Entropy Pooling and View Blending

A jury pool of 100 people gets called in reflecting the town's actual demographics. A judge then rules the seated jury must include at least 30% of some specific group. The clumsy fix is to throw out the whole pool and hand-pick 12 people who satisfy the quota. The careful fix is to re-weight the existing pool — nudge the selection odds up for that group and down for everyone else, just enough to hit 30%, while disturbing everything else about the pool's composition as little as possible. Entropy pooling does the second thing to a set of market scenarios: given a prior collection of possible future outcomes and a view the investor wants imposed — "I think the expected return should be higher than what the base scenarios show" — it finds new probabilities for those same scenarios that satisfy the view while staying as close as mathematically possible to the original.

Minimal distortion, made precise

"As close as possible" is measured by relative entropy — how surprised you'd be, on average, discovering the new probabilities if you'd been expecting the old ones:

$$\min_{\mathbf{p}} \sum_i p_i \ln\!\left(\frac{p_i}{q_i}\right) \quad \text{subject to the investor's view constraints and } \sum_i p_i = 1 .$$

In words: $q_i$ is the prior probability of scenario $i$ (often just $1/N$ if every historical scenario is equally likely to start), $p_i$ is the new, "posterior" probability being solved for. The sum $\sum_i p_i \ln(p_i/q_i)$ is zero only when $p$ and $q$ are identical, and grows the more the new probabilities diverge from the old ones. Minimizing it subject to the view constraints — for example, that the scenario-weighted average return equals some target — finds the least distorted reweighting that still makes the view true. Unlike Black-Litterman, which blends views into a single mean-variance estimate, entropy pooling works on the full scenario set directly, so views can be about anything expressible as a constraint on those scenarios, not just expected returns.

Worked example 1 — three scenarios, one view

A prior scenario set has three equally likely outcomes for next year's return: $-10\%$, $0\%$, $+10\%$, each with prior probability $q_i = 1/3 \approx 0.333$. An analyst's view: the expected return should be $4\%$, not the prior's $0\%$. Two different posterior probability sets both satisfy that view exactly. Candidate A: $p = (0.10,\ 0.40,\ 0.50)$ — check the mean: $-10(0.10) + 0(0.40) + 10(0.50) = -1 + 0 + 5 = 4\%$. Candidate B: $p = (0.25,\ 0.10,\ 0.65)$ — check: $-10(0.25) + 0(0.10) + 10(0.65) = -2.5 + 0 + 6.5 = 4\%$. Both hit the view precisely, but they got there very differently: A barely touches the middle scenario's probability, while B guts it from 0.333 down to 0.10.

Worked example 2 — which one does entropy pooling pick?

Relative entropy scores each candidate against the prior. For A: $0.10\ln(0.10/0.333) + 0.40\ln(0.40/0.333) + 0.50\ln(0.50/0.333) = 0.10(-1.204) + 0.40(0.182) + 0.50(0.405) = -0.120 + 0.073 + 0.203 = 0.155$. For B: $0.25\ln(0.25/0.333) + 0.10\ln(0.10/0.333) + 0.65\ln(0.65/0.333) = 0.25(-0.288) + 0.10(-1.204) + 0.65(0.668) = -0.072 - 0.120 + 0.434 = 0.242$. A's score of $0.155$ is lower than B's $0.242$, meaning A is the less-distorted reweighting — and it's exactly the kind of solution entropy pooling's optimizer converges to, because it barely touches the scenario the investor's view said nothing about, while B distorted probabilities the view never asked it to change.

The mechanics aren't identical to Bayes' rule — entropy pooling minimizes relative entropy under constraints rather than applying the Bayes formula directly — but the shape of the idea is the same one this updater shows: start from a prior belief, fold in new information, and land on a posterior that's been pulled toward the data without being reset to it entirely. Watch how a strong prior resists a small amount of new data, the same way a scenario the view says nothing about resists being reweighted.

What this means in practice

Entropy pooling lets a quant impose qualitative or nonlinear views a mean-variance framework can't express directly — "I think a recession scenario is twice as likely as the historical base rate," or a ranking view across assets without pinning down an exact number — and get a fully updated scenario set back, ready to feed into portfolio construction. It's most useful when the prior comes from a rich historical or simulated scenario set rather than a simple normal distribution, since that's where a single mean-variance update would throw away information the scenarios actually contain.