diff --git a/numpyro/contrib/control_flow/scan.py b/numpyro/contrib/control_flow/scan.py
index 893814f0f..6c0711142 100644
--- a/numpyro/contrib/control_flow/scan.py
+++ b/numpyro/contrib/control_flow/scan.py
@@ -3,7 +3,7 @@
 
 from collections import OrderedDict
 from functools import partial
-from typing import Callable
+from typing import Callable, Optional
 
 import jax
 from jax import device_put, lax, random
@@ -348,7 +348,7 @@ def scan(
     f: Callable,
     init,
     xs,
-    length: int | None = None,
+    length: Optional[int] = None,
     reverse: bool = False,
     history: int = 1,
 ):
diff --git a/numpyro/contrib/stochastic_support/dcc.py b/numpyro/contrib/stochastic_support/dcc.py
index 912ed46f2..02a68acf3 100644
--- a/numpyro/contrib/stochastic_support/dcc.py
+++ b/numpyro/contrib/stochastic_support/dcc.py
@@ -3,7 +3,7 @@
 
 from abc import ABC, abstractmethod
 from collections import OrderedDict, namedtuple
-from typing import Any, Callable, OrderedDict as OrderedDictType
+from typing import Any, Callable, OrderedDict as OrderedDictType, Union
 
 import jax
 from jax import random
@@ -21,9 +21,9 @@
 
 SDVIResult = namedtuple("SDVIResult", ["guides", "slp_weights"])
 
-RunInferenceResult = (
-    dict[str, Any] | tuple[AutoGuide, dict[str, Any]]
-)  # for mcmc or sdvi
+RunInferenceResult = Union[
+    dict[str, Any], tuple[AutoGuide, dict[str, Any]]
+]  # for mcmc or sdvi
 
 
 class StochasticSupportInference(ABC):
@@ -124,12 +124,12 @@ def _combine_inferences(
         branching_traces: dict[str, OrderedDictType],
         *args: Any,
         **kwargs: Any,
-    ) -> DCCResult | SDVIResult:
+    ) -> Union[DCCResult, SDVIResult]:
         raise NotImplementedError
 
     def run(
         self, rng_key: ArrayLike, *args: Any, **kwargs: Any
-    ) -> DCCResult | SDVIResult:
+    ) -> Union[DCCResult, SDVIResult]:
         """
         Run inference on each SLP separately and combine the results.
 
diff --git a/numpyro/diagnostics.py b/numpyro/diagnostics.py
index 527bd9b24..a68631b6a 100644
--- a/numpyro/diagnostics.py
+++ b/numpyro/diagnostics.py
@@ -7,6 +7,7 @@
 
 from collections import OrderedDict
 from itertools import product
+from typing import Union
 
 import numpy as np
 
@@ -230,7 +231,7 @@ def hpdi(x: np.ndarray, prob: float = 0.90, axis: int = 0) -> np.ndarray:
 
 
 def summary(
-    samples: dict | np.ndarray, prob: float = 0.90, group_by_chain: bool = True
+    samples: Union[dict, np.ndarray], prob: float = 0.90, group_by_chain: bool = True
 ) -> dict:
     """
     Returns a summary table displaying diagnostics of ``samples`` from the
@@ -284,7 +285,7 @@ def summary(
 
 
 def print_summary(
-    samples: dict | np.ndarray, prob: float = 0.90, group_by_chain: bool = True
+    samples: Union[dict, np.ndarray], prob: float = 0.90, group_by_chain: bool = True
 ) -> None:
     """
     Prints a summary table displaying diagnostics of ``samples`` from the
diff --git a/numpyro/util.py b/numpyro/util.py
index 828eb1ded..3c710f41a 100644
--- a/numpyro/util.py
+++ b/numpyro/util.py
@@ -10,7 +10,7 @@
 import random
 import re
 from threading import Lock
-from typing import Any, Callable, Generator
+from typing import Any, Callable, Generator, Optional
 import warnings
 
 import numpy as np
@@ -27,7 +27,7 @@
 _CHAIN_RE = re.compile(r"\d+$")  # e.g. get '3' from 'TFRT_CPU_3'
 
 
-def set_rng_seed(rng_seed: int | None = None) -> None:
+def set_rng_seed(rng_seed: Optional[int] = None) -> None:
     """
     Initializes internal state for the Python and NumPy random number generators.
 
@@ -49,7 +49,7 @@ def enable_x64(use_x64: bool = True) -> None:
     jax.config.update("jax_enable_x64", use_x64)
 
 
-def set_platform(platform: str | None = None) -> None:
+def set_platform(platform: Optional[str] = None) -> None:
     """
     Changes platform to CPU, GPU, or TPU. This utility only takes
     effect at the beginning of your program.
@@ -408,7 +408,7 @@ def loop_fn(collection):
 
 
 def soft_vmap(
-    fn: Callable, xs: Any, batch_ndims: int = 1, chunk_size: int | None = None
+    fn: Callable, xs: Any, batch_ndims: int = 1, chunk_size: Optional[int] = None
 ) -> Any:
     """
     Vectorizing map that maps a function `fn` over `batch_ndims` leading axes
@@ -466,7 +466,7 @@ def format_shapes(
     *,
     compute_log_prob: bool = False,
     title: str = "Trace Shapes:",
-    last_site: str | None = None,
+    last_site: Optional[str] = None,
 ):
     """
     Given the trace of a function, returns a string showing a table of the shapes of