intro-bayes.qmd

# Introduction to Bayesian statistical analysis

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{{< include shared-config.qmd >}}

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\usepackage{MnSymbol}

Suppose $X_1,...,X_n \siid N(M,1)$

Suppose $M \sim N(0, 1)$.

Then:
$$
\ba
p(M=\mu|X=x) 
&\propto p(M=\mu, X=x)\\
&= p(X=x|M=\mu) p(M=\mu)\\
&\propto \exp{-\frac{1}{2}n\mu^2 - 2\mu n\bar{x}}
\exp{-\frac{1}{2} \mu^2}\\
&= \exp{-\frac{1}{2}(n+1)\mu^2 - 2\mu n\bar{x}}\\
&\propto \exp{-\frac{1}{2}(n+1)(\mu -  \frac{n}{n+1}\bar{x})^2}
\ea
$$
So: 
$$p(M=\mu|X=x) \sim N(\frac{n}{n+1}\bar{x}, (n+1)^{-1})$$

Let's put this in perspective.

Here's a frequentist CI:

```{r}
set.seed(1)
mu = 2
sigma = 1
n = 20
x = rnorm(n = n, mean = mu, sd = sigma)
xbar = mean(x)
se = sigma/sqrt(n)
CI_freq = xbar + se * qnorm(c(.025, .975))
print(CI_freq)
```

```{r}
lik0 = function(mu) prod(dnorm(x = x , mean = mu, sd = 1))
lik = function(mu) 
  (2*pi*sigma^2)^(-n/2) * 
  exp(
    -1/(2*sigma^2) * 
      (sum(x^2) -2*mu *sum(x) + n *(mu^2)))
# llik2 = function(mu) dnorm(xbar, mean = mu, sd = se)
library(ggplot2)
ngraph = 1001
plot1 = ggplot() +
  geom_function(fun = lik, aes(col = "likelihood"), n = ngraph) +
  xlim(c(-5,10)) +
  theme_bw() +
  labs(col = "") +
  theme(legend.position = "bottom")
print(plot1)
```


Here's a Bayesian CI:

```{r}
mu_prior_mean = 0
mu_prior_sd = 1
mu_post_mean = n/(n+1) * xbar
mu_post_var = 1/(n+1)
mu_post_sd = sqrt(mu_post_var)
CI_bayes = qnorm(
  p = c(.025, .975),
  mean = mu_post_mean,
  sd = mu_post_sd
)
print(CI_bayes)
prior = function(mu) dnorm(mu, mean = mu_prior_mean, sd = mu_prior_sd)
posterior = function(mu) dnorm(mu, mean = mu_post_mean, sd = mu_post_sd)
plot2 = plot1 +
  geom_function(fun = prior, aes(col = "prior"), n = ngraph) +
  geom_function(fun = posterior, aes(col = "posterior"), n = ngraph) 
print(plot2 + scale_y_log10())

```

Here's $p(M \in (l(x),r(x))|X=x)$:

```{r}
pr_in_CI = pnorm(
  CI_freq,
  mean = mu_post_mean,
  sd = mu_post_sd
) |> diff()
print(pr_in_CI)
```

## Other resources

UC Davis courses
- STA 145: Bayesian Statistical Inference
- ECL 298: Bayesian Models - A Statistical Primer

Books
- @rossbayes is a free online textbook