@@ -65,7 +65,7 @@ Thus, we change the specification in [](likelihood_bayes) in the following way.

6565Now, **each period** $t \geq 0$, nature flips a possibly unfair coin that comes up $f$ with probability $\alpha$6666and $g$ with probability $1 -\alpha$.676768-Thus, nature perpetually draws from the **mixture distribution** with c.d.f.68+Thus, nature perpetually draws from the **mixture distribution** with CDF69697070$$7171H(w) = \alpha F(w) + (1-\alpha) G(w), \quad \alpha \in (0,1)

@@ -220,7 +220,7 @@ Here is pseudo code for a direct "method 1" for drawing from our compound lotter

220220221221Our second method uses a uniform distribution and the following fact that we also described and used in [](prob_matrix):222222223

-* If a random variable $X$ has c.d.f. $F$, then a random variable $F^{-1}(U)$ also has c.d.f. $F$, where $U$ is a uniform random variable on $[0,1]$.

223

+* If a random variable $X$ has CDF $F$, then a random variable $F^{-1}(U)$ also has CDF $F$, where $U$ is a uniform random variable on $[0,1]$.

224224225225In other words, if $X \sim F(x)$ we can generate a random sample from $F$ by drawing a random sample from226226a uniform distribution on $[0,1]$ and computing $F^{-1}(U)$.

@@ -267,6 +267,12 @@ def draw_lottery_MC(key, p, N):

267267```268268269269```{code-cell} ipython3270+---271+mystnb:272+ figure:273+ caption: Direct and Monte Carlo draws274+ name: fig-lottery-draws275+---270276# verify271277N = 100000272278α = 0.0

@@ -457,6 +463,12 @@ def plot_π_seq(key, α, π1=0.2, π2=0.8, T=200):

457463```458464459465```{code-cell} ipython3466+---467+mystnb:468+ figure:469+ caption: Belief paths, $\alpha = 0.6$470+ name: fig-pi-seq-1471+---460472plot_π_seq(jax.random.key(42), α=0.6)461473```462474

@@ -467,6 +479,12 @@ sample paths of $\pi_t$ that start from two distinct initial conditions.

467479Let's see what happens when we change $\alpha$.468480469481```{code-cell} ipython3482+---483+mystnb:484+ figure:485+ caption: Belief paths, $\alpha = 0.2$486+ name: fig-pi-seq-2487+---470488plot_π_seq(jax.random.key(42), α=0.2)471489```472490

@@ -568,6 +586,12 @@ def π_lim(key, α, T=5000, π_0=0.4):

568586Let us first plot the KL divergences $KL_g\left(\alpha\right), KL_f\left(\alpha\right)$ for each $\alpha$.569587570588```{code-cell} ipython3589+---590+mystnb:591+ figure:592+ caption: KL divergences against $\alpha$593+ name: fig-kl594+---571595α_arr = np.linspace(0, 1, 100)572596KL_g_arr = KL_g_v(α_arr)573597KL_f_arr = KL_f_v(α_arr)

@@ -598,6 +622,12 @@ recorded on the $x$ axis.

598622Thus, the graph below confirms how a minimum KL divergence governs what our type 1 agent eventually learns.599623600624```{code-cell} ipython3625+---626+mystnb:627+ figure:628+ caption: Limit points and KL divergences629+ name: fig-kl-limit630+---601631α_arr_x = α_arr[(α_arr < discretion) | (α_arr > discretion)]602632keys = jax.random.split(jax.random.key(42), len(α_arr_x))603633π_lim_arr = π_lim_v(keys, α_arr_x)

@@ -710,6 +740,12 @@ def MCMC_run(ws):

710740The following code generates the graph below that displays Bayesian posteriors for $\alpha$ at various history lengths.711741712742```{code-cell} ipython3743+---744+mystnb:745+ figure:746+ caption: Posterior for $\alpha$ as $t$ grows747+ name: fig-posterior-alpha748+---713749fig, ax = plt.subplots()714750715751for i in range(len(sizes)):