|
| 1 | +--- |
| 2 | +layout: proof |
| 3 | +mathjax: true |
| 4 | + |
| 5 | +author: "Joram Soch" |
| 6 | +affiliation: "BCCN Berlin" |
| 7 | +e_mail: "joram.soch@bccn-berlin.de" |
| 8 | +date: 2025-10-24 13:48:56 |
| 9 | + |
| 10 | +title: "Mode of the beta distribution" |
| 11 | +chapter: "Probability Distributions" |
| 12 | +section: "Univariate continuous distributions" |
| 13 | +topic: "Beta distribution" |
| 14 | +theorem: "Mode" |
| 15 | + |
| 16 | +sources: |
| 17 | + - authors: "Wikipedia" |
| 18 | + year: 2025 |
| 19 | + title: "Beta distribution" |
| 20 | + in: "Wikipedia, the free encyclopedia" |
| 21 | + pages: "retrieved on 2025-10-24" |
| 22 | + url: "https://en.wikipedia.org/wiki/Beta_distribution#Mode" |
| 23 | + |
| 24 | +proof_id: "P520" |
| 25 | +shortcut: "beta-mode" |
| 26 | +username: "JoramSoch" |
| 27 | +--- |
| 28 | + |
| 29 | + |
| 30 | +**Theorem:** Let $X$ be a [random variable](/D/rvar) following a [beta distribution](/D/beta): |
| 31 | + |
| 32 | +$$ \label{eq:beta} |
| 33 | +X \sim \mathrm{Bet}(\alpha, \beta) \; . |
| 34 | +$$ |
| 35 | + |
| 36 | +Then, the [mode](/D/mode) of $X$ is |
| 37 | + |
| 38 | +$$ \label{eq:beta-mode-p1} |
| 39 | +\mathrm{mode}(X) \in \left\{ |
| 40 | +\begin{array}{rl} |
| 41 | +\left\lbrace 0, 1 \right\rbrace \; , & \text{if} \quad \alpha < 1 \quad \text{and} \quad \beta < 1 \\ |
| 42 | +\left[ 0, 1 \right] \; , & \text{if} \quad \alpha = 1 \quad \text{and} \quad \beta = 1 |
| 43 | +\end{array} |
| 44 | +\right. |
| 45 | +$$ |
| 46 | + |
| 47 | +and |
| 48 | + |
| 49 | +$$ \label{eq:beta-mode-p2} |
| 50 | +\mathrm{mode}(X) = \left\{ |
| 51 | +\begin{array}{rl} |
| 52 | + 0 \; \text{or} \; 1 \; , |
| 53 | +& \text{if} \quad \alpha < 1 \quad \text{or} \quad \beta < 1 \quad (\text{but not} \; \alpha < 1 \; \text{and} \; \beta < 1) \\ |
| 54 | + \frac{\alpha-1}{\alpha+\beta-2} \; , |
| 55 | +& \text{if} \quad \alpha \geq 1 \quad \text{and} \quad \beta \geq 1 \quad (\text{but not} \; \alpha = 1 \; \text{and} \; \beta = 1) \; . |
| 56 | +\end{array} |
| 57 | +\right. |
| 58 | +$$ |
| 59 | + |
| 60 | + |
| 61 | +**Proof:** The [mode](/D/mode) is the value which maximizes the [probability density function](/D/pdf): |
| 62 | + |
| 63 | +$$ \label{eq:mode} |
| 64 | +\mathrm{mode}(X) = \operatorname*{arg\,max}_x f_X(x) \; . |
| 65 | +$$ |
| 66 | + |
| 67 | +The [probability density function of the beta distribution](/P/beta-pdf) is: |
| 68 | + |
| 69 | +$$ \label{eq:beta-pdf} |
| 70 | +f_X(x) = \frac{1}{\mathrm{B}(\alpha, \beta)} \, x^{\alpha-1} \, (1-x)^{\beta-1} \; . |
| 71 | +$$ |
| 72 | + |
| 73 | +1) If $\alpha < 1$, then 0 is the mode, since |
| 74 | + |
| 75 | +$$ \label{eq:beta-mode-p2a-s1} |
| 76 | +\lim_{\substack{x \rightarrow 0 \\ \alpha < 1}} f_X(x) = \infty \; , |
| 77 | +\quad \text{because} \quad |
| 78 | +\lim_{x \rightarrow 0} x^{\alpha-1} = \infty |
| 79 | +\quad \text{for} \quad |
| 80 | +\alpha < 1 \; , |
| 81 | +$$ |
| 82 | + |
| 83 | +and if $\beta < 1$, then 1 is the mode, since |
| 84 | + |
| 85 | +$$ \label{eq:beta-mode-p2a-s2} |
| 86 | +\lim_{\substack{x \rightarrow 1 \\ \beta < 1}} f_X(x) = \infty \; , |
| 87 | +\quad \text{because} \quad |
| 88 | +\lim_{x \rightarrow 1} (1-x)^{\beta-1} = \infty |
| 89 | +\quad \text{for} \quad |
| 90 | +\beta < 1 \; . |
| 91 | +$$ |
| 92 | + |
| 93 | +2) If both $\alpha < 1$ and $\beta < 1$, then |
| 94 | + |
| 95 | +$$ \label{eq:beta-mode-p1a} |
| 96 | +\lim_{x \rightarrow 0} f_X(x) = \infty |
| 97 | +\quad \text{and} \quad |
| 98 | +\lim_{x \rightarrow 1} f_X(x) = \infty \; , |
| 99 | +$$ |
| 100 | + |
| 101 | +so any value from the set $\left\lbrace 0, 1 \right\rbrace$ may be considered the mode. |
| 102 | + |
| 103 | +3) If both $\alpha = 1$ and $\beta = 1$, then |
| 104 | + |
| 105 | +$$ \label{eq:beta-mode-p1b} |
| 106 | +\begin{split} |
| 107 | + f_X(x) |
| 108 | +&= \frac{1}{\mathrm{B}(1,1)} \, x^{1-1} \, (1-x)^{1-1} \\ |
| 109 | +&= \frac{\Gamma(2)}{\Gamma(1) \Gamma(1)} x^0 \, (1-x)^0 \\ |
| 110 | +&= 1 = \mathrm{const.} \; , |
| 111 | +\end{split} |
| 112 | +$$ |
| 113 | + |
| 114 | +i.e. the distribution becomes equivalent to the (standard) [continuous uniform distribution](/D/cuni) with parameters $a = 0$ and $b = 1$ which has a [constant probability density function](/P/cuni-pdf). Thus, any value from the interval $\left[ 0,1 \right]$ [may be considered the mode](/P/cuni-mode). |
| 115 | + |
| 116 | +4) For the remaining cases, we must analyze the probability density function. The first two deriatives of this function are: |
| 117 | + |
| 118 | +$$ \label{eq:beta-pdf-der1} |
| 119 | +\begin{split} |
| 120 | + f'_X(x) |
| 121 | + = \frac{\mathrm{d}f_X(x)}{\mathrm{d}x} |
| 122 | +&= \frac{1}{\mathrm{B}(\alpha, \beta)} \left[ - (\beta-1) x^{\alpha-1} (1-x)^{\beta-2} + (\alpha-1) x^{\alpha-2} (1-x)^{\beta-1} \right] \\ |
| 123 | +&= \frac{1}{\mathrm{B}(\alpha, \beta)} \left[ (\alpha-1) x^{\alpha-2} (1-x)^{\beta-1} - (\beta-1) x^{\alpha-1} (1-x)^{\beta-2} \right] |
| 124 | +\end{split} |
| 125 | +$$ |
| 126 | + |
| 127 | +$$ \label{eq:beta-pdf-der2} |
| 128 | +\begin{split} |
| 129 | + f''_X(x) |
| 130 | + = \frac{\mathrm{d}^2f_X(x)}{\mathrm{d}x^2} |
| 131 | +&= \frac{1}{\mathrm{B}(\alpha, \beta)} \left[ |
| 132 | + (\alpha-1) \left( (\alpha-2) x^{\alpha-3} (1-x)^{\beta-1} - (\beta-1) x^{\alpha-2} (1-x)^{\beta-2} \right) - \right. \\ |
| 133 | +&\hphantom{=} \quad\quad\quad\quad\; |
| 134 | + \left. (\beta-1) \left( (\alpha-1) x^{\alpha-2} (1-x)^{\beta-2} - (\beta-2) x^{\alpha-1} (1-x)^{\beta-3} \right) |
| 135 | + \right] \\ |
| 136 | +&= \frac{1}{\mathrm{B}(\alpha, \beta)} \left[ |
| 137 | + (\alpha-1) (\alpha-2) x^{\alpha-3} (1-x)^{\beta-1} - 2 (\alpha-1) (\beta-1) x^{\alpha-2} (1-x)^{\beta-2} + \right. \\ |
| 138 | +&\hphantom{=} \quad\quad\quad\quad\; |
| 139 | + \left. (\beta-1) (\beta-2) x^{\alpha-1} (1-x)^{\beta-3} |
| 140 | + \right] \; . |
| 141 | +\end{split} |
| 142 | +$$ |
| 143 | + |
| 144 | +We now calculate the root of the first derivative \eqref{eq:beta-pdf-der1}: |
| 145 | + |
| 146 | +$$ \label{eq:beta-mode-p2b-s1} |
| 147 | +\begin{split} |
| 148 | + f'_X(x) |
| 149 | + = 0 |
| 150 | +&= \frac{1}{\mathrm{B}(\alpha, \beta)} \left[ (\alpha-1) x^{\alpha-2} (1-x)^{\beta-1} - (\beta-1) x^{\alpha-1} (1-x)^{\beta-2} \right] \\ |
| 151 | +&\Leftrightarrow \\ |
| 152 | +(\beta-1) x^{\alpha-1} (1-x)^{\beta-2} &= (\alpha-1) x^{\alpha-2} (1-x)^{\beta-1} \\ |
| 153 | + (\beta-1) x &= (\alpha-1) (1-x) \\ |
| 154 | + x [(\beta-1) + (\alpha-1)] &= \alpha-1 \\ |
| 155 | + x &= \frac{\alpha-1}{\alpha+\beta-2} \; . |
| 156 | +\end{split} |
| 157 | +$$ |
| 158 | + |
| 159 | +Note that for this quantity, we have |
| 160 | + |
| 161 | +$$ \label{eq:beta-mode-p2b-s2} |
| 162 | +\begin{split} |
| 163 | +\frac{\alpha-1}{\alpha+\beta-2} &< 0 \; , |
| 164 | +\quad \text{if} \quad \alpha < 1 |
| 165 | +\quad \text{and} \quad \beta > 2 - \alpha \\ |
| 166 | +\frac{\alpha-1}{\alpha+\beta-2} &> 1 \; , |
| 167 | +\quad \text{if} \quad \beta < 1 |
| 168 | +\quad \text{and} \quad \alpha > 2 - \beta \; . |
| 169 | +\end{split} |
| 170 | +$$ |
| 171 | + |
| 172 | +Also note that the following holds: |
| 173 | + |
| 174 | +$$ \label{eq:beta-mode-p2b-s3} |
| 175 | + 1 - x |
| 176 | += 1 - \frac{\alpha-1}{\alpha+\beta-2} |
| 177 | += \frac{\alpha+\beta-2}{\alpha+\beta-2} - \frac{\alpha-1}{\alpha+\beta-2} |
| 178 | += \frac{\beta-1}{\alpha+\beta-2} \; . |
| 179 | +$$ |
| 180 | + |
| 181 | +By plugging $x$ and $1-x$ into the second deriative \eqref{eq:beta-pdf-der2}, we find: |
| 182 | + |
| 183 | +$$ \label{eq:beta-mode-p2b-s4} |
| 184 | +\begin{split} |
| 185 | + f''_X\left( \frac{\alpha-1}{\alpha+\beta-2} \right) |
| 186 | +&= \frac{1}{\mathrm{B}(\alpha, \beta)} \left[ |
| 187 | + (\alpha-1) (\alpha-2) \left( \frac{\alpha-1}{\alpha+\beta-2} \right)^{\alpha-3} \left( \frac{\beta-1}{\alpha+\beta-2} \right)^{\beta-1} - \right. \\ |
| 188 | +&\hphantom{=} \quad\quad\quad\quad |
| 189 | + \left. 2 (\alpha-1) (\beta-1) \left( \frac{\alpha-1}{\alpha+\beta-2} \right)^{\alpha-2} \left( \frac{\beta-1}{\alpha+\beta-2} \right)^{\beta-2} + \right. \\ |
| 190 | +&\hphantom{=} \quad\quad\quad\quad\; |
| 191 | + \left. (\beta-1) (\beta-2) \left( \frac{\alpha-1}{\alpha+\beta-2} \right)^{\alpha-1} \left( \frac{\beta-1}{\alpha+\beta-2} \right)^{\beta-3} \right] \; . |
| 192 | +\end{split} |
| 193 | +$$ |
| 194 | + |
| 195 | +Multiplying with factors which are certainly positive, we can focus on those parts of the second derivative which determine its sign: |
| 196 | + |
| 197 | +$$ \label{eq:beta-mode-p2b-s5} |
| 198 | +\begin{split} |
| 199 | + \frac{f''_X(x) \cdot \mathrm{B}(\alpha, \beta)}{\left( \frac{\alpha-1}{\alpha+\beta-2} \right)^{\alpha-3} \cdot \left( \frac{\beta-1}{\alpha+\beta-2} \right)^{\beta-3}} |
| 200 | +&= (\alpha-1) (\alpha-2) \left( \frac{\beta-1}{\alpha+\beta-2} \right)^2 - \\ |
| 201 | +&\hphantom{=} 2 (\alpha-1) (\beta-1) \left( \frac{\alpha-1}{\alpha+\beta-2} \right) \left( \frac{\beta-1}{\alpha+\beta-2} \right) + \\ |
| 202 | +&\hphantom{=} (\beta-1) (\beta-2) \left( \frac{\alpha-1}{\alpha+\beta-2} \right)^2 \; . |
| 203 | +\end{split} |
| 204 | +$$ |
| 205 | + |
| 206 | +Further multiplying with or dividing by terms which are necessarily positive and thus do not change the sign of the function value, we get: |
| 207 | + |
| 208 | +$$ \label{eq:beta-mode-p2b-s6} |
| 209 | +\begin{split} |
| 210 | + \frac{f''_X(x) \cdot \mathrm{B}(\alpha, \beta)}{\left( \frac{\alpha-1}{\alpha+\beta-2} \right)^{\alpha-3} \cdot \left( \frac{\beta-1}{\alpha+\beta-2} \right)^{\beta-3}} \cdot \frac{(\alpha+\beta-2)^2}{(\alpha-1) (\beta-1)} |
| 211 | +&= (\alpha-2) (\beta-1) - 2 (\alpha-1) (\beta-1) + (\alpha-1) (\beta-2) \\ |
| 212 | +&= (\alpha-1) [(\beta-2)-(\beta-1)] + (\beta-1) [(\alpha-1)-(\alpha-2)] \\ |
| 213 | +&= - (\alpha-1) - (\beta-1) \\ |
| 214 | +&< 0, |
| 215 | +\quad \text{if} \quad \alpha > 1 |
| 216 | +\quad \text{and} \quad \beta > 1 \; . |
| 217 | +\end{split} |
| 218 | +$$ |
| 219 | + |
| 220 | +Thus, $f''_X(x)$ is negative for $x = \frac{\alpha-1}{\alpha+\beta-2}$, demonstrating that this is a maximum. To summarize: |
| 221 | + |
| 222 | +* If $\alpha < 1$ and $\beta < 1$, then $f_X(x)$ diverges at both ends and both values from the set $\left\lbrace 0, 1 \right\rbrace$ can be seen as the mode of $X$. |
| 223 | + |
| 224 | +* If $\alpha < 1$ or $\beta < 1$ (but not $\alpha < 1$ and $\beta < 1$), then the mode of $X$ is 0 or 1, because $f_X(x)$ tends towards infinity at $x = 0$ or $x = 1$. |
| 225 | + |
| 226 | +* If $\alpha = 1$ and $\beta = 1$, then $f_X(x)$ is constant and any value in the interval $\left[ 0,1 \right]$ can be seen as the mode of $X$. |
| 227 | + |
| 228 | +* If $\alpha \geq 1$ and $\beta \geq 1$ (but not $\alpha = 1$ and $\beta = 1$), then $0 < x = < 1$ and $f'_X(x) = 0$ and $f''_X(x) < 0$, such that $f_X(x)$ reaches its machimum at $\mathrm{mode}(X) = \frac{\alpha-1}{\alpha+\beta-2}$. |
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