{I}_{s, p} (m, s, p) &= \frac{(p - 1) \left( 1 - \exp(-|m| - s) I_0 \left( \sqrt{s^2 + 2 |m| s} \right) \right)}{\sqrt{s^2 + 2 |m| s} \left( p \exp(|m| + s) + (1 - p) I_0 \left( \sqrt{s^2 + 2 |m| s} \right) \right)} \\ & \qquad \times \left( \sqrt{s^2 + 2 |m| s} I_0 \left( \sqrt{s^2 + 2 |m| s} \right) - (|m| + s) I_1 \left( \sqrt{s^2 + 2 |m| s} \right) \right) \\ \mathcal{I}_{p, p} (m, s, p) &= \frac{\exp(|m| + s) - I_0 \left( \sqrt{s^2 + 2 |m| s} \right)}{(1 - p) \left( p \exp(|m| + s) + (1 - p) I_0 \left( \sqrt{s^2 + 2 |m| s} \right) \right)} \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in \mathbb{R}\)
• Variance parameter \(s \in (|m|, \infty)\)
• Inflation parameter \(p \in (0, 1)\)

Probability Mass Function

\[ \begin{aligned} \mathrm{P} [Y = y | m, s, p] &= \begin{cases} p + (1 - p) \exp(-s) I_0 \left( \sqrt{s^2 - m^2} \right) & \text{ for } y = 0 \\ (1 - p) \exp(-s) \left( \frac{s + m}{s - m} \right)^{\frac{y}{2}} I_y \left( \sqrt{s^2 - m^2} \right) & \text{ for } y \neq 0 \\ \end{cases} \\ \end{aligned} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= (1 - p) m \\ \mathrm{var}[Y] &= (1 - p) \left( s + p m^2 \right) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s, p) &= \begin{cases} \frac{m (p - 1) I_{1} \left( \sqrt{s^2 - m^2} \right)}{\sqrt{s^2 - m^2} \left( p \exp(s) + (1 - p) I_{0} \left( \sqrt{s^2 - m^2} \right) \right)} & \text{ for } y = 0 \\ \frac{s y}{s^2 - m^2} - \frac{m}{2 \sqrt{s^2 - m^2}} \frac{ I_{y-1} \left( \sqrt{s^2 - m^2} \right) + I_{y+1} \left( \sqrt{s^2 - m^2} \right) }{ I_y \left( \sqrt{s^2 - m^2} \right) } & \text{ for } y \neq 0 \\ \end{cases} \\ \nabla_{s} (y; m, s, p) &= \begin{cases} \frac{ (p - 1) \left( \sqrt{s^2 - m^2} I_{0} \left( \sqrt{s^2 - m^2} \right) - s I_{1} \left( \sqrt{s^2 - m^2} \right) \right) }{\sqrt{s^2 - m^2} \left( p \exp(s) + (1 - p) I_{0} \left( \sqrt{s^2 - m^2} \right) \right)} & \text{ for } y = 0 \\ -\frac{m y}{s^2 - m^2} + \frac{s}{2 \sqrt{s^2 - m^2}} \frac{ I_{y-1} \left( \sqrt{s^2 - m^2} \right) + I_{y+1} \left( \sqrt{s^2 - m^2} \right) }{ I_y \left( \sqrt{s^2 - m^2} \right) } - 1 & \text{ for } y \neq 0 \\ \end{cases} \\ \nabla_{p} (y; m, s, p) &= \begin{cases} \frac{\exp(s) - I_{0} \left( \sqrt{s^2 - m^2} \right)}{p \exp(s) + (1 - p) I_{0} \left( \sqrt{s^2 - m^2} \right)} & \text{ for } y = 0 \\ \frac{1}{p - 1} & \text{ for } y \neq 0 \\ \end{cases} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s, p) &\approx \frac{m^2 (1 - p) \left( 1 - \exp(-s) I_{0} \left( \sqrt{s^2 - m^2} \right) \right)}{4 \left( s^2 - m^2 \right)} \\ & \qquad \times \left( \frac{2 s}{\sqrt{s^2 - m^2}} - \frac{ I_{m-1} \left( \sqrt{s^2 - m^2} \right) + I_{m+1} \left( \sqrt{s^2 - m^2} \right) }{ I_m \left( \sqrt{s^2 - m^2} \right) } \right)^2 \\ & \qquad + \frac{m^2 (1 - p)^2 \exp(-s) I_{1} \left( \sqrt{s^2 - m^2} \right)^2}{\left( s^2 - m^2 \right) \left( p \exp(s) + (1 - p) I_0 \left( \sqrt{s^2 - m^2} \right) \right)} \\ \mathcal{I}_{m, s} (m, s, p) &\approx \frac{m s (p - 1) \left( 1 - \exp(-s) I_{0} \left( \sqrt{s^2 - m^2} \right) \right) }{4 \left( s^2 - m^2 \right)} \\ & \qquad \times \left( \frac{2 s}{\sqrt{s^2 - m^2}} - \frac{ I_{m-1} \left( \sqrt{s^2 - m^2} \right) + I_{m+1} \left( \sqrt{s^2 - m^2} \right) }{ I_m \left( \sqrt{s^2 - m^2} \right) } \right)^2 \\ & \qquad + \frac{m (1 - p)^2 \exp(-s) I_{1} \left( \sqrt{s^2 - m^2} \right)}{\left( s^2 - m^2 \right) \left( p \exp(s) + (1 - p) I_0 \left( \sqrt{s^2 - m^2} \right) \right)} \\ & \qquad \times \left( \sqrt{s^2 - m^2} I_0 \left( \sqrt{s^2 - m^2} \right) - s I_1 \left( \sqrt{s^2 - m^2} \right) \right) \\ \mathcal{I}_{m, p} (m, s, p) &= \frac{m (p - 1) \left( 1 - \exp(-s) I_0 \left( \sqrt{s^2 - m^2} \right) \right) I_1 \left( \sqrt{s^2 - m^2} \right)}{\sqrt{s^2 - m^2} \left( p \exp(s) + (1 - p) I_0 \left( \sqrt{s^2 - m^2} \right) \right)} \\ \mathcal{I}_{s, s} (m, s, p) &\approx \frac{s^2 (1 - p) \left( 1 - \exp(-s) I_{0} \left( \sqrt{s^2 - m^2} \right) \right)}{4 \left( s^2 - m^2 \right)} \\ & \qquad \times \left( \frac{2 s}{\sqrt{s^2 - m^2}} - \frac{ I_{m-1} \left( \sqrt{s^2 - m^2} \right) + I_{m+1} \left( \sqrt{s^2 - m^2} \right) }{ I_m \left( \sqrt{s^2 - m^2} \right) } \right)^2 \\ & \qquad + \frac{(1 - p)^2 \exp(-s) \left( \sqrt{s^2 - m^2} I_0 \left( \sqrt{s^2 - m^2} \right) - s I_1 \left( \sqrt{s^2 - m^2} \right) \right)^2}{\left( s^2 - m^2 \right) \left( p \exp(s) + (1 - p) I_0 \left( \sqrt{s^2 - m^2} \right) \right)} \\ \mathcal{I}_{s, p} (m, s, p) &= \frac{(p - 1) \left( 1 - \exp(-s) I_0 \left( \sqrt{s^2 - m^2} \right) \right) }{\sqrt{s^2 - m^2} \left( p \exp(s) + (1 - p) I_0 \left( \sqrt{s^2 - m^2} \right) \right)} \\ & \qquad \times \left( \sqrt{s^2 - m^2} I_0 \left( \sqrt{s^2 - m^2} \right) - s I_1 \left( \sqrt{s^2 - m^2} \right) \right) \\ \mathcal{I}_{p, p} (m, s, p) &= \frac{\exp(s) - I_0 \left( \sqrt{s^2 - m^2} \right)}{(1 - p) \left( p \exp(s) + (1 - p) I_0 \left( \sqrt{s^2 - m^2} \right) \right) } \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in \mathbb{R}\)
• Concentration parameter \(v \in (0, \infty)\)

Density Function

\[ f(y | m, v) = \frac{1}{2 \pi I_0(v)} \exp \left( v \cos(y - m) \right) \]

Circular Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m \\ \mathrm{var}[Y] &= 1 - \frac{I_1(v)}{I_0(v)} \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, v) &= v \sin(y - m) \\ \nabla_{v} (y; m, v) &= \cos(y - m) - \frac{I_1(v)}{I_0(v)} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, v) &= v \frac{I_1(v)}{I_0(v)} \\ \mathcal{I}_{m, v} (m, v) &= 0 \\ \mathcal{I}_{v, v} (m, v) &= \frac{1}{2} - \left( \frac{I_1(v)}{I_0(v)} \right)^2 + \frac{I_2(v)}{2 I_0(v)} \\ \end{aligned} \]

Parameters

• First concentration parameter \(a_1 \in (0, \infty)\)
• Second concentration parameter \(a_2 \in (0, \infty)\)

Density Function

\[ f(y | a_1, a_2) = \frac{1}{B(a_1, a_2)} y^{a_1 - 1} (1 - y)^{a_2 - 1} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= \frac{a_1}{a_1 + a_2} \\ \mathrm{var}[Y] &= \frac{a_1 a_2}{(a_1 + a_2)^2 (a_1 + a_2 + 1)} \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{a} (y; a_1, a_2) &= \psi_0(a_1 + a_2) - \psi_0(a_1) + \ln(y) \\ \nabla_{b} (y; a_1, a_2) &= \psi_0(a_1 + a_2) - \psi_0(a_2) + \ln(1 - y) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{a_1, a_1} (a_1, a_2) &= \psi_1(a_1) - \psi_1(a_1 + a_2) \\ \mathcal{I}_{a_1, a_2} (a_1, a_2) &= -\psi_1(a_1 + a_2) \\ \mathcal{I}_{a_2, a_2} (a_1, a_2) &= \psi_1(a_2) - \psi_1(a_1 + a_2) \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in (0, 1)\)
• Size parameter \(v \in (0, \infty)\)

Density Function

\[ f(y | m, v) = \frac{1}{B(m v, (1 - m) v)} y^{m v - 1} (1 - y)^{(1 - m) v - 1} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m \\ \mathrm{var}[Y] &= \frac{m (1 - m)}{v + 1} \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, v) &= \frac{v}{1 - m} (\psi_0(v) - \psi_0(m v) + \ln(y)) \\ \nabla_{v} (y; m, v) &= \psi_0(v) - \psi_0(v - m v) + \ln(1 - y) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, v) &= \frac{v^2}{(1 - m)^2} (\psi_1(m v) - \psi_1(v)) \\ \mathcal{I}_{m, v} (m, v) &= \frac{v}{m - 1} \psi_1(v) \\ \mathcal{I}_{v, v} (m, v) &= \psi_1(v - m v) - \psi_1(v) \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in (0, 1)\)
• Variance parameter \(s \in (0, m (1 - m))\)

Density Function

\[ f(y | m, s) = \frac{1}{B \left( m \left( \frac{m - m^2}{s} - 1 \right), (1 - m) \left( \frac{m - m^2}{s} - 1 \right) \right)} y^{m \left( \frac{m - m^2}{s} - 1 \right) - 1} (1 - y)^{(1 - m) \left( \frac{m - m^2}{s} - 1 \right) - 1} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m \\ \mathrm{var}[Y] &= s \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s) &= \frac{m^2 - m + s}{(m - 1) s} \left( \psi_0 \left( \frac{m - m^2}{s} - 1 \right) - \psi_0 \left( m \left( \frac{m - m^2}{s} - 1 \right) \right) + \ln(y) \right) \\ \nabla_{s} (y; m, s) &= \frac{s^2 (3 m^2 - 2 m + s)}{m (m - 1) (m^2 - m + s)} \left( \psi_0 \left( \frac{m - m^2}{s} - 1 \right) - \psi_0 \left( (1 - m) \left( \frac{m - m^2}{s} - 1 \right) \right) + \ln(1 - y) \right) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s) &= \frac{(m^2 - m + s)^2}{(m - 1)^2 s^2} \left( \psi_1 \left( m \left( \frac{m - m^2}{s} - 1 \right) \right) - \psi_1 \left( \frac{m - m^2}{s} - 1 \right) \right) \\ \mathcal{I}_{m, s} (m, s) &= \frac{s (2 m - 3 m^2 - s)}{m (m^2 - 2 m + 1)} \psi_1 \left( \frac{m - m^2}{s} - 1 \right) \\ \mathcal{I}_{s, s} (m, s) &= \frac{s^2 (3 m^2 - 2 m + s)^2}{(m - 1)^4 m^2} \left( \psi_1 \left( (1 - m) \left( \frac{m - m^2}{s} - 1 \right) \right) - \psi_1 \left( \frac{m - m^2}{s} - 1 \right) \right) \\ \end{aligned} \]

Parameters

• First concentration parameter \(a \in (0, \infty)\)
• Second concentration parameter \(b \in (0, \infty)\)

Density Function

\[ f(y | a, b) = a b y^{a - 1} \left(1 - y^a \right)^{b - 1} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= b B \left(1 + \frac{1}{a}, b \right) \\ \mathrm{var}[Y] &= b B \left(1 + \frac{2}{a}, b \right) - b^2 B \left(1 + \frac{1}{a}, b \right)^2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{a} (y; a, b) &= \frac{(b - 1) \ln(y)}{y^a - 1} + b \ln(y) + \frac{1}{a} \\ \nabla_{b} (y; a, b) &= \ln \left( 1 - y^a \right) + \frac{1}{b} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{a, a} (a, b) &= \frac{1}{a^2} + \frac{b}{a^2 (b - 2)} \left( \left(\psi_0(b) - \psi_0(2) \right)^2 - \left( \psi_1(b) - \psi_1(2) \right) \right) \\ \mathcal{I}_{a, b} (a, b) &= - \frac{\psi_0(b + 1) - \psi_0(2)}{a (b - 1)} \\ \mathcal{I}_{b, b} (a, b) &= \frac{1}{b^2} \\ \end{aligned} \]

Parameters

• Logit-mean parameter \(m \in \mathbb{R}\)
• Logit-variance parameter \(s \in (0, \infty)\)

Density Function

\[ f(y | m, s) = \frac{1}{y (1 - y)}\frac{1}{\sqrt{2 \pi s}} \exp \left( - \frac{1}{2 s} \left( \ln \left( \frac{y}{1-y} \right) - m \right)^2 \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &\approx \frac{1}{K - 1} \sum_{k = 1}^{K - 1} \frac{1}{1 + \exp( - \Phi^{-1}_{m,s} (k / K)} \\ \mathrm{var}[Y] &\approx \frac{1}{K - 1} \sum_{k = 1}^{K - 1} \left( \frac{1}{1 + \exp( - \Phi^{-1}_{m,s} (k / K)} \right)^2 - \left( \frac{1}{K - 1} \sum_{k = 1}^{K - 1} \frac{1}{1 + \exp( - \Phi^{-1}_{m,s} (k / K)} \right)^2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s) &= \frac{1}{s} \left( \ln \left( \frac{y}{1-y} \right) - m \right) \\ \nabla_{s} (y; m, s) &= \frac{1}{2 s^2} \left( \ln \left( \frac{y}{1-y} \right) - m \right)^2 - \frac{1}{2s} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s) &= \frac{1}{s} \\ \mathcal{I}_{m, s} (m, s) &= 0 \\ \mathcal{I}_{s, s} (m, s) &= \frac{1}{2 s^2} \\ \end{aligned} \]

Parameters

• Concentration parameters \(a_i \in (0, \infty)\), \(i = 1,\ldots,n\)

Vector Notation

• Concentration vector \(\boldsymbol{a}\) of length \(n\)

Density Function

\[ f(\boldsymbol{y} | \boldsymbol{a}) = \frac{1}{B(\boldsymbol{a})} \prod_{i=1}^n y_i^{a_i - 1} \]

Moments

\[ \begin{aligned} \mathrm{E}[\boldsymbol{Y}] &= \frac{1}{\sum_{i=1}^n a_i} \boldsymbol{a} \\ \mathrm{var}[\boldsymbol{Y}] &= \frac{1}{1 + \sum_{i=1}^n a_i} \left( \frac{1}{\sum_{i=1}^n a_i} \mathrm{diag}(\boldsymbol{a}) - \frac{1}{\left( \sum_{i=1}^n a_i \right)^2} \boldsymbol{a} \boldsymbol{a}^\intercal \right) \\ \end{aligned} \]

Score

\[ \nabla_{\boldsymbol{a}} (\boldsymbol{y}; \boldsymbol{a}) = \ln(\boldsymbol{y}) - \psi_0 (\boldsymbol{a}) + \psi_0 \left( \sum_{i=1}^n a_i \right) \\ \]

Fisher Information

\[ \mathcal{I}_{\boldsymbol{a}, \boldsymbol{a}} (\boldsymbol{a}) = \mathrm{diag} \left( \psi_1 \left( \boldsymbol{a} \right) \right) - \psi_1 \left( \sum_{i=1}^n a_i \right) \\ \]

Parameters

• Scale parameter \(s \in (0, \infty)\)
• Shape parameter \(a \in (0, \infty)\)

Density Function

\[ f(y | s, a) = \frac{\sqrt{\frac{s}{y}} \left( 1 + \frac{s}{y} \right)}{2 a s \sqrt{2 \pi}} \exp \left( \frac{2 - \frac{y}{s} - \frac{s}{y}}{2 a^2} \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= s \left( 1 + \frac{a^2}{2} \right) \\ \mathrm{var}[Y] &= s^2 a^2 \left( 1 + \frac{5 a^2}{4} \right) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{s} (y; s, a) &= \frac{y}{2 a^2 s^2} - \frac{1}{2 a^2 y} + \frac{1}{s + y} - \frac{1}{2 s} \\ \nabla_{a} (y; s, a) &= \frac{y}{a^3 s} + \frac{s}{a^3 y} - \frac{2 + a^2}{a^3} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{s, s} (s, a) &= \frac{1}{a^2 s^2} \left( 1 + \frac{a}{\sqrt{2 \pi}} \left( a \sqrt{\frac{\pi}{2}} - \pi \exp \left( \frac{2}{a^2} \right) \left(1 - \Phi \left( \frac{2}{a} \right) \right) \right) \right) \\ \mathcal{I}_{s, a} (s, a) &= 0 \\ \mathcal{I}_{a, a} (s, a) &= \frac{2}{a^2} \\ \end{aligned} \]

Parameters

• Scale parameter \(s \in (0, \infty)\)
• First shape parameter \(a \in (0, \infty)\)
• Second shape parameter \(b \in (0, \infty)\)

Density Function

\[ f(y | s, a, b) = \frac{a b}{s} \left( \frac{y}{s} \right)^{a - 1} \left( 1 + \left( \frac{y}{s} \right)^a \right)^{-b - 1} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= s b B \left( b - \frac{1}{a}, 1 + \frac{1}{a} \right), & \quad \text{for } a &> 1 \\ \mathrm{var}[Y] &= s^2 b B \left( b - \frac{2}{a}, 1 + \frac{2}{a} \right) - s^2 b^2 B \left( b - \frac{1}{a}, 1 + \frac{1}{a} \right)^2, & \quad \text{for } a &> 2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{s} (y; s, a, b) &= \frac{a}{s} \left(b \left (\frac{y}{s} \right)^a - 1 \right) \left( \left( \frac{y}{s} \right)^a + 1 \right)^{-1} \\ \nabla_{a} (y; s, a, b) &= \frac{1}{a} - \left( b \left( \frac{y}{s} \right)^a - 1 \right) \ln \left( \frac{y}{s} \right) \left( \left( \frac{y}{s} \right)^a + 1 \right)^{-1} \\ \nabla_{b} (y; s, a, b) &= \frac{1}{b} - \ln \left( \left( \frac{y}{s} \right)^a + 1 \right) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{s, s} (s, a, b) &= \frac{a^2 b}{s^2 (b + 2)} \\ \mathcal{I}_{s, a} (s, a, b) &= - \frac{b ( 1 - \gamma - \psi_0(b + 1))}{s (b + 2)} \\ \mathcal{I}_{s, b} (s, a, b) &= - \frac{a}{s (b + 1)} \\ \mathcal{I}_{a, a} (s, a, b) &= \frac{1}{\alpha^2} \left( 1 + \frac{b}{b + 2} \left( \frac{\pi^2}{6} + \gamma^2 - 2 \gamma + 2 (\gamma - 1) \psi_0(b + 1) + \psi_0(b + 1)^2 + \psi_1(b + 1) \right) \right) \\ \mathcal{I}_{a, b} (s, a, b) &= \frac{1 - \gamma - \psi_0(b)}{a (b + 1)} \\ \mathcal{I}_{b, b} (s, a, b) &= \frac{1}{b^2} \\ \end{aligned} \]

Parameter

• Rate parameter \(r \in (0, \infty)\)

Density Function

\[ f(y | r) = r \exp \left( -r y \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= \frac{1}{r} \\ \mathrm{var}[Y] &= \frac{1}{r^2} \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{r} (y; r) &= \frac{1}{r} - y \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{r, r} (r) &= \frac{1}{r^2} \\ \end{aligned} \]

Parameter

• Scale parameter \(s \in (0, \infty)\)

Density Function

\[ f(y | s) = \frac{1}{s} \exp \left( - \frac{y}{s} \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= s \\ \mathrm{var}[Y] &= s^2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{s} (y; s) &= \frac{y - s}{s^2} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{s, s} (s) &= \frac{1}{s^2} \\ \end{aligned} \]

Parameters

• Rate parameter \(r \in (0, \infty)\)
• Shape parameter \(p \in (0, 1)\)

Density Function

\[ f(y | r, p) = \frac{r}{- \ln(p)} \frac{(1 - p) \exp(-r y)}{1 - (1 - p) \exp(-r y)} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= - \frac{\mathrm{Li}_2(1 - p)}{r \ln(p)} \\ \mathrm{var}[Y] &= - 2 \frac{\mathrm{Li}_3(1 - p)}{r^2 \ln(p)} - \left( \frac{\mathrm{Li}_2(1 - p)}{r \ln(p)} \right)^2 \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{r} (y; r, p) &= \frac{1}{r} - y - \frac{y (1 - p) \exp(-r y)}{1 - (1 - p) \exp(-r y)} \\ \nabla_{p} (y; r, p) &= -\frac{1}{p log(p)} - \frac{1}{1 - p} - \frac{\exp(-r y)}{1 - (1 - p) \exp(-r y)} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{r, r} (r, p) &= -\frac{\mathrm{Li}_2(1 - p)}{r^2 \ln(p)} \\ \mathcal{I}_{r, p} (r, p) &= \frac{1 - p + p \ln(p)}{2 r p (1 - p) \ln(p)} \\ \mathcal{I}_{p, p} (r, p) &= \frac{1}{(1 - p)^2} - \frac{\ln(p) + 1}{(p \ln(p))^2} + \frac{1 - 4 p + 3 p^2 - 2 p^2 \ln(p)}{2 p^2 (1 - p)^2 \ln(p)} \\ \end{aligned} \]

Parameters

• Scale parameter \(s \in (0, \infty)\)
• Shape parameter \(a \in (0, \infty)\)

Density Function

\[ f(y | s, a) = \frac{a}{s} \left( \frac{y}{s} \right)^{a - 1} \left( 1 + \left ( \frac{y}{s} \right)^a \right)^{-2} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= s \frac{\pi / a}{\sin(\pi / a)}, & \quad \text{for } a &> 1 \\ \mathrm{var}[Y] &= s^2 \left( \frac{2 \pi / a}{\sin(2 \pi / a)} - \frac{\pi^2 / a^2}{\sin(\pi / a)^2} \right), & \quad \text{for } a &> 2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{s} (y; s, a) &= \frac{a}{s} \left( \left( \frac{y}{s} \right)^a - 1 \right) \left( \left( \frac{y}{s} \right)^a + 1 \right)^{-1} \\ \nabla_{a} (y; s, a) &= \frac{1}{a} - \left( \left( \frac{y}{s} \right)^a - 1 \right) \ln \left( \frac{y}{s} \right) \left( \left( \frac{y}{s} \right)^a + 1 \right)^{-1} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{s, s} (s, a) &= \frac{a^2}{3 s^2} \\ \mathcal{I}_{s, a} (s, a) &= 0 \\ \mathcal{I}_{a, a} (s, a) &= \frac{\pi^2 + 3}{9 a^2} \\ \end{aligned} \]

Parameters

• Rate parameter \(r \in (0, \infty)\)
• Shape parameter \(a \in (0, \infty)\)

Density Function

\[ f(y | r, a) = \frac{r}{\Gamma(a)} (r y)^{a - 1} \exp \left( -r y \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= \frac{a}{r} \\ \mathrm{var}[Y] &= \frac{a}{r^2} \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{r} (y; r, a) &= \frac{a - r y}{r} \\ \nabla_{a} (y; r, a) &= \ln(r y) - \psi_0(a) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{r, r} (r, a) &= \frac{a}{r^2} \\ \mathcal{I}_{r, a} (r, a) &= - \frac{1}{r} \\ \mathcal{I}_{a, a} (r, a) &= \psi_1(a) \\ \end{aligned} \]

Parameters

• Scale parameter \(s \in (0, \infty)\)
• Shape parameter \(a \in (0, \infty)\)

Density Function

\[ f(y | s, a) = \frac{1}{\Gamma(a)} \frac{1}{s} \left( \frac{y}{s} \right)^{a - 1} \exp \left( - \frac{y}{s} \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= a s \\ \mathrm{var}[Y] &= a s^2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{s} (y; s, a) &= \frac{y - a s}{s^2} \\ \nabla_{a} (y; s, a) &= \ln \left( \frac{y}{s} \right) - \psi_0(a) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{s, s} (s, a) &= \frac{a}{s^2} \\ \mathcal{I}_{s, a} (s, a) &= \frac{1}{s} \\ \mathcal{I}_{a, a} (s, a) &= \psi_1(a) \\ \end{aligned} \]

Parameters

• Rate parameter \(r \in (0, \infty)\)
• First shape parameter \(a \in (0, \infty)\)
• Second shape parameter \(b \in (0, \infty)\)

Density Function

\[ f(y | r, a, b) = \frac{r b}{\Gamma(a)} (r y)^{a b - 1} \exp \left( -(r y)^b \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= \frac{1}{r} \frac{\Gamma \left(a + b^{-1} \right)}{\Gamma \left( a \right) } \\ \mathrm{var}[Y] &= \frac{1}{r^2} \left( \frac{\Gamma \left(a + 2 b^{-1} \right)}{\Gamma \left( a \right) } - \left( \frac{\Gamma \left(a + b^{-1} \right)}{\Gamma \left( a \right) } \right)^2 \right) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{r} (y; r, a, b) &= \frac{b}{r} \left( a - (r y)^b \right) \\ \nabla_{a} (y; r, a, b) &= b \ln(r y) - \psi_0(a) \\ \nabla_{b} (y; r, a, b) &= \left( a - (r y)^b \right) \ln (r y) + \frac{1}{b} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{r, r} (r, a, b) &= \frac{a b^2}{r^2} \\ \mathcal{I}_{r, a} (r, a, b) &= - \frac{b}{r} \\ \mathcal{I}_{r, b} (r, a, b) &= \frac{a \psi_0(a) + 1}{r} \\ \mathcal{I}_{a, a} (r, a, b) &= \psi_1(a) \\ \mathcal{I}_{a, b} (r, a, b) &= - \frac{\psi_0(a)}{b} \\ \mathcal{I}_{b, b} (r, a, b) &= \frac{a \psi_0(a)^2 + 2 \psi_0(a) + a \psi_1(a) + 1}{b^2} \\ \end{aligned} \]

Parameters

• Scale parameter \(s \in (0, \infty)\)
• First shape parameter \(a \in (0, \infty)\)
• Second shape parameter \(b \in (0, \infty)\)

Density Function

\[ f(y | s, a, b) = \frac{1}{\Gamma(a)} \frac{b}{s} \left( \frac{y}{s} \right)^{a b - 1} \exp \left( - \left( \frac{y}{s} \right)^b \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= s \frac{\Gamma \left(a + b^{-1} \right)}{\Gamma \left( a \right) } \\ \mathrm{var}[Y] &= s^2 \left( \frac{\Gamma \left(a + 2 b^{-1} \right)}{\Gamma \left( a \right) } - \left( \frac{\Gamma \left(a + b^{-1} \right)}{\Gamma \left( a \right) } \right)^2 \right) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{s} (y; s, a, b) &= \frac{b}{s} \left( \left( \frac{y}{s} \right)^b - a \right) \\ \nabla_{a} (y; s, a, b) &= b \ln \left( \frac{y}{s} \right) - \psi_0(a) \\ \nabla_{b} (y; s, a, b) &= \left( a - \left( \frac{y}{s} \right)^b \right) \ln \left( \frac{y}{s} \right) + \frac{1}{b} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{s, s} (s, a, b) &= \frac{a b^2}{s^2} \\ \mathcal{I}_{s, a} (s, a, b) &= \frac{b}{s} \\ \mathcal{I}_{s, b} (s, a, b) &= - \frac{a \psi_0(a) + 1}{s} \\ \mathcal{I}_{a, a} (s, a, b) &= \psi_1(a) \\ \mathcal{I}_{a, b} (s, a, b) &= - \frac{\psi_0(a)}{b} \\ \mathcal{I}_{b, b} (s, a, b) &= \frac{a \psi_0(a)^2 + 2 \psi_0(a) + a \psi_1(a) + 1}{b^2} \\ \end{aligned} \]

Parameters

• Log-mean parameter \(m \in \mathbb{R}\)
• Log-variance parameter \(s \in (0, \infty)\)

Density Function

\[ f(y | m, s) = \frac{1}{y}\frac{1}{\sqrt{2 \pi s}} \exp \left( - \frac{\left( \ln(y) - m \right)^2}{2 s} \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= \exp \left( m + \frac{s}{2} \right) \\ \mathrm{var}[Y] &= \left( \exp(s) - 1 \right) \exp \left( 2 m + s \right) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s) &= \frac{\ln(y) - m}{s} \\ \nabla_{s} (y; m, s) &= \frac{\left( \ln(y) - m \right)^2}{2 s^2} - \frac{1}{2s} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s) &= \frac{1}{s} \\ \mathcal{I}_{m, s} (m, s) &= 0 \\ \mathcal{I}_{s, s} (m, s) &= \frac{1}{2 s^2} \\ \end{aligned} \]

Parameters

• Scale parameter \(s \in (0, \infty)\)
• Shape parameter \(b \in (0, \infty)\)

Density Function

\[ f(y | s, b) = \frac{b}{s} \left( 1 + \frac{y}{s} \right)^{-b - 1} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= \frac{s}{b - 1}, & \quad \text{for } a &> 1 \\ \mathrm{var}[Y] &= \frac{s^2 b}{(b - 1)^2 (b - 2)}, & \quad \text{for } a &> 2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{s} (y; s, b) &= \frac{1}{s} \left( b \frac{y}{s} - 1 \right) \left( \frac{y}{s} + 1 \right)^{-1} \\ \nabla_{b} (y; s, b) &= \frac{1}{b} - \ln \left( \frac{y}{s} + 1 \right) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{s, s} (s, b) &= \frac{b}{s^2 (b + 2)} \\ \mathcal{I}_{s, b} (s, b) &= - \frac{1}{s (b + 1)} \\ \mathcal{I}_{b, b} (s, b) &= \frac{1}{b^2} \\ \end{aligned} \]

Parameter

• Scale parameter \(s \in (0, \infty)\)

Density Function

\[ f(y | s) = \frac{y}{s^2} \exp \left(- \frac{y^2}{2 s^2} \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= s \sqrt{\frac{\pi}{2}} \\ \mathrm{var}[Y] &= s^2 \frac{4 - \pi}{2}\\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{s} (y; s) &= \frac{y^2 - 2 s^2}{s^3} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{s, s} (s) &= \frac{4}{s^2} \\ \end{aligned} \]

Parameters

• Rate parameter \(r \in (0, \infty)\)
• Shape parameter \(b \in (0, \infty)\)

Density Function

\[ f(y | r, b) = r b (r y)^{b - 1} \exp \left( -(r y)^b \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= \frac{1}{r} \Gamma \left(1 + b^{-1} \right) \\ \mathrm{var}[Y] &= \frac{1}{r^2} \left( \Gamma \left(1 + 2 b^{-1} \right) - \Gamma \left(1 + b^{-1} \right)^2 \right) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{r} (y; r, b) &= \frac{b}{r} \left( 1 - (r y)^b \right) \\ \nabla_{b} (y; r, b) &= \left( 1 - (r y)^b \right) \ln (r y) + \frac{1}{b} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{r, r} (r, b) &= \frac{b^2}{r^2} \\ \mathcal{I}_{r, b} (r, b) &= \frac{\psi_0(1) + 1}{r} \\ \mathcal{I}_{b, b} (r, b) &= \frac{\psi_0(1)^2 + 2 \psi_0(1) + \psi_1(1) + 1}{b^2} \\ \end{aligned} \]

Parameters

• Scale parameter \(s \in (0, \infty)\)
• Shape parameter \(b \in (0, \infty)\)

Density Function

\[ f(y | s, b) = \frac{b}{s} \left( \frac{y}{s} \right)^{b - 1} \exp \left( - \left( \frac{y}{s} \right)^b \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= s \Gamma \left(1 + b^{-1} \right) \\ \mathrm{var}[Y] &= s^2 \left( \Gamma \left(1 + 2 b^{-1} \right) - \Gamma \left(1 + b^{-1} \right)^2 \right) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{s} (y; s, b) &= \frac{b}{s} \left( \left( \frac{y}{s} \right)^b - 1 \right) \\ \nabla_{b} (y; s, b) &= \left( 1 - \left( \frac{y}{s} \right)^b \right) \ln \left( \frac{y}{s} \right) + \frac{1}{b} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{s, s} (s, b) &= \frac{b^2}{s^2} \\ \mathcal{I}_{s, b} (s, b) &= - \frac{\psi_0(1) + 1}{s} \\ \mathcal{I}_{b, b} (s, b) &= \frac{\psi_0(1)^2 + 2 \psi_0(1) + \psi_1(1) + 1}{b^2} \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in \mathbb{R}\)
• Scale parameter \(s \in (0, \infty)\)
• Asymmetry parameter \(a \in (0, \infty)\)

Density Function

\[ f(y | m, s, a) = \frac{1}{s \left( 1 / a + a\right)} \exp \left\{- \frac{\lvert y - m \rvert}{s} a^{\mathrm{sign}(y - m)} \right\} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m + s (1 / a - a) \\ \mathrm{var}[Y] &= s^2 (1 / a^2 + a^2) \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s, a) &= \frac{\mathrm{sign}(y - m) a^{\mathrm{sign}(y - m)}}{s} \\ \nabla_{s} (y; m, s, a) &= \frac{\lvert y - m \rvert a^{\mathrm{sign}(y - m)}}{s^2} - \frac{1}{s} \\ \nabla_{a} (y; m, s, a) &= -\frac{(y - m) a^{\mathrm{sign}(y - m)}}{s} + \frac{1 - a^2}{a + a^3} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s, a) &= \frac{1}{s^2} \\ \mathcal{I}_{m, s} (m, s, a) &= 0 \\ \mathcal{I}_{m, a} (m, s, a) &= -\frac{2}{s (1 + a^2)} \\ \mathcal{I}_{s, s} (m, s, a) &= \frac{1}{s^2} \\ \mathcal{I}_{s, a} (m, s, a) &= -\frac{1}{s a} \frac{1 - a^2}{1 + a^2} \\ \mathcal{I}_{a, a} (m, s, a) &= \frac{1}{a^2} + \frac{4}{(1 + a^2)^2} \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in \mathbb{R}\)
• Scale parameter \(s \in (0, \infty)\)

Density Function

\[ f(y | m, s) = \frac{1}{2s} \exp \left\{- \frac{\lvert y - m \rvert}{s} \right\} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m \\ \mathrm{var}[Y] &= 2s^2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s) &= \frac{\mathrm{sign}(y - m)}{s} \\ \nabla_{s} (y; m, s) &= \frac{\lvert y - m \rvert}{s^2} - \frac{1}{s} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s) &= \frac{1}{s^2} \\ \mathcal{I}_{m, s} (m, s) &= 0 \\ \mathcal{I}_{s, s} (m, s) &= \frac{1}{s^2} \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in \mathbb{R}\)
• Scale parameter \(s \in (0, \infty)\)

Density Function

\[ f(y | m, s) = \frac{1}{s} \exp \left( - \frac{x - m}{s} \right) \left( 1 + \exp \left( - \frac{x - m}{s} \right) \right)^{-2} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m \\ \mathrm{var}[Y] &= \frac{\pi^2}{3} s^2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s) &= \frac{1}{s} \tanh \left( \frac{y - m}{2 s} \right) \\ \nabla_{s} (y; m, s) &= \frac{y - m}{s^2} \tanh \left( \frac{y - m}{2 s} \right) - \frac{1}{s} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s) &= \frac{1}{3 s^2} \\ \mathcal{I}_{m, s} (m, s) &= 0 \\ \mathcal{I}_{s, s} (m, s) &= \frac{1}{3 s^2} \left( \frac{\pi^2}{3} + 1 \right) \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in \mathbb{R}\)
• Variance parameter \(s \in (0, \infty)\)

Density Function

\[ f(y | m, s) = \frac{1}{\sqrt{2 \pi s}} \exp \left( -\frac{(y - m)^2}{2 s} \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m \\ \mathrm{var}[Y] &= s \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s) &= \frac{y - m}{s} \\ \nabla_{s} (y; m, s) &= \frac{(y - m)^2 - s}{2 s^2} \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s) &= \frac{1}{s} \\ \mathcal{I}_{m, s} (m, s) &= 0 \\ \mathcal{I}_{s, s} (m, s) &= \frac{1}{2 s^2} \\ \end{aligned} \]

Parameters

• Mean parameter \(m \in \mathbb{R}\)
• Variance parameter \(s \in (0, \infty)\)
• Degrees of freedom parameter \(v \in (0, \infty)\)

Density Function

\[ f(y | m, s, v) = \frac{\Gamma \left( \frac{v + 1}{2} \right)}{\Gamma \left( \frac{v}{2} \right) \sqrt{\pi s v}} \left( 1 + \frac{(y - m)^2}{s v} \right)^{-\frac{v + 1}{2}} \]

Moments

\[ \begin{aligned} \mathrm{E}[Y] &= m, & \quad \text{for } v &> 1 \\ \mathrm{var}[Y] &= \frac{v}{v - 2} s, & \quad \text{for } v &> 2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{m} (y; m, s, v) &= \frac{(v + 1) (y - m) }{(y - m)^2 + s v} \\ \nabla_{s} (y; m, s, v) &= \frac{v}{2s} \frac{(y - m)^2 - s}{(y - m)^2 + s v} \\ \nabla_{v} (y; m, s, v) &= \frac{1}{2} \frac{(y - m)^2 - s}{(y - m)^2 + s v} - \frac{1}{2} \ln \left(1 + \frac{1}{v} \frac{(y - m)^2}{s} \right) - \frac{1}{2} \psi_0 \left( \frac{v}{2} \right) + \frac{1}{2} \psi_0 \left( \frac{v + 1}{2} \right) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{m, m} (m, s, v) &= \frac{v + 1}{s (v + 3)} \\ \mathcal{I}_{m, s} (m, s, v) &= 0 \\ \mathcal{I}_{m, v} (m, s, v) &= 0 \\ \mathcal{I}_{s, s} (m, s, v) &= \frac{v}{2 s^2 (v + 3)} \\ \mathcal{I}_{s, v} (m, s, v) &= \frac{-1}{s (v + 1) (v + 3)} \\ \mathcal{I}_{v, v} (m, s, v) &= - \frac{1}{2} \frac{v + 5}{v (v + 1) (v + 3)} + \frac{1}{4} \psi_1 \left( \frac{v}{2} \right) - \frac{1}{4} \psi_1 \left( \frac{v + 1}{2} \right) \\ \end{aligned} \]

Parameters

• Mean parameters \(m_i \in \mathbb{R}, i = 1, \ldots, n\)
• Variance parameters \(s_i \in (0, \infty), i = 1, \ldots, n\)
• Covariance parameters \(c_{ij} \in \mathbb{R}, i = 2, \ldots, n, j = 1, \ldots, i\)

Vector and Matrix Notation

• Mean vector \(\boldsymbol{m}\) of length \(n\)
• Variance-covariance matrix \(\boldsymbol{K}\) of size \(n \times n\)

Density Function

\[ f(\boldsymbol{y} | \boldsymbol{m}, \boldsymbol{K}) = \frac{1}{\sqrt{(2 \pi)^n | \boldsymbol{K}|}} \exp \left( - \frac{1}{2} (\boldsymbol{y} - \boldsymbol{m})^\intercal \boldsymbol{K}^{-1} (\boldsymbol{y} - \boldsymbol{m}) \right) \]

Moments

\[ \begin{aligned} \mathrm{E}[\boldsymbol{Y}] &= \boldsymbol{m} \\ \mathrm{var}[\boldsymbol{Y}] &= \boldsymbol{K} \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{\boldsymbol{m}} (\boldsymbol{y}; \boldsymbol{m}, \boldsymbol{K}) &= \boldsymbol{K}^{-1} \left(\boldsymbol{y} - \boldsymbol{m} \right) \\ \nabla_{\mathrm{vec}(\boldsymbol{K})} (\boldsymbol{y}; \boldsymbol{m}, \boldsymbol{K}) &= \mathrm{vec} \left( \frac{1}{2} \boldsymbol{K}^{-1} \left(\boldsymbol{y} - \boldsymbol{m} \right) \left(\boldsymbol{y} - \boldsymbol{m} \right)^\intercal \boldsymbol{K}^{-1} - \frac{1}{2} \boldsymbol{K}^{-1} \right) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{\boldsymbol{m}, \boldsymbol{m}} (\boldsymbol{m}, \boldsymbol{K}) &= \boldsymbol{K}^{-1} \\ \mathcal{I}_{\boldsymbol{m}, \mathrm{vec}(\boldsymbol{K})} (\boldsymbol{m}, \boldsymbol{K}) &= \boldsymbol{0} \\ \mathcal{I}_{\mathrm{vec}(\boldsymbol{K}), \mathrm{vec}(\boldsymbol{K})} (\boldsymbol{m}, \boldsymbol{K}) &= \frac{1}{4} \boldsymbol{K}^{-1} \otimes \boldsymbol{K}^{-1} + \frac{1}{4} \mathrm{vec}\left(\boldsymbol{K}^{-1} \right) \mathrm{vec}\left(\boldsymbol{K}^{-1} \right)^\intercal \\ \end{aligned} \]

Parameters

• Mean parameters \(m_i \in \mathbb{R}, i = 1, \ldots, n\)
• Variance parameters \(s_i \in (0, \infty), i = 1, \ldots, n\)
• Covariance parameters \(c_{ij} \in \mathbb{R}, i = 2, \ldots, n, j = 1, \ldots, i\)
• Degrees of freedom parameter \(v \in (0, \infty)\)

Vector and Matrix Notation

• Mean vector \(\boldsymbol{m}\) of length \(n\)
• Variance-covariance matrix \(\boldsymbol{K}\) of size \(n \times n\)

Density Function

\[ f(\boldsymbol{y} | \boldsymbol{m}, \boldsymbol{K}, v) = \frac{\Gamma \left( \frac{v + n}{2} \right)}{\Gamma \left( \frac{v}{2} \right) \sqrt{(v \pi)^n | \boldsymbol{K}|}} \left( 1 + \frac{1}{v} (\boldsymbol{y} - \boldsymbol{m})^\intercal \boldsymbol{K}^{-1} (\boldsymbol{y} - \boldsymbol{m}) \right)^{-\frac{v + n}{2}} \]

Moments

\[ \begin{aligned} \mathrm{E}[\boldsymbol{Y}] &= \boldsymbol{m}, & \quad \text{for } v &> 1 \\ \mathrm{var}[\boldsymbol{Y}] &= \frac{v}{v - 2} \boldsymbol{K}, & \quad \text{for } v &> 2 \\ \end{aligned} \]

Score

\[ \begin{aligned} \nabla_{\boldsymbol{m}} (\boldsymbol{y}; \boldsymbol{m}, \boldsymbol{K}, v) &= \frac{v + n}{v + \left(\boldsymbol{y} - \boldsymbol{m} \right)^\intercal \boldsymbol{K}^{-1} \left(\boldsymbol{y} - \boldsymbol{m} \right)} \boldsymbol{K}^{-1} \left(\boldsymbol{y} - \boldsymbol{m} \right) \\ \nabla_{\mathrm{vec}(\boldsymbol{K})} (\boldsymbol{y}; \boldsymbol{m}, \boldsymbol{K}, v) &= \mathrm{vec} \left( \frac{1}{2} \frac{v + n}{v + \left(\boldsymbol{y} - \boldsymbol{m} \right)^\intercal \boldsymbol{K}^{-1} \left(\boldsymbol{y} - \boldsymbol{m} \right)} \boldsymbol{K}^{-1} \left(\boldsymbol{y} - \boldsymbol{m} \right) \left(\boldsymbol{y} - \boldsymbol{m} \right)^\intercal \boldsymbol{K}^{-1} - \frac{1}{2} \boldsymbol{K}^{-1} \right) \\ \nabla_{v} (\boldsymbol{y}; \boldsymbol{m}, \boldsymbol{K}, v) &= \frac{1}{2} \frac{ \left(\boldsymbol{y} - \boldsymbol{m} \right)^\intercal \boldsymbol{K}^{-1} \left(\boldsymbol{y} - \boldsymbol{m} \right) - n }{ \left(\boldsymbol{y} - \boldsymbol{m} \right)^\intercal \boldsymbol{K}^{-1} \left(\boldsymbol{y} - \boldsymbol{m} \right)) + v} - \frac{1}{2} \ln \left( 1 + \frac{1}{v} \left(\boldsymbol{y} - \boldsymbol{m} \right)^\intercal \boldsymbol{K}^{-1} \left(\boldsymbol{y} - \boldsymbol{m} \right) \right) \\ & \qquad - \frac{1}{2} \psi_0 \left( \frac{v}{2} \right) + \frac{1}{2} \psi_0 \left( \frac{v + n}{2} \right) \\ \end{aligned} \]

Fisher Information

\[ \begin{aligned} \mathcal{I}_{\boldsymbol{m}, \boldsymbol{m}} (\boldsymbol{m}, \boldsymbol{K}, v) &= \frac{v + n}{v + n + 2} \boldsymbol{K}^{-1} \\ \mathcal{I}_{\boldsymbol{m}, \mathrm{vec}(\boldsymbol{K})} (\boldsymbol{m}, \boldsymbol{K}, v) &= \boldsymbol{0} \\ \mathcal{I}_{\boldsymbol{m}, v} (\boldsymbol{m}, \boldsymbol{K}, v) &= \boldsymbol{0} \\ \mathcal{I}_{\mathrm{vec}(\boldsymbol{K}), \mathrm{vec}(\boldsymbol{K})} (\boldsymbol{m}, \boldsymbol{K}, v) &= \frac{1}{4} \frac{v + n}{v + n + 2} \boldsymbol{K}^{-1} \otimes \boldsymbol{K}^{-1} + \frac{1}{4} \frac{v + n - 2}{v + n + 2} \mathrm{vec}\left(\boldsymbol{K}^{-1} \right) \mathrm{vec}\left(\boldsymbol{K}^{-1} \right)^\intercal \\ \mathcal{I}_{\mathrm{vec}(\boldsymbol{K}), v} (\boldsymbol{m}, \boldsymbol{K}, v) &= - \frac{1}{(v + n +2)(v + n)} \mathrm{vec}\left(\boldsymbol{K}^{-1} \right) \\ \mathcal{I}_{v, v} (\boldsymbol{m}, \boldsymbol{K}, v) &= ) - \frac{1}{2} \frac{n (v + n + 4)}{v (v + n + 2)(v + n)} + \frac{1}{4} \psi_1 \left( \frac{v}{2} \right) - \frac{1}{4} \psi_1 \left( \frac{v + n}{2} \right) \\ \end{aligned} \]