# How do you define a good approximation for a probability distributions?

We know a series of probability distribution approximations that are considered good as long as some condition holds. A few examples are:

• Binomial can be approximated by Normal if $$np(1-p) > 10$$ and
• Beta can be approximated by Normal if $$\alpha,\beta$$ are large and similar
• Binomial can be approximated by Poisson if $$n$$ is large and $$p$$ is small

Now my question is, how can we quantify if an approximation is "good"? I guess those thresholds are somehow arbitrary, but also that I can measure some quantity that can indicate how robust is the given approximation. What would be the correct metric to check when looking for a good approximation?

For example, let's say that I want to approximate a Beta with a Normal. I know that $$\mu = \frac{\alpha}{\alpha+\beta}$$ and $$\sigma = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$$.

Let's say $$\alpha = 100, \beta = 10$$, then I can write:

alpha = 100
beta = 10

mu = alpha/(alpha+beta)
sd = sqrt((alpha*beta)/((alpha+beta)^2 * (alpha+beta+1)))

bb = dbeta(seq(0, 1, 0.001), alpha, beta)
nn = dnorm(seq(0, 1, 0.001), mu, sd)

plot(bb, type = "l", col = "red")
lines(nn, col = "blue")



This seems to be fairly good. Switching to $$\alpha = 10, \beta = 1$$ leads to a very poor approximation instead.

So I wonder whether there are established procedures to quantify this. I imagine that an approximation can be good in some contexts and not in others, so I would like to understand what is the go-to general quantitative framework to evaluate this in each case.

What would you check and how would you decide if an approximation of a PDF is good or not?

Thank you

EDIT

As there is no universal criterion to define a good approximation, I define my particular focus in the deconvolution problem.

Taking the Beta/Normal case as an example, I am concerned about the goodness of approximation that I would obtain using two normals as Beta approximations and convolving them, compared to directly computing the convolution of Betas. I think this could be addressed with some simulations.

• A small Kolmogorov–Smirnov statistic is one possible measure: if the two cumulative distribution functions are $F(x)$ and $G(x)$ then you want $\max\limits_x |F(x)-G(x)|$ to be small. It usual use is to compare sample data to a specified distribution, but you can also use it for two distributions. Comparing densities does not work: Poisson or binomial distributions do not have a density Jul 10, 2022 at 17:42
• The answer depends on your needs: what are you going to use the approximation for? For example, approximations used to compute p-values need to give accurate tail values between the $0.0001$ and $0.1$ and $0.9$ and $0.9999$ quantiles, usually--but accuracy in the middle would be irrelevant. Approximations for other purposes require other measures of accuracy. There is no "go-to general quantitative framework."
– whuber
Jul 10, 2022 at 19:54
• Thank you for the comment @whuber. One application is for sure the p-value. The one that made me ask the question is the convolution problem: for example, the convolution of two Beta distributions is not a Beta itself and it's harder to deal with, but if the two can be approximated to normal then the convolution becomes an easy task. So if there is not a general approach, the strategy could be to compute the real convolution numerically and the approx convolution with different parameters, and see how well they match. I will update the post with this information. Jul 10, 2022 at 20:10
• But again--we need you to tell us in what way the approximation needs to work well and how well it needs to work.
– whuber
Jul 10, 2022 at 21:21