What is the expected value of the logarithm of Gamma distribution?

If the expected value of $$\mathsf{Gamma}(\alpha, \beta)$$ is $$\frac{\alpha}{\beta}$$, what is the expected value of $$\log(\mathsf{Gamma}(\alpha, \beta))$$? Can it be calculated analytically?

The parametrisation I am using is shape-rate.

• If $X \sim \text{Gamma}(a,b)$, then according to mathStatica/Mathematica, $E[ \log(X) ] = \log(b)$ + PolyGamma[a], where PolyGamma denotes the digamma function – wolfies Oct 9 '18 at 4:49
• I should add that you do not provide the pdf form of your Gamma variable, and since you report that the mean is $\alpha/\beta$ (whereas for me it would be $a b$, it appears you are using different notation than I am, where your $\beta= 1/b$ – wolfies Oct 9 '18 at 4:51
• True, sorry. The parametrisation I am using is shape-rate. ${\displaystyle {\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}$ I will try to find it for this parametrisation. Could you please suggest the query for Mathematica/WolframAlpha? – Stefano Vespucci Oct 9 '18 at 6:01
• See also Johnson, Lotz and Balakrishna (1994) continuous univariate distributions Vol 1 2nd Ed. pp. 337-349. – Björn Oct 9 '18 at 6:31
• Also see Wikipedia: Gamma Distribution # Logarithmic expectation and variance – Glen_b Oct 9 '18 at 22:45

This one (maybe surprisingly) can be done with easy elementary operations (employing Richard Feynman's favorite trick of differentiating under the integral sign with respect to a parameter).

We are supposing $$X$$ has a $$\Gamma(\alpha,\beta)$$ distribution and we wish to find the expectation of $$Y=\log(X).$$ First, because $$\beta$$ is a scale parameter, its effect will be to shift the logarithm by $$\log\beta.$$ (If you use $$\beta$$ as a rate parameter, as in the question, it will shift the logarithm by $$-\log\beta.$$) This permits us to work with the case $$\beta=1.$$

After this simplification, the probability element of $$X$$ is

$$f_X(x) = \frac{1}{\Gamma(\alpha)} x^\alpha e^{-x} \frac{\mathrm{d}x}{x}$$

where $$\Gamma(\alpha)$$ is the normalizing constant

$$\Gamma(\alpha) = \int_0^\infty x^\alpha e^{-x} \frac{\mathrm{d}x}{x}.$$

Substituting $$x=e^y,$$ which entails $$\mathrm{d}x/x = \mathrm{d}y,$$ gives the probability element of $$Y$$,

$$f_Y(y) = \frac{1}{\Gamma(\alpha)} e^{\alpha y - e^y} \mathrm{d}y.$$

The possible values of $$Y$$ now range over all the real numbers $$\mathbb{R}.$$

Because $$f_Y$$ must integrate to unity, we obtain (trivially)

$$\Gamma(\alpha) = \int_\mathbb{R} e^{\alpha y - e^y} \mathrm{d}y.\tag{1}$$

Notice $$f_Y(y)$$ is a differentiable function of $$\alpha.$$ An easy calculation gives

$$\frac{\mathrm{d}}{\mathrm{d}\alpha}e^{\alpha y - e^y} \mathrm{d}y = y\, e^{\alpha y - e^y} \mathrm{d}y = \Gamma(\alpha) y\,f_Y(y).$$

The next step exploits the relation obtained by dividing both sides of this identity by $$\Gamma(\alpha),$$ thereby exposing the very object we need to integrate to find the expectation; namely, $$y f_Y(y):$$

\eqalign{ \mathbb{E}(Y) &= \int_\mathbb{R} y\, f_Y(y) = \frac{1}{\Gamma(\alpha)} \int_\mathbb{R} \frac{\mathrm{d}}{\mathrm{d}\alpha}e^{\alpha y - e^y} \mathrm{d}y \\ &= \frac{1}{\Gamma(\alpha)} \frac{\mathrm{d}}{\mathrm{d}\alpha}\int_\mathbb{R} e^{\alpha y - e^y} \mathrm{d}y\\ &= \frac{1}{\Gamma(\alpha)} \frac{\mathrm{d}}{\mathrm{d}\alpha}\Gamma(\alpha)\\ &= \frac{\mathrm{d}}{\mathrm{d}\alpha}\log\Gamma(\alpha)\\ &=\psi(\alpha), }

the logarithmic derivative of the gamma function (aka "polygamma"). The integral was computed using identity $$(1).$$

Re-introducing the factor $$\beta$$ shows the general result is

$$\mathbb{E}(\log(X)) = \log\beta + \psi(\alpha)$$

for a scale parameterization (where the density function depends on $$x/\beta$$) or

$$\mathbb{E}(\log(X)) = -\log\beta + \psi(\alpha)$$

for a rate parameterization (where the density function depends on $$x\beta$$).

• With polygamma function do you mean of which order (e.g., 0,1) being a digamma (As @wolfies pointed out), trigamma? – Stefano Vespucci Oct 15 '18 at 23:16
• @Stefano I mean the logarithmic derivative of gamma, as stated. That means $\psi(z) = \Gamma^\prime(z)/\Gamma(z).$ – whuber Oct 15 '18 at 23:18

The answer by @whuber is quite nice; I will essentially restate his answer in a more general form which connects (in my opinion) better with statistical theory, and which makes clear the power of the overall technique.

Consider a family of distributions $$\{F_\theta : \theta \in \Theta\}$$ which consitute an exponential family, meaning they admit a density $$f_\theta(x) = \exp\left\{s(x)\theta - A(\theta) + h(x)\right\}$$ with respect to some common dominating measure (usually, Lebesgue or counting measure). Differentiating both sides of
$$\int f_\theta(x) \ dx = 1$$ with respect to $$\theta$$ we arrive at the score equation $$\int f'_\theta(x) = \int \frac{f'_\theta(x)}{f_\theta(x)} f_\theta(x) = \int u_\theta(x) \, f_\theta(x) \ dx = 0 \tag{\dagger}$$ where $$u_\theta(x) = \frac d {d\theta} \log f_\theta(x)$$ is the score function and we have defined $$f'_\theta(x) = \frac{d}{d\theta} f_\theta(x)$$. In the case of an exponential family, we have $$u_\theta(x) = s(x) - A'(\theta)$$ where $$A'(\theta) = \frac d {d\theta} A(\theta)$$; this is sometimes called the cumulant function, as it is evidently very closely related to the cumulant-generating function. It follows now from $$(\dagger)$$ that $$E_\theta[s(X)] = A'(\theta)$$.

We now show this helps us compute the require expectation. We can write the gamma density with fixed $$\beta$$ as an exponential family $$f_\theta(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x} = \exp\left\{\log(x) \alpha + \alpha \log \beta - \log \Gamma(\alpha) - \beta x \right\}.$$ This is an exponential family in $$\alpha$$ alone with $$s(x) = \log x$$ and $$A(\alpha) = \log \Gamma(\alpha) - \alpha \log \beta$$. It now follows immediately by computing $$\frac d {d\alpha} A(\alpha)$$ that $$E[\log X] = \psi(\alpha) - \log \beta.$$

• +1 Thank you for pointing out this nice generalization. – whuber Oct 9 '18 at 19:02