# Information Matrix

## Filter Information matrix

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### The fundamental 'atom' of mortality modelling

In a recent blog, I looked at the most fundamental unit of observation in a mortality study, namely an individual life. But is there such a thing as a fundamental unit of modelling mortality? In Macdonald & Richards (2024) we argue that there is, namely an infinitesimal Bernoulli trial based on the mortality hazard.

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: survival models, Filter information matrix by tag: product integral

### The product integral in practice

In a (much) earlier blog, Angus introduced the product-integral representation of the survival function:

\[{}_tp_x = \prod_0^t(1-\mu_{x+s}ds),\qquad(1)\]

**Written by:**Stephen Richards

**Tags:**Filter information matrix by tag: product integral, Filter information matrix by tag: Kaplan-Meier

### The Karma of Kaplan-Meier

Our new book, *Modelling Mortality with Actuarial Applications*, describes several non-parametric estimators of two quantities:

**Written by:**Angus Macdonald

**Tags:**Filter information matrix by tag: Kaplan-Meier, Filter information matrix by tag: Nelson-Aalen, Filter information matrix by tag: Fleming-Harrington, Filter information matrix by tag: product integral

### Introducing the Product Integral

Of all the actuary's standard formulae derived from the life table, none is more important in survival modelling than:

\[{}_tp_x = \exp\left(-\int_0^t\mu_{s+s}ds\right).\qquad(1)\]

**Written by:**Angus Macdonald

**Tags:**Filter information matrix by tag: survival models, Filter information matrix by tag: survival probability, Filter information matrix by tag: force of mortality, Filter information matrix by tag: product integral