# Statistical Inference

Suppose we have a variable $$X$$ (may or may not be a random variable) that represents the state of nature. We observe a variable $$Y$$ which is obtained by some model of the world $$P_{Y|X}$$.

![Figure 2: Inference Setup](https://3896527066-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F-MQYqMkpf2WPiP6MKWRe%2Fuploads%2Fgit-blob-e93342754737cd58776c6c399bee170148535542%2Fc5e8e0b02debea1574c6bd75bf14879564123188.png?alt=media)

Suppose we know that $$X\sim \pi$$ where $$\pi$$ is a probability distribution. If we observe $$Y=y$$, then the _a posteriori_ estimate of $$X$$ is given by Bayes Rule

$$\text{Pr}\left\{X=x | Y=y\right\} = \frac{P_{Y|X}(y|x)\pi(x)}{\sum_{\tilde{x}}P_{Y|X}(y|\tilde{x})\pi(\tilde{x})} \propto P_{Y|X}(y|x)\pi(x).$$

Since the estimate is only dependent on the model and the prior, we don’t actually need to compute the probabilities to figure out the most likely $$X$$.

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#### Definition 77

The Maximum A Posteriori (MAP) estimate is given by

$$\hat{X}_{MAP}(y) = \text{argmax}_x P_{Y|X}(y|x)\pi(x)$$
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If we have no prior information on $$X$$, then we can assume $$\pi$$ is uniform, reducing Definition 77 to only optimize over the model.

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#### Definition 78

The Maximum Likelihood (ML) estimate is given by

$$\hat{X}_{ML}(y) = \text{argmax}_x P_{Y|X}(y|x)$$
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## Binary Hypothesis Testing

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#### Definition 79

A Binary Hypothesis Test is a type of statistical inference where the unknown variable $$X\in\{ 0, 1 \}$$.
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Since there are only two possible values of $$X$$ in a binary test, there are two “hypotheses” that we have, and we want to accept the more likely one.

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#### Definition 80

The Null Hypothesis $$H_0$$ says that $$Y\sim P_{Y|X=0}$$
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#### Definition 81

The Alternate Hypothesis $$H_1$$ says that $$Y\sim P_{Y|X=1}$$
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With two possible hypotheses, there are two kinds of errors we can make.

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#### Definition 82

A Type I error (false positive) is when we incorrectly reject the null hypothesis. The Type I error probability is then

$$\text{Pr}\left\{\hat{X}(Y) = 1 | X = 0\right\}$$
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#### Definition 83

A Type II error (false negative) is when we incorrectly accept the null hypothesis. The Type II error probability is then

$$\text{Pr}\left\{\hat{X}(Y) = 0 | X = 1\right\}$$
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Our goal is to create a decision rule $$\hat{X}: \mathcal{Y} \to \{0, 1\}$$ that we can use to predict $$X$$. Based on what the decision rule is used for, there will be requirements on how large the probability of Type I and Type II errors can be. We can formulate the search for a hypothesis test as an optimization. For some $$\beta \in [0, 1]$$, we want to find

$$\hat{X}_\beta(Y) = \text{argmin} \text{Pr}\left\{\hat{X}(Y)=0 | X=1\right\} \quad : \quad \text{Pr}\left\{\hat{X}(Y)=1|X=0\right\} \leq \beta. \qquad (1)$$

Intuitively, our test should depend on $$p_{Y|X}(y|1)$$ and $$p_{Y|X}(y|0)$$ since these quantities give us how likely we are to get our observations if we knew the ground truth. We can define a ratio that formally compares these two quantities.

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#### Definition 84

The likelihood ratio is given by

$$L(y) = \frac{p_{Y|X}(y|1)}{p_{Y|X}(y|0)}$$
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Notice that we can write MLE as a threshold on the likelihood ratio since if $$L(y) \geq 1$$, then we say $$X=1$$, and vice versa. However, there is no particular reason that $$1$$ must always be the number at which we threshold our likelihood ratio, and so we can generalize this idea to form different forms of tests.

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#### Definition 85

For some threshold $$c$$ and randomization probability $$\gamma$$, a threshold test is of the form

$$\hat{X}(y) = \begin{cases} 1 & \text{ if } L(y) > c\\ 0 & \text{ if } L(y) < c\\ \text{ Bernoulli}(\gamma) & \text { if } L(y) = c. \end{cases}$$
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MAP fits into the framework of a threshold test since we can write

$$\hat{X}_{MAP} = \begin{cases} 1 & \text{ if } L(y) \geq \frac{\pi_0}{\pi_1}\\ 0 & \text{ if } L(y) < \frac{\pi_0}{\pi_1} \end{cases}$$

It turns out that threshold tests are optimal with respect to solving Equation 1.

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#### Theorem 44 (Neyman Pearson Lemma) <a href="#theorem-44" id="theorem-44"></a>

Given $$\beta\in[0, 1]$$, the optimal decision rule to

$$\hat{X}_\beta(Y) = \text{argmin} \text{Pr}\left\{\hat{X}(Y)=0 | X=1\right\} \quad : \quad \text{Pr}\left\{\hat{X}(Y)=1|X=0\right\} \leq \beta$$

is a threshold test.
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When $$L(y)$$ is monotonically increasing or decreasing, we can make the decision rule even simpler since it can be turned into a threshold on $$y$$. For example, if $$L(y)$$ is monotonically inreasing, then an optimal decision rule might be

$$\hat{X}(y) = \begin{cases} 1 & \text{ if } y > c\\ 0 & \text{ if } y < c\\ \text{Bernoulli}(\gamma) & \text{ if } y = c. \end{cases}$$
