How to plot $x^{1700}(1-x)^{300}$?

Question

I'm trying to plot a Bernoulli likelihood function on R:
$$x^{1700}(1-x)^{300}$$
But when I try to plot this function on R it looks like this:

I think the maximum should be at 0.85, but it shows me a completely flat graph with max at 0.001
It looks just okay when I try to plot a likelihood function with z=1800, N=2000

What do you think is the problem? Thanks in advance!

Sycorax · Accepted Answer

Stephan's answer about floating point is correct. As a work-around, you could plot the data on a logarithmic scale. Instead of plotting
$$
x ^{1700} (1-x)^{300}
$$
you would plot
$$
1700log(x) + 300log(1-x)
$$
Working on a logarithmic scale can be nice when it keeps the data in a reasonable range for floating point arithmetic. Because $log$ is monotonic increasing, values will retain the same ordering (any maxima occur at the same values of $x$), even though they're reported on a different scale.

kjetil b halvorsen · Answer

That likelihood function is proportional to a beta density with parameters $alpha=1701, beta=301$ so can be plotted as a beta density, as a likelihood function is only defined up to proportionality: What does "likelihood is only defined up to a multiplicative constant of proportionality" mean in practice?  resulting in

It might be more informative to plot the log likelihood function:

For reference, the R code used below:
plot( function(x) dbeta(x, 1701, 301), from=0, to=1, col="red", n=1001, main="Beta likelihood function")
plot( function(x) dbeta(x, 1701, 301, log=TRUE), from=0, to=1, col="red", n=1001, main="Beta loglikelihood function")

Stephan Kolassa · Answer

The $y$ value of your maximum (which indeed is at $x=0.85$) is $exp(-845.42)approx 10^{-367.16}$. The smallest double numbers R can work with are about $2times 10^{-308}$. You are simply running out of number space. If you really want to plot this, use a dedicated package for high precision arithmetic.

Answered by Stephan Kolassa on December 30, 2020

How to plot $x^{1700}(1-x)^{300}$?

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