In [1]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
np.random.seed(1789)
from IPython.core.display import HTML
def css_styling():
styles = open("styles/custom.css", "r").read()
return HTML(styles)
css_styling()
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Though many of you will have taken a statistics course or two during your undergraduate (or graduate education, most of those who have will likely not have had a course in Bayesian statistics. Most introductory courses, particularly for non-statisticians, still do not cover Bayesian methods at all. Even today, Bayesian courses (similarly to statistical computing courses!) are typically tacked onto the curriculum, rather than being integrated into the program.
In fact, Bayesian statistics is not just a particular method, or even a class of methods; it is an entirely different paradigm for doing statistical analysis.
Practical methods for making inferences from data using probability models for quantities we observe and about which we wish to learn. -- Gelman et al. 2013
A Bayesian model is described by parameters, uncertainty in those parameters is described using probability distributions.
All conclusions from Bayesian statistical procedures are stated in terms of probability statements
This confers several benefits to the analyst, including:
Any statistical paradigm, Bayesian or otherwise, involves at least the following:
In mathematical notation, this implies a (very) general model of the following form:
Here, the model $f$ accepts data values $y$ as an argument, conditional on particular values of $\theta$.
Frequentist inference typically involves deriving estimators for the unknown parameters. Estimators are formulae that return estimates for particular estimands, as a function of data. They are selected based on some chosen optimality criterion, such as unbiasedness, variance minimization, or efficiency.
For example, lets say that we have collected some data on the prevalence of autism spectrum disorder (ASD) in some defined population. Our sample includes $n$ sampled children, $y$ of them having been diagnosed with autism. A frequentist estimator of the prevalence $p$ is:
\\[\hat{p} = \frac{y}{n}\\]Why this particular function? Because it can be shown to be unbiased and minimum-variance.
It is important to note that, in a frequentist world, new estimators need to be derived for every estimand that is introduced.
This implies the following form:
This formulation used to be referred to as inverse probability, because it infers from observations to parameters, or from effects to causes.
Bayesians do not seek new estimators for every estimation problem they encounter. There is only one estimator for Bayesian inference: Bayes' Formula.
Gelman et al. (2013) describe the process of conducting Bayesian statistical analysis in 3 steps.
As was noted above, Bayesian statistics involves using probability models to solve problems. So, the first task is to completely specify the model in terms of probability distributions. This includes everything: unknown parameters, data, covariates, missing data, predictions. All must be assigned some probability density.
This step involves making choices.
The mathematical form $p(\theta | y)$ that we associated with the Bayesian approach is referred to as a posterior distribution.
posterior /pos·ter·i·or/ (pos-tēr´e-er) later in time; subsequent.
Why posterior? Because it tells us what we know about the unknown $\theta$ after having observed $y$.
This posterior distribution is formulated as a function of the probability model that was specified in Step 1. Usually, we can write it down but we cannot calculate it analytically. In fact, the difficulty inherent in calculating the posterior distribution for most models of interest is perhaps the major contributing factor for the lack of widespread adoption of Bayesian methods for data analysis. Various strategies for doing so comprise this tutorial.
But, once the posterior distribution is calculated, you get a lot for free:
Though frequently ignored in practice, it is critical that the model and its outputs be assessed before using the outputs for inference. Models are specified based on assumptions that are largely unverifiable, so the least we can do is examine the output in detail, relative to the specified model and the data that were used to fit the model.
Specifically, we must ask:
Misunderstanding of probability may be the greatest of all impediments to scientific literacy. — Stephen Jay Gould
Because of its reliance on probabilty models, its worth talking a little bit about probability. There are different ways to define probability, depending on how it is being used. In fact, Bayesian statistics invokes an additional definition of probability that is not used elsewhere.
Classical probability is an assessment of possible outcomes of elementary events. Elementary events are assumed to be equally likely.
Unlike classical probability, frequentist probability is an EMPIRICAL definition. It is an objective statement desribing events that have occurred.
Subjective probability is a measure of one's uncertainty in the value of $X$. It characterizes the state of knowledge regarding some unknown quantity using probability.
It is not associated with long-term frequencies nor with equal-probability events.
For example:
Now that we have some probability under our belt, we turn to Bayes' formula. While frequentist statistics uses different estimators for different problems, Bayes formula is the only estimator that Bayesians need to obtain estimates of unknown quantities that we care about. It turns out to be straightforward to derive Bayes' formula directly from the definition of conditional probability.
Recall that the goal in Bayesian inference is to calculate the posterior distribution of our unknowns:
This expression is a conditional probability. It is the probability of $\theta$ given the observed values of $Y=y$.
In general, the conditional probability of A given B is defined as follows:
$$Pr(B|A) = \frac{Pr(A \cap B)}{Pr(A)}$$To gain an intuition for this, it is helpful to use a Venn diagram:
Notice from this diagram that the following conditional probability is also true:
$$Pr(A|B) = \frac{Pr(A \cap B)}{Pr(B)}$$These can both be rearranged to be expressions of the joint probability of A and B. Setting these equal to one another:
$$Pr(B|A)Pr(A) = Pr(A|B)Pr(B)$$Then rearranging:
$$Pr(B|A) = \frac{Pr(A|B)Pr(B)}{Pr(A)}$$This is Bayes' formula. Replacing the generic A and B with things we care about reveals why Bayes' formula is so important:
The equation expresses how our belief about the value of $\theta$, as expressed by the prior distribution $P(\theta)$ is reallocated following the observation of the data $y$.
The innocuous denominator $P(y)$ usuallt cannot be computed directly, and is actually the expression in the numerator, integrated over all $\theta$:
The intractability of this integral is one of the factors that has contributed to the under-utilization of Bayesian methods by statisticians.
Once considered a controversial aspect of Bayesian analysis, the prior distribution characterizes what is known about an unknown quantity before observing the data from the present study. Thus, it represents the information state of that parameter. It can be used to reflect the information obtained in previous studies, to constrain the parameter to plausible values, or to represent the population of possible parameter values, of which the current study's parameter value can be considered a sample.
The likelihood represents the information in the observed data, and is used to update prior distributions to posterior distributions. This updating of belief is justified becuase of the likelihood principle, which states:
Following observation of $y$, the likelihood $L(\theta|y)$ contains all experimental information from $y$ about the unknown $\theta$.
Bayesian analysis satisfies the likelihood principle because the posterior distribution's dependence on the data is only through the likelihood. In comparison, most frequentist inference procedures violate the likelihood principle, because inference will depend on the design of the trial or experiment.
Remember from the density estimation section that the likelihood is closely related to the probability density (or mass) function. The difference is that the likelihood varies the parameter while holding the observations constant, rather than vice versa.
In [2]:
from scipy.stats import binom
# Binomial probability mass function
yvals = range(10+1)
plt.plot(yvals, binom.pmf(yvals, 10, 0.5), 'ro')
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In [3]:
# Binomial likelhood function
pvals = np.linspace(0, 1)
y = 4
plt.plot(pvals, binom.pmf(y, 10, pvals));
Bayesian analysis usually requires integration over multiple dimensions that is intractable both via analytic methods or via standard methods of numerical integration.
Following the steps for setting up a Bayesian model, after observing data $y$ that we hypothesize as having being obtained from a sampling model $f(y|\theta)$, we then place a prior distribution $p(\theta)$ on the parameters to describe the uncertainty in their true values. We then obtain inference by calculating the posterior distribution, which is proportional to the product of these quantities:
$$p(\theta | y) \propto f(y|\theta) p(\theta)$$unfortunately, for most problems of interest, the normalizing constant cannot be calculated because it involves multi-dimensional integration over $\theta$.
One approach is to work with non-normalized posteriors, and obtain approximate inference.
Tsutakawa et al. (1985) provides mortality data for stomach cancer among men aged 45-64 in several cities in Missouri. The file cancer.csv contains deaths $y_i$ and subjects at risk $n_i$ for 20 cities from this dataset.
In [4]:
cancer = pd.read_csv('../data/cancer.csv')
cancer
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If we use a simple binomial model, which assumes independent samples from a binomial distribution with probability of mortality $p$, we can use MLE to obtain an estimate of this probability.
$$L(p \mid n, y) = \frac{n!}{y!(n-y)!} p^y (1-p)^{n-y}$$$$\hat{p} = \frac{y}{n}$$
In [5]:
ytotal, ntotal = cancer.sum().astype(float)
p_hat = ytotal/ntotal
p_hat
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However, if we compare the variation of $y$ under this model, it is to small relative to the observed variation:
In [6]:
cancer.y.var()
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In [7]:
p_hat*(1.-p_hat)*ntotal
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Hence, the data are strongly overdispersed relative to what is predicted under a model with a fixed probability of death. A more realistic model would allow for these probabilities to vary among the cities.
One way of representing this is conjugating the binomial distribution with another distribution that describes the variation in the binomial probability. A sensible choice for this is the beta distribution:
$$f(p \mid \alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) \Gamma(\beta)} p^{\alpha - 1} (1 - p)^{\beta - 1}$$Conjugating this with the binomial distribution, and reparameterizing such that $\alpha = K\eta$ and $\beta = K(1-\eta)$ for $K > 0$ and $\eta \in (0,1)$ results in the beta-binomial distribution:
$$f(y \mid K, \eta) = \frac{n!}{y!(n-y)!} \frac{B(K\eta+y, K(1-\eta) + n - y)}{B(K\eta, K(1-\eta))}$$where $B$ is the beta function.
What remains is to place priors over the parameters $K$ and $\eta$. Common choices for diffuse (i.e. vague or uninformative) priors are:
$$\begin{aligned} p(K) &\propto \frac{1}{(1+K)^2} \cr p(\eta) &\propto \frac{1}{\eta(1-\eta)} \end{aligned}$$These are not normalized, but our posterior will not be normalized anyhow, so this is not an issue.
In [8]:
K_x = np.linspace(0, 10)
K_prior = lambda K: 1./(1. + K)**2
plt.plot(K_x, K_prior(K_x))
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In [9]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
eta_x = np.linspace(0, 1)
eta_prior = lambda eta: 1./(eta*(1.-eta))
plt.plot(eta_x, eta_prior(eta_x))
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Now, by multiplying these quantities together, we can obtain a non-normalized posterior.
$$p(K, \eta | \mathbf{y}) \propto \frac{1}{(1+K)^2} \frac{1}{\eta(1-\eta)} \prod_i \frac{B(K\eta+y_i, K(1-\eta) + n_i - y_i)}{B(K\eta, K(1-\eta))}$$This can be calculated in Python as follows (log-transformed):
In [10]:
from scipy.special import betaln
def betabin_post(params, n, y):
K, eta = params
post = betaln(K*eta + y, K*(1.-eta) + n - y).sum()
post -= len(y)*betaln(K*eta, K*(1.-eta))
post -= np.log(eta*(1.-eta))
post -= 2.*np.log(1.+K)
return post
betabin_post((15000, 0.003), cancer.n, cancer.y)
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An easy (though computationally expensive) way of getting the joint posterior distribution of the parameters is to evaluate betabin_post on a grid of parameter values.
In [11]:
# Create grid
K_x = np.linspace(1, 20000)
eta_x = np.linspace(0.0001, 0.003)
# Calculate posterior on grid
z = np.array([[betabin_post((K, eta), cancer.n, cancer.y)
for eta in eta_x] for K in K_x])
# Plot posterior
x, y = np.meshgrid(eta_x, K_x)
cplot = plt.contour(x, y, z-z.max(), [-0.5, -1, -2, -3, -4], cmap=plt.cm.RdBu)
plt.ylabel('K');plt.xlabel('$\eta$');
This is fine, but the precision parameter $K$ is heavily skewed.
To deal with the skewness in $K$ and to facilitate modeling, we can transform the beta-binomial parameters to the real line via:
$$\begin{aligned} \theta_1 &= \log(K) \cr \theta_2 &= \log\left(\frac{\eta}{1-\eta}\right) \end{aligned}$$which we can easily implement by wrapping betabin_post:
In [12]:
def betabin_trans(theta, n, y):
K = np.exp(theta[0])
eta = 1./(1. + np.exp(-theta[1]))
# Jacobians for transformation
J = theta[0] + theta[1]
return betabin_post((K, eta), n, y) + J
betabin_trans((10, -7.5), cancer.n, cancer.y)
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In [13]:
# Create grid
log_K_x = np.linspace(0, 20)
logit_eta_x = np.linspace(-8, -5)
# Calculate posterior on grid
z = np.array([[betabin_trans((t1, t2), cancer.n, cancer.y)
for t2 in logit_eta_x] for t1 in log_K_x])
# Plot posterior
x, y = np.meshgrid(logit_eta_x, log_K_x)
cplot = plt.contour(x, y, z - z.max(), [-0.5, -1, -2, -4, -8], cmap=plt.cm.RdBu)
plt.clabel(cplot, inline=1, fontsize=10, fmt='%1.1f')
plt.ylabel('log(K)');plt.xlabel('logit($\eta$)');
An alternative approach to summarizing a $p$-dimensional posterior distribution involves estimating the mode of the posterior, and approximating the density as having a multivariate normal distribution.
$$p(x \mid \theta, \Sigma) = (2\pi |\Sigma|)^{-1/2} \exp\left\{ -\frac{1}{2} (x-\theta)^{\prime}\Sigma^{-1}(x-\theta) \right\}$$which, when log-transformed, becomes:
$$l(x \mid \theta, \Sigma) \propto - \frac{1}{2} \log|\Sigma| -\frac{1}{2} (x-\theta)^{\prime}\Sigma^{-1}(x-\theta)$$If we consider the logarithm of the unnormalized joint posterior:
$$h(\theta | y) = \log[f(y|\theta) p(\theta)]$$one way to approximate this function is to use a second-order Taylor series expansion around the mode $\hat{\theta}$:
$$h(\theta | y) \approx h(\hat{\theta} | y) + \frac{1}{2}(\theta-\hat{\theta})' h''(\hat{\theta} | y) (\theta-\hat{\theta})$$This form is simply the multivariate normal distribution with $\hat{\theta}$ as the mean and the inverse negative Hessian as the covariance matrix:
$$\Sigma = -h''(\hat{\theta} | y)^{-1}$$We can apply one of several numerical methods for multivariate optimization to numerically estimate the mode of the posterior. Here, we will use the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm that is provided by SciPy's fmin_bfgs function. In addition to returning an estimate of the mode, it returns the estimated variance-covariance matrix, which we will need to parameterize the multivariate normal distribution.
Applying this to the beta-binomial posterior estimation problem, we simply provide an initial guess for the mode:
In [14]:
from scipy.optimize import fmin_bfgs
betabin_trans_min = lambda *args: -betabin_trans(*args)
init_value = (10, -7.5)
opt = fmin_bfgs(betabin_trans_min, init_value,
args=(cancer.n, cancer.y), full_output=True)
mode, var = opt[0], opt[3]
mode, var
Out[14]:
Thus, our approximated mode is $\log(K)=7.6$, $\text{logit}(\eta)=-6.8$. We can plug this value, along with the variance-covariance matrix, into a function that returns the kernel of a multivariate normal distribution, and use this to plot the approximate posterior:
In [15]:
det = np.linalg.det
inv = np.linalg.inv
def lmvn(value, mu, Sigma):
# Log kernel of multivariate normal
delta = np.array(value) - mu
return 1 / (2. * (np.log(det(Sigma))) - np.dot(delta.T, np.dot(inv(Sigma), delta)))
In [16]:
z = np.array([[lmvn((t1, t2), mode, var)
for t2 in logit_eta_x] for t1 in log_K_x])
x, y = np.meshgrid(logit_eta_x, log_K_x)
cplot = plt.contour(x, y, z - z.max(), cmap=plt.cm.RdBu)
plt.ylabel('log(K)');plt.xlabel('logit($\eta$)');
Along with this, we can estimate a 95% probability interval for the estimated mode:
In [17]:
from scipy.stats.distributions import norm
se = np.sqrt(np.diag(var))
mode[0] + norm.ppf(0.025)*se[0], mode[0] + norm.ppf(0.975)*se[0]
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In [18]:
mode[1] + norm.ppf(0.025)*se[1], mode[1] + norm.ppf(0.975)*se[1]
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Of course, this approximation is only reasonable for posteriors that are not strongly skewed, bimodal, or leptokurtic (heavy-tailed).
It is often possible to compute integrals required by Bayesian analysis by simulating (drawing samples) from posterior distributions. For example, consider the expected value of a random variable $\mathbf{x}$:
$$\begin{gathered} \begin{split}E[{\bf x}] = \int {\bf x} f({\bf x}) d{\bf x}, \qquad {\bf x} = \{x_1,...,x_k\}\end{split}\notag\\\begin{split}\end{split}\notag\end{gathered}$$where $k$ (the dimension of vector $x$) is perhaps very large. If we can produce a reasonable number of random vectors $\{{\bf x_i}\}$, we can use these values to approximate the unknown integral. This process is known as Monte Carlo integration. In general, MC integration allows integrals over probability density functions ...
$$\begin{gathered} \begin{split}I = \int h(\mathbf{x}) f(\mathbf{x}) \mathbf{dx}\end{split}\notag\\\begin{split}\end{split}\notag\end{gathered}$$... to be estimated by finite sums:
$$\begin{gathered} \begin{split}\hat{I} = \frac{1}{n}\sum_{i=1}^n h(\mathbf{x}_i),\end{split}\notag\\\begin{split}\end{split}\notag\end{gathered}$$where $\mathbf{x}_i$ is a sample from $f$. This estimate is valid and useful because:
In the integral $I$ above, if we replace $f(\mathbf{x})$ with a posterior, $p(\theta|y)$ and make $h(\theta)$ an interesting function of the unknown parameter, the resulting expectation is that of the posterior of $h(\theta)$:
$$E[h(\theta)|y] = \int h(\theta) p(\theta|y) d\theta \approx \frac{1}{n}\sum_{i=1}^n h(\theta)$$We also require integrals to obtain marginal estimates from a joint model. If $\theta$ is of length $K$, then inference about any particular parameter is obtained by:
$$p(\theta_i|y) \propto \int p(\theta|y) d\theta_{-i}$$where the -i subscript indicates all elements except the $i^{th}$.
Though Monte Carlo integration allows us to estimate integrals that are unassailable by analysis and standard numerical methods, it relies on the ability to draw samples from the posterior distribution. For known parametric forms, this is not a problem; probability integral transforms or bivariate techniques (e.g Box-Muller method) may be used to obtain samples from uniform pseudo-random variates generated from a computer. Often, however, we cannot readily generate random values from non-standard posteriors. In such instances, we can use rejection sampling to generate samples.
Consider a function, $f(x)$ which can be evaluated for any value on the support of $x:S_x = [A,B]$, but may not be integrable or easily sampled from. If we can calculate the maximum value of $f(x)$, we can then define a rectangle that is guaranteed to contain all possible values $(x,f(x))$. It is then trivial to generate points over the box and enumerate the values that fall under the curve.
$$\begin{gathered} \begin{split}\frac{\mbox{Points under curve}}{\mbox{Points generated}} \times \mbox{box area} = \lim_{n \to \infty} \int_A^B f(x) dx\end{split}\notag\\\begin{split}\end{split}\notag\end{gathered}$$
In [19]:
def rtriangle(low, high, mode):
alpha = -1
# Run until accepted
while np.random.random() > alpha:
u = np.random.uniform(low, high)
if u < mode:
alpha = (u - low) / (mode - low)
else:
alpha = (high - u) / (high - mode)
return(u)
In [20]:
_ = plt.hist([rtriangle(0, 7, 2) for t in range(10000)], bins=100)
This approach is useful, for example, in estimating the normalizing constant for posterior distributions.
If $f(x)$ has unbounded support (i.e. infinite tails), such as a Gaussian distribution, a bounding box is no longer appropriate. We must specify a majorizing (or, enveloping) function, $g(x)$, which implies:
$$\begin{gathered} \begin{split}cg(x) \ge f(x) \qquad\forall x \in (-\infty,\infty)\end{split}\notag\\\begin{split}\end{split}\notag\end{gathered}$$Having done this, we can now sample ${x_i}$ from $g(x)$ and accept or reject each of these values based upon $f(x_i)$. Specifically, for each draw $x_i$, we also draw a uniform random variate $u_i$ and accept $x_i$ if $u_i < f(x_i)/cg(x_i)$, where $c$ is a constant. This procedure is repeated until a sufficient number of samples is obtained. This approach is made more efficient by choosing an enveloping distribution that is “close” to the target distribution, thus maximizing the number of accepted points.
To apply rejection sampling to the beta-binomial example, we first need to find a majorizing function $g(x)$ from which we can easily draw samples. We have seen in the previous section that the multivariate normal might serve as a suitable candidate, if multiplied by an appropriately large value of $c$. However, the thinness of the normal tails makes it difficult to use as a majorizing function. Instead, a multivariate Student's T distribution offers heavier tails for a suitably-small value for the degrees of freedom $\nu$:
$$f(\mathbf{x}| \nu,\mu,\Sigma) = \frac{\Gamma\left[(\nu+p)/2\right]}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}\left|{\boldsymbol\Sigma}\right|^{1/2}\left[1+\frac{1}{\nu}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})\right]^{(\nu+p)/2}}$$We can draw samples from a multivariate-T density by combining mutlivariate normal and $\chi^2$ random variates:
Generating multivariate-T samples
If $X$ is distributed multivariate normal $\text{MVN}(\mathbf{0},\Sigma)$ and $S$ is a $\chi^2$ random variable with $\mu$ degrees of freedom, then a multivariate Student's-T random variable $T = T_1,\ldots,T_p$ can be generated by $T_i = \frac{\sqrt{\nu}X_i}{S} + \mu_i$, where $\mu = \mu_1,\ldots,\mu$ is a mean vector.
This is implemented in Python by:
In [21]:
chi2 = np.random.chisquare
mvn = np.random.multivariate_normal
rmvt = lambda nu, S, mu=0, size=1: (np.sqrt(nu) * (mvn(np.zeros(len(S)), S, size).T
/ chi2(nu, size))).T + mu
Finally, we need an implementation of the multivariate T probability distribution function, which is as follows:
In [22]:
from scipy.special import gammaln
def mvt(x, nu, S, mu=0):
d = len(S)
n = len(x)
X = np.atleast_2d(x) - mu
Q = X.dot(np.linalg.inv(S)).dot(X.T).sum()
log_det = np.log(np.linalg.det(S))
log_pdf = gammaln((nu + d)/2.) - 0.5 * (d*np.log(np.pi*nu) + log_det) - gammaln(nu/2.)
log_pdf -= 0.5*(nu + d)*np.log(1 + Q/nu)
return(np.exp(log_pdf))
The next step is to find the constant $c$ that ensures:
$$cg(\theta) \ge f(\theta|y) \qquad\forall \theta \in (-\infty,\infty)$$Alternatively, we want to ensure:
$$\log[f(\theta|y)] - \log[g(\theta)] \le c'$$
In [23]:
def calc_diff(theta, n, y, nu, S, mu):
return betabin_trans(theta, n, y) - np.log(mvt(theta, nu, S, mu))
calc_diff_min = lambda *args: -calc_diff(*args)
We can calculate an appropriate value of $c'$ by simply using the approximation method described above on calc_diff (tweaked to produce a negative value for minimization):
In [24]:
opt = fmin_bfgs(calc_diff_min,
(12, -7),
args=(cancer.n, cancer.y, 4, 2*var, mode),
full_output=True)
In [25]:
c = opt[1]
c
Out[25]:
Now we can execute a rejection sampling algorithm:
In [26]:
def reject(post, nu, S, mu, n, data, c):
k = len(mode)
# Draw samples from g(theta)
theta = rmvt(nu, S, mu, size=n)
# Calculate probability under g(theta)
gvals = np.array([np.log(mvt(t, nu, S, mu)) for t in theta])
# Calculate probability under f(theta)
fvals = np.array([post(t, data.n, data.y) for t in theta])
# Calculate acceptance probability
p = np.exp(fvals - gvals + c)
return theta[np.random.random(n) < p]
In [27]:
nsamples = 1000
sample = reject(betabin_trans, 4, var, mode, nsamples, cancer, c)
In [28]:
z = np.array([[betabin_trans((t1, t2), cancer.n, cancer.y)
for t2 in logit_eta_x] for t1 in log_K_x])
x, y = np.meshgrid(logit_eta_x, log_K_x)
cplot = plt.contour(x, y, z - z.max(), [-0.5, -1, -2, -4, -8], cmap=plt.cm.RdBu)
plt.clabel(cplot, inline=1, fontsize=10, fmt='%1.1f')
plt.ylabel('log(K)');plt.xlabel('logit($\eta$)')
plt.scatter(*sample.T[[1,0]])
Out[28]:
Notice that the efficiency of rejection sampling is not very high for this problem.
In [29]:
float(sample.size)/nsamples
Out[29]:
Rejection sampling is usually subject to declining performance as the dimension of the parameter space increases. Further improvement is gained by using optimized algorithms such as importance sampling which, as the name implies, samples more frequently from important areas of the distribution.
As we have seen, the primary difficulty in Bayesian inference is calculating the posterior density for models of moderate-to-high dimension. For example, calculating the posterior mean of some function $h$ requires two difficult integration steps:
$$E[h(\theta) | y] = \frac{\int h(\theta)f(y|\theta) p(\theta) d\theta}{\int f(y|\theta) p(\theta) d\theta} = \frac{\int h(\theta)p(\theta | y) d\theta}{\int p(\theta|y) d\theta}$$If the posterior $p(\theta|y)$ is a density from which it is easy to sample, we could approximiate these integrals using Monte Carlo simulation, but too often it is not.
Instead, assume that we can draw from a tractable probability density $q(\theta)$ that is some approximation of $p$. We could then write:
$$E[h(\theta) | y] = \frac{\int h(\theta) \frac{p(\theta|y)}{q(\theta)} q(\theta) d\theta}{\int \frac{p(\theta|y)}{q(\theta)} q(\theta) d\theta}$$Expressed this way, $w(\theta) = p(\theta|y) / q(\theta)$ can be regarded as weights for the $M$ values of $\theta$ sampled from $q$ that we can use to correct the sample so that it approximates $h(\theta)$. Specifically, the importance sampling estimate of $E[h(\theta) | y]$ is:
$$\hat{h}_{is} = \frac{\sum_{i=1}^{M} h(\theta^{(i)})w(\theta^{(i)})}{\sum_{i=1}^{M} w(\theta^{(i)})}$$where $\theta^{(i)}$ is the $i^{th}$ sample simulated from $q(\theta)$. The standard error for the importance sampling estimate is:
$$\text{SE}_{is} = \frac{\sqrt{\sum_{i=1}^{M} [(h(\theta^{(i)}) - \hat{h}_{is}) w(\theta^{(i)})]^2}}{\sum_{i=1}^{M} w(\theta^{(i)})}$$The efficiency of importance sampling is related to the selection of the importance sampling distribution $q$.
As a simple illustration of importance sampling, let's consider again the problem of estimating the parameters of the beta-binomial example. Here, we will use a multivariate T density as the simulation distribution $q$.
Here are 1000 sampled values to use for approximating the posterior:
In [30]:
theta = rmvt(4, var, mode, size=1000)
We can obtain the probability of these values under the posterior density:
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f_theta = np.array([betabin_trans(t, cancer.n, cancer.y) for t in theta])
and under the T distribution:
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q_theta = np.array([mvt(t, 4, var, mode) for t in theta])
This allows us to calculate the importance weights:
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w = np.exp(f_theta - q_theta - max(f_theta - q_theta))
notice that we have subtracted the maximum value of the differences, which normalizes the weights.
Now, we can obtain estimates of the parameters:
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theta_si = [(w*t).sum()/w.sum() for t in theta.T]
theta_si
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Finally, the standard error of the estimates:
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se = [np.sqrt((((theta.T[i] - theta_si[i])* w)**2).sum()/w.sum()) for i in (0,1)]
se
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The importance sampling method can be modified to incorporate weighted bootstrapping, in a procedure called sampling importance resampling (SIR). As previously, we obtain a sample of size $M$ from an importance sampling distribution $q$ and calculate the corresponding weights $w(\theta_i) = p(\theta|y) / q(\theta)$.
Instead of directly re-weighting the samples from $q$, SIR instead transforms the weights into probabilities via:
$$p_i = \frac{w(\theta_i)}{\sum_{i=1}^M w(\theta_i)}$$These probabilities are then used to resample their respective $\theta_i$ values, with replacement. This implies that the resulting resamples $\theta_i^{\prime}$ will be distributed approximately as the posterior $p(\theta|y)$.
Using again the beta-binomial example, we can take the weights calculated above, and convert them to probabilities:
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p_sir = w/w.sum()
The choice function in numpy.random can be used to generate a random sample from an arbitrary 1-D array.
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theta_sir = theta[np.random.choice(range(len(theta)), size=10000, p=p_sir)]
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fig, axes = plt.subplots(2)
_ = axes[0].hist(theta_sir.T[0], bins=30)
_ = axes[1].hist(theta_sir.T[1], bins=30)
One advantage of this approach is that one can easily extract a posterior probability interval for each parameter, simply by extracting quantiles from the resampled values.
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logK_sample = theta_sir[:,0]
logK_sample.sort()
logK_sample[[250, 9750]]
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Perform a Bayesian sensitivity analysis by performing SIR on the stomach cancer dataset $N$ times, with one observation (a city) removed from the dataset each time. Calculate and plot posterior medians and 95% posterior intervals for each $f(\theta|y_{(-i)})$ to visually analyze the influence of each observation.
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