Are probabilities of certain events (e.g. amount of rainfall over a period, probability of a Fed rate hike) inferred from derivatives on non-tradeable random variables (e.g. Weather Futures, Fed Funds Futures) stated in the risk-neutral measure (with the money market numeraire) or real-world measure?
St Louis Fed published in 2006 a very nice paper: What Are the Odds? Option-Based Forecasts of FOMC Target Changes by William Emmons, Aeimit Lakdawala, and Christopher Neely, which discusses futures on fed funds, options on such futures, implied risk-netural probabilities, and how they differ from objective real-world probabilities.
Excellent question to grasp the concepts. Basically prices NEVER indicate anything about real-world probabilities. Prices are formed by the interaction of supply and demand from economic agents who might not know (or even care) of the underlying’s future behaviour.
Risk-neutral measures’ existence is a consequence of the FTAP, based on possible replication strategies for contingent claims. Either you can risklessly replicate the payments for such an asset, and the risk-neutral measure used to price it is unique because the no-arbitrage price, which is the value of the replicating portfolio, is unique. If you cannot replicate the claim, there is a range of NA prices (e.g. for a call option, it is $\left[ \left[PV\left(F_T - K\right)\right]^+, PV(F_T) \right]$, $F_T$ being the underlying’s forward price and $PV$ the present value) and infinitely many risk-neutral measures. For no reason should any one of them be particularly informative about real-world probabilities.
For non-traded assets, the “market” (investment banks offering tradeable prices) will use historic data to infer the behaviour of the underlying’s price, and allow for a good “margin of error”. E.g., if the historical standard deviation of temperatures is 15%, they might price using models with standard deviation ranging between 20% and 22%. But the future standard deviation could well be 30% due to climate instability, they wouldn’t know!