Lidstone's law of succession

The Lidstone's law of succession is also commonly used in statistical practice. A parameter $\lambda$ controls the formula of this law of succession given by:

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

It has been shown that this class of probability estimates is in fact a linear interpolation between the maximum likelihood estimate given in section 5.3.3.2 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 :

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

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

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if $\lambda=0$ we have

$$P_{\lambda=0}(s)=\frac{n_s}{N_s}$$

The Lidstone's law of succession is then equal to the maximum likelihood estimate.

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if $\lambda=1$ we have

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

The Lidstone's law of succession is then equal to the Laplace's law of succession.

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if $\lambda$ tends to $\infty$ then we have

$$P_{\lambda \to \infty}(s)=\frac{1}{\vert\Sigma\vert}$$

The lidstone's law of succession is then equal to the uniform estimate.

We can then see that the parameter $\lambda$ has a meaning. It represents the trust we have in relative frequencies. $\lambda<1$ represents more trust in relative frequencies than the Laplace's law of succession while $\lambda>1$ represents less trust in relative frequencies. In practice, $\lambda$ varies between $\frac{1}{32}$ and $1$, a common value being $\lambda=\frac{1}{2}$ [51].