How do we know that Janet Yellen isn't holding down interest rates, as Donald Trump claims?

AFAIK Australian bonds are denominated in Australian dollars, and Chinese in Renminbi / Yuan, so if you need your money back in USD at the end you could easily lose money if the currencies fall in value vs the USD. More risk = more reward.

Since a post election tightening would cause the stock market to fall, if someone believed that the Fed was going to tighten after the election it would be easy enough to short the market. If they believed that the Fed was being too loose with policy and would continue to do so after the election this would be reflected in a higher inflation rate. There are ways to make money via speculation over inflation rates such as currency trading, investing in inflation hedges such as gold, or using TIPS spreads.
From my limited knowledge of finance it does not appear that lots of people are shorting the entire market or engaging in currency speculation. Given the current anemic growth it would appear that the Fed has not been too loose with monetary policy.
In Trump’s defense it would not be the first time the Fed has intervened in markets to influence an election. Greenspan is widely thought to have backed off inflation fighting to help Carter out in 1980 and we know Nixon pressured Burns to keep rates low in 1972.

That’s correct, besides which there’s significant perceived greater credit* risk in China at least compared to the US, which would be a more minor factor if at all with Australia. This could seen more specifically by looking not at the yield of own currency bonds, but what the net yield would be if you added currency derivative transactions to hedge the FX risk. On that basis the yields of various govt bonds ‘magically’ get a lot closer to one another than they appear in own currency terms.

And in general the rate comparison given by Kunilou is interesting but not all that relevant to the question, and it is a serious one even in realistic rather than exaggerated Trumpian terms, of whether the Fed is keeping US interest rates too low. For one major thing, other central banks’ rate decisions are seriously influenced by the Fed’s actions. The US is only 20-some % of world GDP but the USD has an out sized influence relative to that in world trade and finance. Of course there’s also feedback in other direction: the Fed is partly constrained in raising US rates when other central banks keep them so low, which they’ve pretty much admitted in recent years, albeit indirectly.

But in general it’s not a crazy tinfoil hat to think that the Fed’s rate decisions near elections might be somewhat influenced by politics. It’s not necessarily partisan bias, since right now if the Fed were to take a significant** new action it could only go in one direction, upset markets and hurt the de facto incumbent party the Democrats. But it’s still not fantastic to suppose the liberal Democrat Yellens’ views and outlook might be less than 100% divorced from partisan politics. Again, first ask yourself before dismissing that whether you would as absolutely if it were a Republican chairman, especially a very conservative one (Bernanke was only nominally a Republican, others in the past have been more solidly Republican than he).

*not to be thought of narrowly as just ‘if China absolutely can’t come up with Yuan to pay the bonds’, since they can print Yuan like the US or Australia can print their respective Dollars. Rather it would rule of law issues in China as they relate to foreigners especially, or that China could choose to tell all investors they have to take a ‘hair cut’ on their RMB bonds even though it’s technically possible for China to pay. Russia did that in 1996: holders of Ruble denominated bonds didn’t get all their money back. It’s a myth that you have zero nominal credit risk buying bonds of a govt which issues in its own currency. It might be technically possible for them print the money to pay you, but they might have good reasons not to. Besides which of course getting fully nominally paid back in devalued currency is a loss anyway in real terms.
**arguably it could go back to Quantitative Easing or newly go to outright negative rates but one of the legit questions is whether these extraordinary actions by central banks generally are really having much effect anymore. Whereas a bold change in course to more historically normal rates would definitely have an effect.

Greenspan wasn’t on the Fed in 1980. Did you mean Volcker?