Lidstone's law of succession

The Lidstone's law of succession is given by:

$$P_\lambda(s)=\frac{n_s+\lambda}{N_s+\vert\Sigma\vert\lambda}$$

where the parameter $\lambda$ is in the range $\left[ 0 , + \infty \right[$. It has been shown that this class of probability estimates is in fact a linear interpolation between the maximum likelihood estimate given in section 6.3.3.2 and the uniform prior $\frac{1}{\vert\Sigma\vert}$. Indeed, we can define a new constant $\mu$ by:

$$\mu=\frac{N_s}{N_s+\vert\Sigma\vert\lambda}$$

We then have a new form for the estimate given by Lidstone's law of succession:

$$P_\mu(s)=\mu\frac{n_s}{N_s}+(1-\mu)\frac{1}{\vert\Sigma\vert}$$

It is interesting to consider particular cases of the Lidstone's law of succession:

-

$\lambda=0$ gives the maximum likelihood estimate.

-

$\lambda=1$ gives Laplace's law of succession.

-

if $\lambda$ tends to $\infty$ then we have the uniform estimate $\frac{1}{\vert\Sigma\vert}$.

$\lambda$ thus represents the trust we have in relative frequencies. $\lambda<1$ implies more trust in relative frequencies than the Laplace's law of succession while $\lambda>1$ represents less trust in relative frequencies. In practice, people use values of $\lambda$ in the range $\left[\frac{1}{32},1\right]$, a common value being $\lambda=\frac{1}{2}$ [81].