diff --git a/lectures/likelihood_bayes.md b/lectures/likelihood_bayes.md index c9b203f11..7bf5d1d00 100644 --- a/lectures/likelihood_bayes.md +++ b/lectures/likelihood_bayes.md @@ -11,7 +11,7 @@ kernelspec: name: python3 --- -(likelihood_ratio_process)= +(likelihood_bayes)= ```{raw} jupyter
@@ -26,11 +26,11 @@ kernelspec: ## Overview -This lecture describes the role that **likelihood ratio processes** play in **Bayesian learning**. +This lecture describes the role that *likelihood ratio processes* play in *Bayesian learning*. -As in {doc}`this lecture `, we'll use a simple statistical setting from {doc}`this lecture `. +As in {doc}`likelihood_ratio_process`, we'll use a simple statistical setting from {doc}`exchangeable`. -We'll focus on how a likelihood ratio process and a **prior** probability determine a **posterior** probability. +We'll focus on how a likelihood ratio process and a *prior* probability determine a *posterior* probability. We'll derive a convenient recursion for today's posterior as a function of yesterday's posterior and today's multiplicative increment to a likelihood process. @@ -38,13 +38,12 @@ today's multiplicative increment to a likelihood process. We'll also present a useful generalization of that formula that represents today's posterior in terms of an initial prior and today's realization of the likelihood ratio process. -We'll study how, at least in our setting, a Bayesian eventually learns the probability distribution that generates the data, an outcome that -rests on the asymptotic behavior of likelihood ratio processes studied in {doc}`this lecture `. +We'll study how, at least in our setting, a Bayesian eventually learns the probability distribution that generates the data, an outcome that +rests on the asymptotic behavior of likelihood ratio processes studied in {doc}`likelihood_ratio_process`. -We'll also drill down into the psychology of our Bayesian learner and study dynamics under his subjective beliefs. +We'll also drill down into the psychology of our Bayesian learner and study dynamics under his subjective beliefs. -This lecture provides technical results that underly outcomes to be studied in {doc}`this lecture ` -and {doc}`this lecture ` and {doc}`this lecture `. +This lecture provides technical results that underlie outcomes to be studied in {doc}`odu`, {doc}`wald_friedman`, and {doc}`navy_captain`. We'll begin by loading some Python modules. @@ -68,16 +67,16 @@ def set_seed(): set_seed() ``` -## The Setting +## The setting -We begin by reviewing the setting in {doc}`this lecture `, which we adopt here too. +We begin by reviewing the setting in {doc}`likelihood_ratio_process`, which we adopt here too. A nonnegative random variable $W$ has one of two probability density functions, either $f$ or $g$. Before the beginning of time, nature once and for all decides whether she will draw a sequence of IID draws from $f$ or from $g$. -We will sometimes let $q$ be the density that nature chose once and for all, so +We let $q$ be the density that nature chose once and for all, so that $q$ is either $f$ or $g$, permanently. Nature knows which density it permanently draws from, but we the observers do not. @@ -95,7 +94,7 @@ from either $f$ or $g$. We want to use these observations to infer whether nature chose $f$ or $g$. -A **likelihood ratio process** is a useful tool for this task. +A likelihood ratio process is a useful tool for this task. To begin, we define the key component of a likelihood ratio process, namely, the time $t$ likelihood ratio as the random variable @@ -110,7 +109,7 @@ That means that under the $g$ density, $\ell (w_t)= \frac{f\left(w_{t}\right)}{g\left(w_{t}\right)}$ is evidently a nonnegative random variable with mean $1$. -A **likelihood ratio process** for sequence +A likelihood ratio process for sequence $\left\{ w_{t}\right\} _{t=1}^{\infty}$ is defined as $$ @@ -140,7 +139,7 @@ The likelihood ratio and its logarithm are key tools for making inferences using a classic frequentist approach due to Neyman and Pearson {cite}`Neyman_Pearson`. -We'll again deploy the following Python code from {doc}`this lecture ` that +We'll again deploy the following Python code from {doc}`likelihood_ratio_process` that evaluates $f$ and $g$ as two different beta distributions, then computes and simulates an associated likelihood ratio process by generating a sequence $w^t$ from *some* @@ -154,7 +153,7 @@ G_a, G_b = 3, 1.2 @vectorize def p(x, a, b): r = gamma(a + b) / (gamma(a) * gamma(b)) - return r * x** (a-1) * (1 - x) ** (b-1) + return r * x**(a-1) * (1 - x)**(b-1) # The two density functions. f = jit(lambda x: p(x, F_a, F_b)) @@ -167,7 +166,6 @@ def simulate(a, b, T=50, N=500): ''' Generate N sets of T observations of the likelihood ratio, return as N x T matrix. - ''' l_arr = np.empty((N, T)) @@ -193,13 +191,13 @@ l_arr_f = simulate(F_a, F_b, N=50000) l_seq_f = np.cumprod(l_arr_f, axis=1) ``` -## Likelihood Ratio Processes and Bayes’ Law +## Likelihood ratio processes and Bayes’ law -Let $\pi_0 \in [0,1]$ be a Bayesian statistician's prior probability that nature generates $w^t$ as a sequence of i.i.d. draws from +Let $\pi_0 \in [0,1]$ be a Bayesian statistician's prior probability that nature generates $w^t$ as a sequence of IID draws from distribution $f$. -* here "probability" is to be interpreted as a way to summarize or express a subjective opinion -* it does **not** mean an anticipated relative frequency as sample size grows without limit +* here "probability" is to be interpreted as a way to summarize or express a subjective opinion +* it does *not* mean an anticipated relative frequency as sample size grows without limit Let $\pi_{t+1}$ be a Bayesian posterior probability defined as @@ -208,9 +206,9 @@ $$ $$ (eq:defbayesposterior) The likelihood ratio process is a principal actor in the formula that governs the evolution -of the posterior probability $\pi_t$, an instance of **Bayes' Law**. +of the posterior probability $\pi_t$, an instance of *Bayes’ law*. -Let's derive a couple of formulas for $\pi_{t+1}$, one in terms of likelihood ratio $l(w_t)$, the other in terms of +Let's derive a couple of formulas for $\pi_{t+1}$, one in terms of likelihood ratio $\ell(w_t)$, the other in terms of $L(w^t)$. To begin, we use the notational conventions @@ -311,7 +309,7 @@ Dividing both the numerator and the denominator on the right side of the equat ```{math} :label: eq_recur1 -\pi_{t+1}=\frac{\pi_{t} l_t(w_{t+1})}{\pi_{t} l_t(w_{t+1})+1-\pi_{t}} +\pi_{t+1}=\frac{\pi_{t} \ell(w_{t+1})}{\pi_{t} \ell(w_{t+1})+1-\pi_{t}} ``` with $\pi_{0}$ being a Bayesian prior probability that $q = f$, @@ -339,7 +337,7 @@ As $t \rightarrow + \infty$, the likelihood ratio process dominates the initial limiting behavior of $\pi_t$. -To illustrate this insight, below we will plot graphs showing **one** simulated +To illustrate this insight, below we will plot graphs showing *one* simulated path of the likelihood ratio process $L_t$ along with two paths of $\pi_t$ that are associated with the *same* realization of the likelihood ratio process but *different* initial prior probabilities $\pi_{0}$. @@ -363,19 +361,25 @@ for t in range(T): ``` ```{code-cell} ipython3 +--- +mystnb: + figure: + caption: Posterior paths and log likelihood + name: fig-posterior-lratio-f +--- fig, ax1 = plt.subplots() for i in range(2): - ax1.plot(range(T+1), π_seq_f[i, :], label=fr"$\pi_0$={π_seq_f[i, 0]}") + ax1.plot(range(T+1), π_seq_f[i, :], + label=fr"$\pi_0$={π_seq_f[i, 0]}", lw=2) ax1.set_ylabel(r"$\pi_t$") -ax1.set_xlabel("t") +ax1.set_xlabel(r"$t$") ax1.legend() -ax1.set_title("when f governs data") ax2 = ax1.twinx() -ax2.plot(range(1, T+1), np.log(l_seq_f[0, :]), '--', color='b') -ax2.set_ylabel("$log(L(w^{t}))$") +ax2.plot(range(1, T+1), np.log(l_seq_f[0, :]), '--', color='b', lw=2) +ax2.set_ylabel(r"$\log(L(w^{t}))$") plt.show() ``` @@ -397,19 +401,25 @@ for t in range(T): ``` ```{code-cell} ipython3 +--- +mystnb: + figure: + caption: Posterior paths and log likelihood + name: fig-posterior-lratio-g +--- fig, ax1 = plt.subplots() for i in range(2): - ax1.plot(range(T+1), π_seq_g[i, :], label=fr"$\pi_0$={π_seq_g[i, 0]}") + ax1.plot(range(T+1), π_seq_g[i, :], + label=fr"$\pi_0$={π_seq_g[i, 0]}", lw=2) ax1.set_ylabel(r"$\pi_t$") -ax1.set_xlabel("t") +ax1.set_xlabel(r"$t$") ax1.legend() -ax1.set_title("when g governs data") ax2 = ax1.twinx() -ax2.plot(range(1, T+1), np.log(l_seq_g[0, :]), '--', color='b') -ax2.set_ylabel("$log(L(w^{t}))$") +ax2.plot(range(1, T+1), np.log(l_seq_g[0, :]), '--', color='b', lw=2) +ax2.set_ylabel(r"$\log(L(w^{t}))$") plt.show() ``` @@ -439,25 +449,25 @@ $f$. Let's study how the posterior probability $\pi_t = {\rm Prob}(q=f|w^{t}) $ behaves when nature generates the history $w^t = \{w_1, w_2, \dots, w_t\}$ under a different timing protocol. -Until now we assumed that before time $1$ nature somehow chose to draw $w^t$ as an iid sequence from **either** $f$ **or** $g$. +Until now we assumed that before time $1$ nature somehow chose to draw $w^t$ as an IID sequence from either $f$ or $g$. -Nature's decision about whether to draw from $f$ or $g$ was thus **permanent**. +Nature's decision about whether to draw from $f$ or $g$ was thus *permanent*. -We now assume a different timing protocol in which before **each period** $t =1, 2, \ldots$ nature +We now assume a different timing protocol in which before *each period* $t =1, 2, \ldots$ nature * flips an $x$-weighted coin, then * draws from $f$ if it has drawn a "head" * draws from $g$ if it has drawn a "tail". -Under this timing protocol, nature draws permanently from **neither** $f$ **nor** $g$, so a statistician who thinks that nature is drawing -i.i.d. draws **permanently** from one of them is mistaken. +Under this timing protocol, nature draws permanently from *neither* $f$ *nor* $g$, so a statistician who thinks that nature is drawing +IID draws *permanently* from one of them is mistaken. -* in truth, nature actually draws **permanently** from an $x$-mixture of $f$ and $g$ -- a distribution that is neither $f$ nor $g$ when +* in truth, nature actually draws *permanently* from an $x$-mixture of $f$ and $g$ — a distribution that is neither $f$ nor $g$ when $x \in (0,1)$ -Thus, the Bayesian prior $\pi_0$ and the sequence of posterior probabilities described by equation {eq}`eq_Bayeslaw1033` should **not** be interpreted as the statistician's opinion about the mixing parameter $x$ under the alternative timing protocol in which nature draws from an $x$-mixture of $f$ and $g$. +Thus, the Bayesian prior $\pi_0$ and the sequence of posterior probabilities described by equation {eq}`eq_Bayeslaw1033` should *not* be interpreted as the statistician's opinion about the mixing parameter $x$ under the alternative timing protocol in which nature draws from an $x$-mixture of $f$ and $g$. This is clear when we remember the definition of $\pi_t$ in equation {eq}`eq:defbayesposterior`, which for convenience we repeat here: @@ -467,11 +477,11 @@ $$ -Let's write some Python code to study how $\pi_t$ behaves when nature actually generates data as i.i.d. draws from neither $f$ nor from $g$ -but instead as i.i.d. draws from an $x$-mixture of two beta distributions. +Let's write some Python code to study how $\pi_t$ behaves when nature actually generates data as IID draws from neither $f$ nor from $g$ +but instead as IID draws from an $x$-mixture of two beta distributions. ```{note} -This is a situation in which the statistician's model is misspecified, so we should anticipate that a Kullback-Liebler divergence with respect to an $x$-mixture distribution will shape outcomes. +This is a situation in which the statistician's model is misspecified, so we should anticipate that a Kullback–Leibler divergence with respect to an $x$-mixture distribution will shape outcomes. ``` We can study how $\pi_t$ would behave for various values of nature's mixing probability $x$. @@ -522,6 +532,12 @@ Let's study how the posterior probability $\pi_t$ that nature permanently draws an $x$-mixture of $f$ and $g$. ```{code-cell} ipython3 +--- +mystnb: + figure: + caption: Posterior under mixture model + name: fig-posterior-mixture +--- fig, ax = plt.subplots(figsize=(10, 6)) T_plot = 200 @@ -546,7 +562,7 @@ ax.legend() plt.show() ``` -Evidently, $\pi_t$ converges to 1. +Evidently, $\pi_t$ converges to 1. This indicates that the model concludes that the data is generated by $f$. @@ -554,7 +570,7 @@ Why does this happen? Given $x = 0.5$, the data generating process is a mixture of $f$ and $g$: $m(w) = \frac{1}{2}f(w) + \frac{1}{2}g(w)$. -Let's check the [KL divergence](rel_entropy) of the mixture distribution $m$ from both $f$ and $g$. +Let's check the {ref}`KL divergence ` of the mixture distribution $m$ from both $f$ and $g$. ```{code-cell} ipython3 def compute_KL(f, g): @@ -568,7 +584,7 @@ def compute_KL(f, g): def compute_div_m(f, g): """ - Compute Jensen-Shannon divergence + Compute KL(m, f) and KL(m, g) """ def m(w): return 0.5 * (f(w) + g(w)) @@ -586,7 +602,7 @@ Since $KL(m, f) < KL(m, g)$, $f$ is "closer" to the mixture distribution $m$. Hence by our discussion on KL divergence and likelihood ratio process in {doc}`likelihood_ratio_process`, $\log(L_t) \to \infty$ as $t \to \infty$. -Now looking back to the key equation {eq}`eq_Bayeslaw1033`. +Now look back at the key equation {eq}`eq_Bayeslaw1033`. Consider the function @@ -607,12 +623,12 @@ This topic is taken up in {doc}`mix_model`. We explore how to learn the true mixing parameter $x$ in the exercise of {doc}`mix_model`. -## Behavior of Posterior Probability $\{\pi_t\}$ Under Subjective Probability Distribution +## Behavior of posterior probability $\{\pi_t\}$ under subjective probability distribution We'll end this lecture by briefly studying what our Bayesian learner expects to learn under the -subjective beliefs $\pi_t$ cranked out by Bayes' law. +subjective beliefs $\pi_t$ cranked out by Bayes’ law. -This will provide us with some perspective on our application of Bayes's law as a theory of learning. +This will provide us with some perspective on our application of Bayes’ law as a theory of learning. As we shall see, at each time $t$, the Bayesian learner knows that he will be surprised. @@ -629,19 +645,19 @@ We'll review and reiterate and rearrange some formulas that we have encountered The worker's initial beliefs induce a joint probability distribution over a potentially infinite sequence of draws $w_0, w_1, \ldots $. -Bayes' law is simply an application of laws of +Bayes’ law is simply an application of laws of probability to compute the conditional distribution of the $t$th draw $w_t$ conditional on $[w_0, \ldots, w_{t-1}]$. -After our worker puts a subjective probability $\pi_{-1}$ on nature having selected distribution $F$, we have in effect assumed from the start that the decision maker **knows** the joint distribution for the process $\{w_t\}_{t=0}$. +After our worker puts a subjective probability $\pi_{-1}$ on nature having selected distribution $F$, we have in effect assumed from the start that the decision maker *knows* the joint distribution for the process $\{w_t\}_{t=0}$. We assume that the worker also knows the laws of probability theory. -A respectable view is that Bayes' law is less a theory of learning than a statement about the consequences of information inflows for a decision maker who thinks he knows the truth (i.e., a joint probability distribution) from the beginning. +A respectable view is that Bayes’ law is less a theory of learning than a statement about the consequences of information inflows for a decision maker who thinks he knows the truth (i.e., a joint probability distribution) from the beginning. ### Mechanical details again -At time $0$ **before** drawing a wage offer, the worker attaches probability $\pi_{-1} \in (0,1)$ to the distribution being $F$. +At time $0$ *before* drawing a wage offer, the worker attaches probability $\pi_{-1} \in (0,1)$ to the distribution being $F$. Before drawing a wage at time $0$, the worker thus believes that the density of $w_0$ is @@ -652,7 +668,7 @@ $$ Let $a \in \{ f, g\} $ be an index that indicates whether nature chose permanently to draw from distribution $f$ or from distribution $g$. -After drawing $w_0$, the worker uses Bayes' law to deduce that +After drawing $w_0$, the worker uses Bayes’ law to deduce that the posterior probability $\pi_0 = {\rm Prob}({a = f | w_0}) $ that the density is $f(w)$ is @@ -673,7 +689,7 @@ or $$ -\pi_t=\frac{\pi_{t-1} l_t(w_t)}{\pi_{t-1} l_t(w_t)+1-\pi_{t-1}} +\pi_t=\frac{\pi_{t-1} \ell(w_t)}{\pi_{t-1} \ell(w_t)+1-\pi_{t-1}} $$ @@ -694,9 +710,9 @@ E(\pi_t | \pi_{t-1}) & = \int \Bigl[ { \pi_{t-1} f(w) \over \pi_{t-1} f(w) + (1 \end{aligned} $$ -so that the process $\pi_t$ is a **martingale**. +so that the process $\pi_t$ is a *martingale*. -Indeed, it is a **bounded martingale** because each $\pi_t$, being a probability, +Indeed, it is a *bounded martingale* because each $\pi_t$, being a probability, is between $0$ and $1$. @@ -704,10 +720,10 @@ In the first line in the above string of equalities, the term in the first set o is just $\pi_t$ as a function of $w_{t}$, while the term in the second set of brackets is the density of $w_{t}$ conditional on $w_{t-1}, \ldots , w_0$ or equivalently conditional on the *sufficient statistic* $\pi_{t-1}$ for $w_{t-1}, \ldots , w_0$. -Notice that here we are computing $E(\pi_t | \pi_{t-1})$ under the **subjective** density described in the second +Notice that here we are computing $E(\pi_t | \pi_{t-1})$ under the *subjective* density described in the second term in brackets. -Because $\{\pi_t\}$ is a bounded martingale sequence, it follows from the **martingale convergence theorem** that $\pi_t$ converges almost surely to a random variable in $[0,1]$. +Because $\{\pi_t\}$ is a bounded martingale sequence, it follows from the *martingale convergence theorem* that $\pi_t$ converges almost surely to a random variable in $[0,1]$. Practically, this means that probability one is attached to sample paths $\{\pi_t\}_{t=0}^\infty$ that converge. @@ -718,12 +734,12 @@ Thus, let $\{\pi_t(\omega)\}_{t=0}^\infty$ denote a particular sample path index \in \Omega$. We can think of nature as drawing an $\omega \in \Omega$ from a probability distribution -${\textrm{Prob}} \Omega$ and then generating a single realization (or _simulation_) $\{\pi_t(\omega)\}_{t=0}^\infty$ of the process. +${\textrm{Prob}}(\Omega)$ and then generating a single realization (or _simulation_) $\{\pi_t(\omega)\}_{t=0}^\infty$ of the process. The limit points of $\{\pi_t(\omega)\}_{t=0}^\infty$ as $t \rightarrow +\infty$ are realizations of a random variable that is swept out as we sample $\omega$ from $\Omega$ and construct repeated draws of $\{\pi_t(\omega)\}_{t=0}^\infty$. -By staring at law of motion {eq}`eq_recur1` or {eq}`eq:like44` , we can figure out some things about the probability distribution of the limit points +By staring at the law of motion {eq}`eq_recur1` or {eq}`eq:like44`, we can figure out some things about the probability distribution of the limit points @@ -749,8 +765,8 @@ $$ $$ -Thus, for some realizations, $\lim_{\rightarrow + \infty} \pi_t(\omega) =1$ -while for other realizations, $\lim_{\rightarrow + \infty} \pi_t(\omega) =0$. +Thus, for some realizations, $\lim_{t \rightarrow + \infty} \pi_t(\omega) =1$ +while for other realizations, $\lim_{t \rightarrow + \infty} \pi_t(\omega) =0$. Now let's remember that $\{\pi_t\}_{t=0}^\infty$ is a martingale and apply the law of iterated expectations. @@ -771,7 +787,7 @@ Applying the above formula to $\pi_\infty$, we obtain $$ E_{-1} \pi_\infty(\omega) = \pi_{-1} -$$ +$$ (eq:expect_pi_infty) where the mathematical expectation $E_{-1}$ here is taken with respect to the probability measure ${\textrm{Prob}(\Omega)}$. @@ -789,7 +805,7 @@ E_{-1} \pi_\infty(\omega) = \lambda \cdot 1 + (1-\lambda) \cdot 0 = \lambda $$ -Combining this equation with equation (20), we deduce that +Combining this equation with equation {eq}`eq:expect_pi_infty`, we deduce that the probability that ${\textrm{Prob}(\Omega)}$ attaches to $\pi_\infty(\omega)$ being $1$ must be $\pi_{-1}$. @@ -837,7 +853,9 @@ def martingale_simulate(π0, N=5000, T=200): def fraction_0_1(π0, N, T, decimals): π_path, w_path = martingale_simulate(π0, N=N, T=T) - values, counts = np.unique(np.round(π_path[:,-1], decimals=decimals), return_counts=True) + values, counts = np.unique( + np.round(π_path[:,-1], decimals=decimals), + return_counts=True) return values, counts def create_table(π0s, N=10000, T=500, decimals=2): @@ -861,16 +879,16 @@ T = 200 ```{code-cell} ipython3 fig, ax = plt.subplots() for i in range(100): - ax.plot(range(T+1), π_path[i, :]) + ax.plot(range(T+1), π_path[i, :], lw=2) -ax.set_xlabel('$t$') +ax.set_xlabel('time') ax.set_ylabel(r'$\pi_t$') plt.show() ``` The above graph indicates that -* each of paths converges +* each of the paths converges * some of the paths converge to $1$ @@ -895,7 +913,7 @@ Evidently, by $t = 199$, $\pi_t$ has converged to either $0$ or $1$. The fraction of paths that have converged to $1$ is $.5$ -The fractions of paths that have converged to $0$ is also $.5$. +The fraction of paths that have converged to $0$ is also $.5$. Does the fraction $.5$ ring a bell? @@ -930,23 +948,25 @@ $w_t$'s and the $\pi_t$ sequences that gave rise to them. Notice that one of the paths involves systematically higher $w_t$'s, outcomes that push $\pi_t$ upward. The luck of the draw early in a simulation pushes the subjective distribution to draw from -$F$ more frequently along a sample path, and this pushes $\pi_t$ toward $0$. +$F$ more frequently along a sample path, and this pushes $\pi_t$ toward $1$. ```{code-cell} ipython3 fig, ax = plt.subplots() for i, j in enumerate([10, 100]): - ax.plot(range(T+1), π_path[j,:], color=colors[i], label=fr'$\pi$_path, {j}-th simulation') - ax.plot(range(1,T+1), w_path[j,:], color=colors[i], label=fr'$w$_path, {j}-th simulation', alpha=0.3) + ax.plot(range(T+1), π_path[j,:], color=colors[i], + label=fr'${{\pi_t}}$, {j}-th simulation', lw=2) + ax.plot(range(1,T+1), w_path[j,:], color=colors[i], + label=fr'${{w_t}}$, {j}-th simulation', alpha=0.3, lw=2) ax.legend(loc='upper right') -ax.set_xlabel('$t$') +ax.set_xlabel('time') ax.set_ylabel(r'$\pi_t$') ax2 = ax.twinx() -ax2.set_ylabel("$w_t$") +ax2.set_ylabel(r"$w_t$") plt.show() ``` -## Initial Prior is Verified by Paths Drawn from Subjective Conditional Densities +## Initial prior is verified by paths drawn from subjective conditional densities @@ -962,7 +982,7 @@ The second column reports the fraction of $N = 10000$ simulations for which $\pi The third column reports the fraction of $N = 10000$ simulations for which $\pi_{t}$ had converged to $1$ at the terminal date $T=500$ for each simulation. ```{code-cell} ipython3 -# create table +# Create table table = create_table(list(np.linspace(0,1,11)), N=10000, T=500) table ``` @@ -970,7 +990,7 @@ table The fraction of simulations for which $\pi_{t}$ had converged to $1$ is indeed always close to $\pi_{-1}$, as anticipated. -## Drilling Down a Little Bit +## Drilling down a little bit To understand how the local dynamics of $\pi_t$ behaves, it is enlightening to consult the variance of $\pi_{t}$ conditional on $\pi_{t-1}$. @@ -981,7 +1001,7 @@ $$ \Bigl[ \pi_{t-1} f(w) + (1-\pi_{t-1})g(w) \Bigr] d w $$ -We can use a Monte Carlo simulation to approximate this conditional variance. +We can use a Monte Carlo simulation to approximate this conditional variance. We approximate it for a grid of points $\pi_{t-1} \in [0,1]$. @@ -989,27 +1009,27 @@ Then we'll plot it. ```{code-cell} ipython3 @jit -def compute_cond_var(pi, mc_size=int(1e6)): - # create monte carlo draws +def compute_cond_var(π, mc_size=int(1e6)): + # Create Monte Carlo draws mc_draws = np.zeros(mc_size) for i in prange(mc_size): - if np.random.rand() <= pi: + if np.random.rand() <= π: mc_draws[i] = np.random.beta(F_a, F_b) else: mc_draws[i] = np.random.beta(G_a, G_b) - dev = pi*f(mc_draws)/(pi*f(mc_draws) + (1-pi)*g(mc_draws)) - pi + dev = π*f(mc_draws)/(π*f(mc_draws) + (1-π)*g(mc_draws)) - π return np.mean(dev**2) -pi_array = np.linspace(0, 1, 40) +π_array = np.linspace(0, 1, 40) cond_var_array = [] -for pi in pi_array: - cond_var_array.append(compute_cond_var(pi)) +for π in π_array: + cond_var_array.append(compute_cond_var(π)) fig, ax = plt.subplots() -ax.plot(pi_array, cond_var_array) +ax.plot(π_array, cond_var_array, lw=2) ax.set_xlabel(r'$\pi_{t-1}$') ax.set_ylabel(r'$\sigma^{2}(\pi_{t}\vert \pi_{t-1})$') plt.show() @@ -1021,7 +1041,7 @@ Notice how the conditional variance approaches $0$ for $\pi_{t-1}$ near either The conditional variance is nearly zero only when the agent is almost sure that $w_t$ is drawn from $F$, or is almost sure it is drawn from $G$. -## Related Lectures +## Related lectures This lecture has been devoted to building some useful infrastructure that will help us understand inferences that are the foundations of -results described in {doc}`this lecture ` and {doc}`this lecture ` and {doc}`this lecture `. +results described in {doc}`odu`, {doc}`wald_friedman`, and {doc}`navy_captain`. diff --git a/lectures/likelihood_ratio_process.md b/lectures/likelihood_ratio_process.md index 26c315dbb..06537cda6 100644 --- a/lectures/likelihood_ratio_process.md +++ b/lectures/likelihood_ratio_process.md @@ -73,7 +73,7 @@ $f$ or $g$. Before the beginning of time, nature once and for all decides whether she will draw a sequence of IID draws from either $f$ or $g$. -We will sometimes let $q$ be the density that nature chose once and for all, so +We let $q$ be the density that nature chose once and for all, so that $q$ is either $f$ or $g$, permanently. Nature knows which density it permanently draws from, but we the observers do not. diff --git a/lectures/likelihood_ratio_process_2.md b/lectures/likelihood_ratio_process_2.md index 6eb9a8c8c..2851fd4cb 100644 --- a/lectures/likelihood_ratio_process_2.md +++ b/lectures/likelihood_ratio_process_2.md @@ -75,7 +75,7 @@ $f$ or $g$. Before the beginning of time, nature once and for all decides whether she will draw a sequence of IID draws from either $f$ or $g$. -We will sometimes let $q$ be the density that nature chose once and for all, so +We let $q$ be the density that nature chose once and for all, so that $q$ is either $f$ or $g$, permanently. Nature knows which density it permanently draws from, but we the observers do not.