## Likelihood Ratio Process 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

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

Let $\pi_{t+1}$ be a Bayesian posterior probability defined as

$$

{\rm Prob}(w^{t+1} |\emptyset) = \pi_0 f(w^{t+1})+ (1 - \pi_0)

{\rm Prob}(w^{t+1} |\emptyset) = \pi_0 f(w^{t+1})+ (1 - \pi_0) g(w^{t+1})

$$

Laws of probability say that the joint distribution ${\rm Prob}(AB)$ of events $A$ and $B$ are connected to the conditional distributions

${\rm Prob}(A |B)$ and ${\rm Prob}(B |A)$ by

$$

{\rm Prob}(AB) = {\rm Prob}(A |B) {\rm Prob}(B) = {\rm Prob}(B |A) {\rm Prob}(A) .

$$ (eq:problawAB)

We are interested in events

$$

A = \{q=f\}, \quad B = \{w^{t+1}\}, \quad

$$

Probability laws connecting joint probability distributions and conditional probability distributions imply that

where braces $\{\cdot\}$ are our shorthand for "event".

So in our setting, probability laws {eq}`eq:problawAB` imply that

$$

{\rm Prob}(q=f |w^{t+1}) {\rm Prob}(w^{t+1} |\emptyset) = {\rm Prob}(w^{t+1} |q=f) {\rm Prob}(q=f | \emptyset)

$$

```{math}

:label: eq_recur1

\pi_{t+1}=\frac{\pi_{t} l_t(w_{t+1})}{\pi_{t} l_t(w_t)+1-\pi_{t}}

\pi_{t+1}=\frac{\pi_{t} l_t(w_{t+1})}{\pi_{t} l_t(w_{t+1})+1-\pi_{t}}

```

with $\pi_{0}$ being a Bayesian prior probability that $q = f$,

```

We thus conclude that the likelihood ratio process is a key ingredient of the formula {eq}`eq_Bayeslaw1033` for

a Bayesian's posterior probabilty that nature has drawn history $w^t$ as repeated draws from density

a Bayesian's posterior probability that nature has drawn history $w^t$ as repeated draws from density

$f$.

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 flips an $x$-weighted coin and with probability

$x \in (0,1)$ draws from $f$ in period $t$ and with probability $1 - x $ draws from $g$.

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.

We will first use this sequence to study how $\pi_t$ behaves.

```{note}

Later, we can use it to study how a statistician who knows that an $x$-mixture of $f$ and $g$ could construct maximum likelihood or Bayesian estimators of $x$ along with the free parameters of $f$ and $g$.

Later, we can use it to study how a statistician who knows that nature generates data from an $x$-mixture of $f$ and $g$ could construct maximum likelihood or Bayesian estimators of $x$ along with the free parameters of $f$ and $g$.

```

```{code-cell} ipython3

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 $t \to \infty$.

{doc}`likelihood_ratio_process`, $\log(L_t) \to \infty$ as $t \to \infty$.

Now looking back to the key equation {eq}`eq_Bayeslaw1033`.

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 assumes 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.

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

the posterior probability $\pi_0 = {\rm Prob} ({a = f | w_0}) $

the posterior probability $\pi_0 = {\rm Prob}({a = f | w_0}) $

that the density is $f(w)$ is

$$

Practically, this means that probability one is attached to sample paths

$\{\pi_t\}_{t=0}^\infty$ that converge.

According to the theorem, it different sample paths can converge to different limiting values.

According to the theorem, different sample paths can converge to different limiting values.

Thus, let $\{\pi_t(\omega)\}_{t=0}^\infty$ denote a particular sample path indexed by a particular $\omega

\in \Omega$.

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 push the subjective distribution to draw from

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 $0$.

```{code-cell} ipython3

The second column reports the fraction of $N = 10000$ simulations for which $\pi_{t}$ had converged to $0$ at the terminal date $T=500$ for each simulation.

The third column reports the fraction of $N = 10000$ simulations for which $\pi_{t}$ had converged to $1$ as the terminal date $T=500$ for each simulation.

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

plt.show()

```

The shape of the the conditional variance as a function of $\pi_{t-1}$ is informative about the behavior of sample paths of $\{\pi_t\}$.

The shape of the conditional variance as a function of $\pi_{t-1}$ is informative about the behavior of sample paths of $\{\pi_t\}$.

Notice how the conditional variance approaches $0$ for $\pi_{t-1}$ near either $0$ or $1$.