# The Huggins-Roy family of effective sample sizes

When we work with weighted samples, it’s essential to introduce adjustments for the sample size. Indeed, let’s consider two following weighted samples:

$$ \mathbf{x}_1 = \{ x_1, x_2, \ldots, x_n \}, \quad \mathbf{w}_1 = \{ w_1, w_2, \ldots, w_n \}, $$ $$ \mathbf{x}_2 = \{ x_1, x_2, \ldots, x_n, x_{n+1} \}, \quad \mathbf{w}_2 = \{ w_1, w_2, \ldots, w_n, 0 \}. $$Since the weight of $x_{n+1}$ in the second sample is zero,
it’s natural to expect that both samples have the same set of properties.
However, there is a major difference between $\mathbf{x}_1$ and $\mathbf{x}_2$: their sample sizes which are
$n$ and $n+1$.
In order to eliminate this difference, we typically introduce the *effective sample size* (ESS)
which is estimated based on the list of weights.

There are various ways to estimate the ESS. In this post, we briefly discuss the Huggins-Roy’s family of ESS.

For a list of weights $\mathbf{w} = \{ w_1, w_2, \ldots, w_n \}$, we can consider the corresponding normalized weights (or standardized weights):

$$ \overline{\mathbf{w}} = \frac{\mathbf{w}}{\sum_{i=1}^n w_i}. $$For any non-degenerate weighted sample, the sum of all weights is always positive so that the normalized weights are defined.

The Huggins-Roy’s family is given by:

$$ \operatorname{ESS}_\beta(\overline{\mathbf{w}}) = \Bigg( \frac{1}{\sum_{i=1}^n \overline{w}_i^\beta } \Bigg)^{\frac{1}{\beta - 1}} = \Bigg( \sum_{i=1}^n \overline{w}_i^\beta \Bigg)^{\frac{1}{1 - \beta}}. $$This family is proposed in huggins2019 and discussed in [Elvira2021] and elvira2022.

In order to understand this approach, let’s consider several special cases.

- $\beta = 0$:

where $n_z(\overline{\mathbf{w}})$ is the number of zeros in $\overline{\mathbf{w}}$. This approach is quite straightforward: we just omit elements with zero weights.

- $\beta = 1/2$:

- $\beta = 1$:

This approach is also known as *perplexity*

- $\beta = 2$:

This approach is often referenced as the Kish’s effective sample size (see kish1965).

- $\beta = \infty$:

This approach is also popular and straightforward: we define the sample size based on the maximum weight.

## References

**[Elvira2022]**

Víctor Elvira, Luca Martino, and Christian P. Robert. “Rethinking the effective sample size.” International Statistical Review (2022).

https://arxiv.org/pdf/1809.04129.pdf**[Elvira2021]**

Víctor Elvira, Luca Martino. “Effective sample size approximations as entropy measures.” (2021)

https://vixra.org/pdf/2111.0145v1.pdf**[Huggins2019]**

Huggins, Jonathan H., and Daniel M. Roy. “Sequential Monte Carlo as approximate sampling: bounds, adaptive resampling via $\infty$-ESS, and an application to particle Gibbs.” Bernoulli 25, no. 1 (2019): 584-622.

https://arxiv.org/pdf/1503.00966.pdf**[Kish1965]**

Kish, Leslie. Survey sampling. Chichester., 1965.

https://doi.org/10.1002/bimj.19680100122