How did the value of lambda represent caution in Rudolf Carnap’s inductive logic?

In the inductive logic developed by Rudolf Carnap, the parameter $\lambda$ (lambda) serves as an index of caution regarding how an inquirer updates their beliefs based on observed evidence.1

Carnap’s formula for the probability that the next individual belongs to a specific category is expressed as:
$$\frac{n_i+(\lambda /k)}{n+\lambda }$$
In this model, $\lambda$ represents the weight given to an imaginary set of individuals distributed evenly across all $k$ cells, which acts as a buffer against relying solely on observed data.1

The value of $\lambda$ dictates the level of caution:
* Low Caution ($\lambda =0$): The inquirer relies strictly on the observed relative frequencies ($n_i/n$), making them highly sensitive to current data.1
* High Caution (Large $\lambda$): The inquirer is more skeptical of the observed sample, allowing experience to alter their initial a priori probabilities only very slowly.1

Ultimately, Carnap suggested that the degree of caution should correspond to the perceived orderliness of the universe; in less orderly environments, greater caution is required, whereas in orderly ones, smaller samples may be considered more reliable.1