Interpretation of conditional probability

As repeated throughout these notes, and illustrated with many examples, probability is always conditioned probability. Absolute probability makes no sense. Nevertheless, there is still something in the primer which can be misleading and that needs to be clarified, namely the so-called `formula of conditional probability' (Section ):

$$P(E\,\vert\,H) = \frac{P(E\cap H)}{P(H)}\hspace{1.0cm}(P(H)\ne 0)\,.$$ (8.1)

What does it mean? Textbooks present it as a definition (a kind of 4th axiom), although very often, a few lines later in the same book, the formula $P(E\cap H)=P(E\,\vert\,H)\cdot P(H)$ is presented as a theorem (!).

In the subjective approach, one is allowed to talk about $P(E\,\vert\,H)$ independently of $P(E\cap H)$ and $P(H)$. In fact, $P(E\,\vert\,H)$ is just the assessment of the probability of $E$, under the condition that $H$ is true. Then it cannot depend on the probability of $H$. It is easy to show with an example that this point of view is rather natural, whilst that of considering () as a definition is artificial. Let us take

• $H$ = Higgs mass of 250 GeV;
• $E$ = the decay products which are detected in a LHC detector;
• the evaluation of $P(E\,\vert\,H)$ is a standard PhD student task. He chooses $m_H = 250\,$GeV in the Monte Carlo and counts how many events pass the cuts (for the interpretation of this operation, see the previous section). No one would think that $P(E\,\vert\,H)$ must be evaluated only from $P(E\cap H)$ and $P(H)$, as the definition () would imply. Moreover, the procedure is legitimate even if we knew with certainty that the Higgs mass was below 200 GeV and, therefore, $P(H)=0$.

In the subjective approach, () is a true theorem required by coherence. It means that although one can speak of each of the three probabilities independently of the others, once two of them have been elicited, the third is constrained. It is interesting to demonstrate the theorem to show that it has nothing to do with the kind of heuristic derivation of Section :

• Let us imagine a coherent bet on the conditional event $E\,\vert\,H$ to win a unitary amount of money ($B=1$, as the scale factor is inessential). Remembering the meaning of conditional probability in terms of bets (see Section ), this means that
  • we pay (with certainty) $A=P(E\,\vert\,H)$;
  • we win 1 if $E$ and $H$ are both verified (with probability $P(E\cap H)$);
  • we get our money back (i.e. $A$) if $H$ does not happen (with probability $P(\overline{H})$).
• The expected value of the `gain' $G$ is given by the probability of each event multiplied by the gain associated with each event:
  

  $$(G) = 1\cdot (-P(E\,\vert\,H)) + P(E\cap H)\cdot 1 + P(\overline{H})\cdot P(E\,\vert\,H) \,,$$

  
  where the first factors of the products on the right-hand side of the formula stand for probability, the second for the amount of money. It follows that
  

  $$(G) = -P(E\,\vert\,H) + P(E\cap H) + (1-P(H))\cdot P(E\,\vert\,H)$$

  $$= P(E\cap H) - P(E\,\vert\,H)\cdot P(H)\,.$$ (8.2)

  
  
• Coherence requires the rational better to be indifferent to the direction of the bet, i.e. E$(G)=0$. Applying this condition to () we obtain ().