I’m kind of having trouble with coming up with an example/thought experiment that could explain this. I tried to look around for a simpler explanation but wasn’t really able to find any. I do know that two independent variables have a covariance of 0 but why is it that the other way around does not work?
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Covariance is a measure of how linearly related two variables are. Independent variables are not related and therefore have a covariance of 0; however, the covariance can also be 0 if the variables are dependent but do not have a linear relationship, for example the points that make a circle. For that reason, you can not assume two variables are independent given that their covariance is 0.
Independence is an extremely strong property. It means that knowing any information about one variable do not give you any information about another variable. In particular, that means *any* 2 functions on each variable are uncorrelated, their covariance is 0.
Covariance 0 merely means that that they 2 variables are linearly uncorrelated, it says nothing about other non-linear functions of these variables. For example, from Cov(X,Y)=0 you don’t know anything about Cov(X^2 , Y^2 ). For example, let’s say X^2 =Y^2 =Z, where Z is a random positive number with positive variance, there are 50% chance of each variable being positive and negative independently for any given value of Z, and their variance is positive. Then X and Y are not independent, since knowing one variable tell you a lot about the other variable, and in fact Cov(X^2 ,Y^2 ) is positive. But Cov(X,Y)=0 because knowing one variable does not tell you whether the other is positive or negative.
The other answers are good. If you haven’t seen it, might I suggest [the datasaurus rex](https://blog.revolutionanalytics.com/2017/05/the-datasaurus-dozen.html)?
This isn’t very realistic but suppose that you live in a city where some people have one dog and others have two dogs. All the people with one dog also own one cat. Meanwhile those with two dogs are split half and half between having no cats and having two cats.
If you knew how many dogs somebody has, then on average, they’d always have one cat. This implies the covariance is 0 since the value of one variable doesn’t effect the average of the other variable. But they are not independent, since knowing the number of dogs someone has does effect the probabilities of the number of cats they have.
[This picture from wikipedia](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Correlation_examples2.svg/1920px-Correlation_examples2.svg.png) ([article](https://en.wikipedia.org/wiki/Correlation)) explains it nicely.
On the bottom row are a series of 2D distributions with 0 covariance that are very clearly not independent (ie – knowing the position of a point on the x-axis tells you a lot about where that point might lie on the y-axis (except the one on the bottom-right – that one may actually be independent))
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