On the Relation between Robust and Bayesian Decision Making

x and the state of the world is given by s. If x is the real interest rate, then
the factor s represents the sensitivity of the economy’s inflation rate to the real
interest rate, a number likely to be unknown to the policy maker. Moreover,
the policy maker might be unable to assign probabilities to the various values
of s.

For simplicity, suppose that the desired target inflation rate is π* = 2 and
that there are only two potential multipliers si < s⅛ with si = 1 and s⅛ = 3.

The loss functions associated for each of these multipliers are shown in figure
1. The dotted line in the graph indicates the maximum loss associated with
each action. Figure 1 clearly shows that the robust decision that minimizes the
maximum loss is given by x* = 1.⁴

Suppose a Bayesian central bank assigns equal probability to each of the
two multipliers s (i = l,h). The optimal Bayesian decision is then given by
x* = 0.8.

The Bayesian decision maker reacts less aggressively than the robust decision
maker. This is the case because the Bayesian trades off the gains and losses
across the different realizations of s. At the robust decision (x = 1) the loss
functions in figure 1 have different absolute slope coefficients depending on the
value of s^l (i = l,h). Therefore, the Bayesian has an incentive to decrease the
interest rate below 1 since the gains made for the realization s⅛ will exceed the
losses for realizations s_i, given the prior probabilities assigned to these states.

When the Bayesian’s objective function is subjected to increasingly convex
transformations through T^k (∙) risk aversion increases. This implies that the
gains for the state s⅛ will be appreciated less relative to the potential losses
for state s_i. Graphically one can interpret this as figure 1 being scaled in the
direction of the у-axis with each point being scaled by a factor that is increasing
with its distance from the X-axis. As a result, the slope of T(x, si) to the left
of x = 1 increases much faster than the absolute value of the slope of T(x, s⅛)
to the left of this point. This pushes the Bayesian decision into the direction of
the robust decision.

Figure 2 shows how the Bayesian decision approaches the robust decision
as к increases, which suggests a convenient way to calculate (approximately)
robust decisions.⁵

⁴Since there is no uncertainty about the sign of the parameters s the optimal robust decision
coincides with the optimal decision under certainty equivalence, as noted by Onatski (2000).

⁵The speed of convergence will depend, amongst other things, on the Bayesian’s prior.

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